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# algebraic formulas
## Online Tutoring Is The Easiest, Most Cost-Effective Way For Students To Get The Help They Need Whenever They Need It.
In the algebraic formula sheets we have all algebraic formulas which are used in solving algebraic expressions. Algebra is a major part of mathematics, in algebra we includes all relations and operations. Algebraic formulas are used in quadratic equations, binomial theorems, difference of squares, law of exponents etc. Following is the list of some important Algebraic Formulas that we use in solving the Algebraic expression.
1. (a+b) ^2 = a^2 + 2ab + b^2
2. (a-b) ^2 = a^2 -2ab + b^2
3. (a + b) ^3 = a^3 + 3a^2b + 3ab^2 + b^3
4. (a – b) ^3 = a^3 – 3a^2b + 3ab^2 – b^3
5. a^3 – b^3 = (a – b) (a^2 + ab + b^2)
6. a^3 + b^3 = (a + b) (a^2 – ab + b^2)
7. x^m . x^n = x^ (m+n)
8. x^m / x^n = x^(m – n)
9. (x^m) ^n = x^mn
10. (xy) ^m = x^m y^m)
Following are the some examples based on the algebraic formulas
Example 1: - Solve given expression x^2 – 49
Solution: - Given equation is:-
x^2 – 49
ð (x+7) (x-7) [ As a^2 – b^2 = (a+b) (a-b)]
Hence x^2 – 49 = (x+7) (x-7)
Example 2: Solve given expression b^2 (a^4) ^2 / a^6
Solution:- Given equation is
b^2 (a^4) ^2 / a^6
In order to solve these types of questions we have to use law of exponents.
= b^2 a^8/a^6
= b^2 a^ (8-6)
= b^2 a^2
Hence the solution for the algebraic expression b^2 (a^4)^2 / a^6 is b^2 a^2.<|endoftext|>
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One of the key differences between humans and non-human animals, it is thought, is the ability to flexibly communicate our thoughts to others. The consensus has long been that animal communication, such as the food call of a chimpanzee or the alarm call of a lemur, is the result of an automatic reflex guided primarily by the inner physiological state of the animal. Chimpanzees, for example, can't "lie" by producing a food call when there's no food around and, it is thought, they can't not emit a food call in an effort to hoard it all for themselves. By contrast, human communication via language is far more flexible and intentional.
But recent research from across the animal kingdom has cast some doubt on the idea that animal communication always operates below the level of conscious control. Male chickens, for example, call more when females are around, and male Thomas langurs (a monkey native to Indonesia) continue shrieking their alarm calls until all females in their group have responded. Similarly, vervet monkeys are more likely sound their alarm calls when their are other vervet monkeys around, and less likely when they're alone. The same goes for meerkats. And possibly chimps, as well.
Still, these sorts of "audience effects" can be explained by lower-level physiological factors. In yellow-bellied marmots, small ground squirrels native to the western US and southwestern Canada, the production of an alarm call correlates with glucocorticoid production, a physiological measurement of stress. And when researchers experimentally altered the synthesis of glucocorticoids in rhesus macaques, they found a change in the probability of alarm call production.
The wolf's howl is commonly thought of as indicating social separation. In popular culture, it is often used to reflect sadness, or loss. It was therefore not surprising that scientists at Austria's Wolf Science Center noticed that wolves howled when one of their packmates was separated from the pack.
The wolves that live at the Wolf Science Center, just forty kilometers north of Vienna, are hand-raised by humans for the first five months of their lives, before being introduced into a pack. This makes it easier for biologists and psychologists to safely interact with them once they become mature. The scientists at the Wolf Science Center are trying to understand how wolves - mythical animals who are wary of humans - transformed into the curly-tailed floppy-eared balls of fur that sleep curled up at the feet of our beds. What makes a wolf more wolf-like and a dog more dog-like? They realized that the howls of their wolves could contribute to the question of whether animal communication can be flexible or intentional.
Since the wolves live in large enclosures in packs of just 2-3 individuals, the keepers, trainers, and scientists who work with them regularly take them out on leashes for extra exercise. The remaining wolves always howled. New research published by Francesco Mazzini and colleagues in the journal Current Biology explains why.
To see whether howling could be the result of a physiological stress response associated with social separation, twenty minutes after a wolf was removed from a group, saliva was collected from its packmates. Previous research had demonstrated that the canine physiological stress response peaks roughly twenty minutes after a stressful situation. For the entire twenty minutes, the researchers also recorded all vocalizations made by the remaining wolves.
When the wolf that was removed was socially dominant, the remaining wolves howled. This isn't entirely surprising, given the centrality of social dominance to wolf life. However, when the wolf that was removed was a close friend, dominance notwithstanding, the remaining wolves howled even more.
Stress alone couldn't explain this pattern of results. While this sort of social separation was reflected in a salivary cortisol increase, the physiological stress response did not vary in sync with the wolves' howling response. While the separation was stressful in general, the howling itself was indicative of social dynamics rather than a more basic physiological reflex.
Mazzini writes that "social partner preference," or friendship, "is a more dynamic and flexible feature of wolf life [than dominance] and thus is more likely to be modulated by cognition." This explains why wolves howled more when their friends were removed than when dominant individuals, who perhaps were not their closest friends, were removed, despite the equivalent change in cortisol levels. "This provides strong support for the hypothesis that wolf howling is potentially a strategically employed vocalization with the goal of ultimately promoting contact with important individuals."
This study provides further evidence that not all animal communication is the result of automatic, inflexible physiological events, but can be intentional and voluntary. For wolves, it is important to maintain contact with allies, even if when they're out of visual range. One way they do this, apparently, is by playing a wolfy version of "Marco Polo."
Mazzini F., Townsend S., Virányi Z. & Range F. (2013). Wolf Howling Is Mediated by Relationship Quality Rather Than Underlying Emotional Stress, Current Biology, DOI: 10.1016/j.cub.2013.06.066
Elsewhere on SciAm
For more on animal communication:
For more on wolves:
Photo via Mazzini et al.<|endoftext|>
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Vitamins and Their Role in Good Health
Vitamins are organic substances contained in various natural foodstuffs in minute amounts. Because of the crucial role these substances play in normal metabolism, a lack of them can cause a whole range of medical conditions.
Carbon is a main component of vitamins, being organic compounds; and because the body produces insufficient amounts of them, it is necessary to obtain them from food. But in contrast to proteins, fats and carbohydrates, vitamins supply no energy, although they are do help the body work and grow at optimal levels.
There are thirteen essential vitamins that provide a whole range of health benefits, including better eyesight, a stronger immune system, stronger bones, faster wound healing process, and several others. Inadequate vitamin intake can make you more likely to develop illness, from mild to life-threatening.
Types of Vitamins
Vitamins are either fat soluble or water-soluble, depending on body storage. There are four fat-soluble vitamins – A, D, E and K – all stored in fat tissue for up to as long as half a year.
On the other hand, water-soluble vitamins, namely vitamin C and the vitamin B series (B6, B12, pantothenic acid, folate, biotin, thiamine and niacin) are all distributed all over the body through blood circulation. As water-soluble vitamins are not stored in the body, it is important to replenish your stores regularly.
All the thirteen vitamins have their own individual functions, but they can work as a group as well in improving your health. Vitamin A gives you better skin, bones and teeth, aside form good eyesight and immunity.
Vitamin C also strengthens immunity, encourages good tissue development and helps the body in absorbing iron. Vitamin, D coupled with calcium (another mineral), is vital to bone health and immunity as well. Vitamin E aids in your body’s use of vitamin K, which affects bone health and blood-clotting mechanisms, and contributes to optimal production of red blood cells.
The B vitamins, for their part, play a role in optimal metabolism, brain function, hormone production, cardiac activity, central nervous system functions, and cellular maintenance.
Effects of Vitamin Deficiencies
Insufficient vitamin intake puts your health at risk, specifically in relation to heart disease, osteoporosis and cancer. Insufficient vitamin B intake sets the stage for anemia and irreversible nerve damage.
Without enough vitamin C in your diet, you will have limited stores of collagen, which makes up your body’s primary tissue. In extreme vitamin C deficiency cases, people can be afflicted with scurvy, which is characterized by overall weakness, gingivitis, anemia and skin hemorrhage.
Lastly, vitamin D deficiency leads to rickets, which manifests as bone pain and deformation, and overall poor growth in children, and as poor bone health, hypertension, and autoimmune diseases in adults.<|endoftext|>
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# What is 3x + -5.44=29
It's about the process of elimination in algebra but I just can't figure it out
Ryan,
In 3x + -5.44=29 we want to find the value of x. The two operators + - together are not quite proper notation but I believe the equation should be 3x + (-5.44)=29 or more simply 3x -5.44=29.
1st: 3x -5.44=29 on both sides of the = sign, add +5.44 and get:
3x -5.44 + 5.44 =29 +5.44 or 3x = 34.44
2nd: Divide both sides of the equation by 3 and get:
3x /3 = 34.44 /3 or x= 11.48
3rd: Since we are dealing with no more that 3 significant digits, the answer is reduced to:
x=11.5
Good Luck! BruceS
3x+ -5.44=29
So, a positive sign and a negative sign make the sign negative so rewrite the problem as
3x-5.44=29
then, move 5.44 to the other side of the problem to get 3x alone. To to this, ADD 5.44 to both sides. This will cancel out the -5.44 on the left because -5.44+5.44=0
So the problem now looks like this: 3x=29+5.44
Add 29 and 5.44 which gives you 3x=34.44
Then, divide both sides by 3 and this will tell you what x is equal to.
3 divided by 3 is one. So the right side of the problem is 1x or just x. 34.44 divided by 3 is 11.48.
So, 1x or x (it is the same thing) is equal to 11.48.
I hope this helped :)
3x + -5.44 = 29
First, you want to add the inverse of -5.44 to both sides of the equation.
3x=29 + 5.44
3x = 34.44
Now you want to multiply the invers of 3 (which is 1/3) to both sides of equation. 3 x 1/3=1, so that leaves you with just x on one side.
x = 34.44/3
x = 11.48<|endoftext|>
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# Probability and Simulation: The Study of Randomness
## Presentation on theme: "Probability and Simulation: The Study of Randomness"— Presentation transcript:
Probability and Simulation: The Study of Randomness
Chapter 6 Probability and Simulation: The Study of Randomness
6.1 Objectives Students will be able to: Define Simulation.
List the five steps involved in a simulation. Explain what is meant by independent trials. Use a table of random digits to carry out a simulation. Given a probability problem, conduct a simulation in order to estimate the probability desired. Use a calculator or a computer to conduct a simulation of a probability problem.
6.1 Simulation The imitation of chance behavior, based on a model that accurately reflects the phenomenon under consideration, is called simulation.
Simulation Steps Step 1: State the problem or describe the random phenomenon Toss a coin 10 times. What is the likelihood of a run of at least 3 consecutive heads or 3 consecutive tails? Step 2: State the assumptions. There are two: A head or tail is equally likely to occur on each toss. Tosses are independent of each other (that is, what happens on one toss will not influence the next toss)
Simulation Steps Step 3: Assign digits to represent outcomes.
One digit simulates one toss of the coin. Odd digits represent heads; even digits represents tails. Step 4: Simulate many repetitions. We will complete 25 repetitions for this simulation. In a random number table, even and odd digits occur with the same long-term relative frequency, 50%. This is just one assignment of digits for coin tossing. Successive digits in the table simulate independent tosses.
Simulation Steps Step 5: State your conclusions.
We estimate the probability of a run of size 3 by the proportion Estimated probability = 23/25 = 0.92. 25 repetitions is not enough to be confident that our estimate is accurate. We can use the computer to do thousands of trials for us. A long simulation (mathematical analysis) finds that the true probability is about
Assigning Digits Choose a person at random from a group of which 70% are employed. 0, 1, 2, 3, 4, 5, 6 = employed 7, 8, 9 not employed 00, 01, …, 69 employed 70, 71,…, 99 not employed Choose a person at random from a group of which 73% are employed. 00, 01, …, 72 employed 73, 74, …, 99 not employed 0, 1, 2, 3, 4, 5, 6 = employed 7, 8, 9 not employed 00, 01, …, 69 employed 70, 71,…, 99 not employed
Assigning Digits Choose a person at random from a group of which 50% are employed, 20% are unemployed, and 30% are not in the labor force. 0, 1, 2, 3, 4 = employed 5, 6 unemployed 7, 8, 9 not in the labor force 0,1 = unemployed 2, 3, 4 = not in the labor force 5, 6, 7, 8, 9, = employed 0, 1, 2, 3, 4 = employed 5, 6 unemployed 7, 8, 9 not in the labor force 0,1 = unemployed 2, 3, 4 = not in the labor force 5, 6, 7, 8, 9, = employed
Example Page 397 #6.1 Establishing Correspondence
State how you would use the following aids to establish a correspondence in a simulation that involves a 75% chance: A coin A six-sided dice A random digit table A standard deck of playing cards a. Flip the coin twice. Let HH represent failure, and let the other outcomes (HT, TH, TT) represent success. Let 1, 2, 3, represent a success, and let 4 represent a failure. If 5 or 6 come up, ignore them and roll again. Peel off two consecutive digits from the table; let 00 through 74 represent a success, and let 75 through 99 represent a failure. Let diamonds, spades, and clubs represent a success, and let hearts represent a failure.
Example Page 404 #6.15 The birthday problem
Use your calculator and a simulation method to determine the chances of at least 2 students with the same birthday in a class of 23 unrelated students. Determine the chances of at least 2 people having the same birthday in a room of 41 people. What assumptions are you making in your simulations?
Types of Simulations Situations in which our interest is in the number of successes out of a fixed number of trials (assuming equal probabilities and independence from trial to trial) are often solved using the binomial distribution. Situations in which our interest is in how many trials it takes for an event to occur (again assuming equal probabilities and independence from trial to trial) are often solved using the geometric distribution.
6.2 Objectives Students will be able to
Explain how the behavior of a chance event differs in the short and long run. Explain what is meant by a random phenomenon. Explain what it means to say that the idea of probability is empirical. Define probability in terms of relative frequency. Define sample space. Define event. Explain what is meant by probability model. Construct a tree diagram. Use the multiplication principle to determine the number of outcomes in a sample space. Explain what is meant by sampling with replacement and sampling without replacement.
6.2 Objectives List the four rules that must be true for any assignment of probabilities. Explain what is meant by {A U B} and {A ∩ B}. Explain what is meant by each of the regions in a Venn diagram Give an example of two events A and B where A ∩ B = Ø. Use a Venn diagram to illustrate the intersection of two events A and B. Compute the probability of an event, given the probabilities of the outcomes that make up the event. Explain what is meant by equally likely outcomes. Compute the probability of an event in the cases of equally likely outcomes. Define what it means for two events to be independent. Give the multiplication rule for independent events. Given two events, determine if they are independent.
Randomness and Probability
We call a phenomenon random if individual outcomes are uncertain but there is nonetheless a regular distribution of outcomes in a large number of repetitions. The probability of any outcome of a random phenomenon is the proportion of times the outcome would occur in a very long series of repetitions. That is, the probability is long-term relative frequency.
Types of Probability Theoretical probability (Classical)
Empirical probability is based on observations rather than theorizing. Subjective probability
Example Page 410 #6.21 Pennies Spinning
Hold a penny upright on its edge under your forefinger on a hard surface, then snap it with your forefinger so that it spins for some time before falling. Based on 50 spins, estimate the probability of heads.
Probability Models The sample space is the set of all possible outcomes. An event is any outcome or a set of outcomes of a random phenomenon. That is, an event is a subset of the sample space. A probability model is a mathematical description of a random phenomenon consisting of two parts: A sample space, S and A way of assigning probabilities to events.
Types of models Discrete models have a countable number of outcomes.
Continuous models correspond to intervals on the number line. Discrete: number of heads observed on a flip of 3 coins Continuous: heights of a sample of 15 ninth-graders
There are 36 possible outcomes.
Tree Diagrams Two dice are rolled. Describe the sample space. Start 1st roll 2nd roll 1 2 3 4 5 6 There are 36 possible outcomes.
Two dice are rolled and the sum is noted.
1,1 1,2 1,3 1,4 1,5 1,6 2,1 2,2 2,3 2,4 2,5 2,6 3,1 3,2 3,3 3,4 3,5 3,6 4,1 4,2 4,3 4,4 4,5 4,6 5,1 5,2 5,3 5,4 5,5 5,6 6,1 6,2 6,3 6,4 6,5 6,6
Find the probability the sum is 4.
Find the probability the sum is 4 or 11.
Multiplication Principle
If you can do one task in n1 ways and a second task in n2 ways, then both tasks can be done in n1 x n2 number of ways. How many ways can you flip 3 coins? How many ways can you flip a coin and roll a die?
Replacement Sampling with replacement Sampling without replacement
How many 4 digit pin numbers can you make? How many 4 digit pin numbers can you make if all numbers are distinct?
Example Number of ways Sum Outcomes 1 2 1, 1 3 1,2 2,1 4 5 6 7 8 9 10
11 12 Page 417 #6.35 Rolling Two Dice In how many ways can you get an even sum? In how many ways can you get a sum of 5? Of 8? Describe a pattern you see in the table.
Probability Rules Rule 1: Any probability is a number between 0 and 1.
Rule 2: The sum of the probabilities of all possible outcomes must equal 1. Rule 3: If two events have no outcomes in common, the probability that one or the other occurs is the sum of their individual probabilities. Rule 4: The probability that an event will occur is 1 minus the probability that the even does occur.
Venn diagrams Mutually exclusive (disjoint) Union (A U B)
Intersect (A ∩ B)
Venn diagrams Empty Set Complement A U Ac = S A and Ac = empty set
Equally Likely If a random phenomenon has k possible outcomes that are all equally likely, then each individual outcome has probability 1/k.
Example Page 423 #6.38 Distribution of M&M colors
If you draw an M&M candy at random from a bag of candies, the candy you draw will have one of six colors. The probability of drawing each color depends on the proportion of each color amoung the candies made. The table below gives the probability of each color for a randomly chosen milk chocolate M&M: What must be the probability of drawing a blue candy? Color: Brown Red Yellow Green Orange Blue Probability: 0.13 0.14 0.16 0.20 ?
The probabilities for peanut M&M’s are different.
What is the probability that a peanut M&M is blue? What is the probability that a milk chocolate M&M is red, yellow or orange? What is the probability that a peanut M&M is one of these colors? Color: Brown Red Yellow Green Orange Blue Probability: 0.12 0.15 0.23 ?
Multiplication Rule for Independent Events
Rule 5: Two events are independent if knowing that one occurs does not change the probability that the other occurs. If A and B are independent, P(A and B) = P(A)P(B) **A and B are independent iff P(A|B) = P(A) Independence and disjoint are not the same thing… if two events are disjoint then they cannot be independent because knowing that one occurred would tell us that the other cannot.
Caution… The multiplication rule applies only to independent events. You cannot use it if events are not independent! The addition rule applies only to disjoint events. Disjoint does not mean independent.
Example Page 430 #6.46 Defective Chips
An automobile manufacturer buys computer chips from a supplier. The supplier sends a shipment containing 5% defective chips. Each chip chosen from the shipment has probability 0.05 of being defective, and each automobile uses 12 chips selected independently. What is the probability that all 12 chips in a car will work properly?
Example Page 430 #6.48 College Student Demographics
Choose a random college student at least 15 years of age. We are interested in the events A = {The person chosen is male} B = {The person chosen is 25 years or older} Government data recorded in Table 4.5 (page 292) allow us to assign probabilities to these events. Explain why P(A) = 0.44 Find P(B) Find the probability that the person chosen is a male at least 25 years old, P(A and B). Are these events A and B independent?
Example Page 430 #6.51 Telephone Success
Most sample surveys use random digit dialing equipment to call residential telephone numbers at random. The telephone polling firm Zogby International reports that the probability that a call reaches a live person is Calls are independent. A polling firm places 5 calls. What is the probability that none of them reaches a person? When calls are made to New York City, the probability of reaching a person is What is the probability that none of 5 calls made to New York City reaches a person?
Example Page 432 #6.53 Student Survey
Choose a student at random from a large statistics class. Give a reasonable sample space S for answers to each of the following questions. (In some cases you may have the freedom to specify S.) Are you right or left handed? What is your height in centimeters? (1 inch = 2.54 cm) How much money in coins (not bills) are you carrying? How many minutes did you study last night?
Example Page 434 #6.62 Roulette
A roulette wheel has 38 slots, numbers 0, 00, and 1 to 36. The slots 0 and 00 are colored green, 18 of the others are red, and 18 are black. The dealer spins the wheel and at the same time rolls a small ball along the wheel in the opposite direction. The wheel is carefully balanced so that the ball is equally likely to land in any slot when the wheel slows. Gamblers can bet various combinations of numbers and colors. What is the probability that the ball will land in any one slot? If you bet on “red”, you win if the ball lands in a red slot. What is the probability of winning? The slot numbers are laid out on a board on which gamblers place their bets. One column of numbers on the board contains multiples of 3, that is 3, 6, 9, …, 36. You place a ”column bet” that wins if any of these numbers comes up. What is your probability of winning?
6.3 Objectives Students will be able to
State the Addition Rule for disjoint events. State the general addition rule for union of two sets. Given two events A and B, compute P(A U B). Define what is meant by a joint event and joint probability. Given two events, compute their joint probability. Explain what is meant by the conditional probability P(A|B). State the general multiplication rule to define P(B|A). Explain what is meant by Bayes’s rule. Define independent events in terms of a conditional probability.
Union The Union of any collection of events is the event that at least one of the collection occurs.
If events A, B, and C are disjoint, then P(A or B or C) = P(A) + P(B) + P(C)
General Addition Rule for Unions of Two Events
P(A or B) = P(A) + P(B) – P(A and B)
Example Page 441 #6.70 Tastes in Music I
Musical styles other than rock and pop are becoming more popular. A survey of college students finds that 40% like country music, 30% like gospel music, and 10% like both. Make a Venn diagram with these results. What percent of college students like country but not gospel? What percent like neither country nor gospel?
Conditional Probability
P(A|B) “probability of A, given B”
General Multiplication Rule
The joint probability that events A and B will both happen can be found by P(A and B) = P(A) x P(B|A) Here P(B|A) is the conditional probability that B occurs, given the information that A occurs.
Conditional Probability
When P(A) > 0, the conditional probability of B, given A, is `P(B|A) = P(A and B) P(A) We require that the probability of A occurring be greater than 0 because the probability of B given A makes no sense if A cannon occur. Notice this comes from rearranging the multiplication rule
Example Page 446 #6.72 Pay at the Pump
At a self-service gas station, 40% of the customers pump regular gas, 35% pump midgrade, and 25% pump premium gas. Of those who pump regular, 30% pay at least \$20. Of those who pump midgrade, 50% pay at least \$20. And of those who pump premium, 60% pay at least \$20. What is the probability that the next customer pays at least \$20?
Example Page 447 #6.76 The probability of a flush
A poker player holds a flush when all 5 cards in the hand belong to the same suit. We will find the probability of a flush when all the cards are dealt. Remember that a deck contains 52 cards,13 of each suit, and that when the deck is well shuffled, each card dealt is equally likely to be any of those that remain in the deck. We will concentrate on spades, what is the probability that the first card dealt is a spade? What is the conditional probability that the second card is a spade, given that the first is a spade?
Continue to count the remaining cards to find the conditional probabilities of a spade on the third, fourth, and fifth card, given in each case that all previous cards are spades. The probability of being dealt 5 spades is the product of the five probabilities that you have found. Why? What is this probability? The probability of being dealt 5 hearts or 5 diamonds or 5 clubs is the same as the probability of being dealt 5 spades. What is the probability of being dealt a flush?
Intersection The intersection of any collection of events is the probability that all of the events occur.
Example Page 448 Example 6.30
Independent Events Two events A and B that both have positive probability are independent if P(B|A) = P(B)<|endoftext|>
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TOEFL PBT Test / Reading Comprehension Practice Questions
The Reading Comprehension section contains reading passages and questions about the passages. The questions are about information that is stated or implied in the passage and about some of the specific words in the passages. Because many English words have more than one meaning, it is important to remember that these questions concern the meaning of a word or phrase within the context of the passage.
Before completing these practice questions, you might wish to print out an answer sheet.
Directions and Practice Questions for Reading Comprehension
Directions and examples of the types of questions you will find in the Reading Comprehension section of the TOEFL® test follow. Use the answer key to see the correct answers for the Reading Comprehension questions.
Section 3 measures your ability to read and understand short passages similar in topic and style to those that students are likely to encounter in North American universities and colleges. This section contains reading passages and questions about the passages.
Directions: In the Reading Comprehension section you will read several passages. Each one is followed by a number of questions about it. You are to choose the one best answer, A, B, C or D, to each question. Then, on your answer sheet, find the number of the question and fill in the space that corresponds to the letter of the answer you have chosen.
از لینک زیر ادامه مطلب و پاسخ سوالات را دانلود کنیددانلود نمونه سوالات تافل TOEFL PBT Test به همراه پاسخ<|endoftext|>
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# Order of Operations: 360 or 354?
Last Saturday, I received Facebook message from a student asking help to simplify $[5(4)^3 + 6(11-4)] - 36 / 9 (2)$. He got $354$ but his teacher’s answer was $360$.
Although the problem above seems simple, a lot of students get confused by it, and in this case, even the teacher too. The expression above simplifies to
$[5(64) + 6(7)] - 36 / 9*2$
$= [320 + 42] - 4(2)$
$=[320 + 42]- 4(2)$
$=362 - 8$
$= 354$.
The misconception about the order of operations usually arises from acronyms like PEMDAS (parenthesis, exponent, multiplication, division, addition, and subtraction). Mathematical operations should be performed in that order: simplify the expression within the parenthesis first, and then simplify the expressions with exponent, perform multiplication, and so on. In multiplication and division, however, even though multiplication comes first, if the two operations are adjacent and without parenthesis, we perform the operation from left to right. In the example above, the INCORRECT way to simplify $36/9*2$ is to multiply $9$ by $2$ first before dividing; that is, $36/18=2$. The CORRECT way is dividing $36$ by $9$ first (which equals $4$), and then multiplying it by $2$, which equals $8$.
Note, however, that $36 / 9*2 = 8$, but $36/(9*2)$ is $2$ since you have to simplify the operation within the parenthesis first.
Hence $16/2*2 + 1 = 17$ and NOT $5$, and $6/2*3 = 9$ and NOT $1$.
The rule is you perform mathematical operations in the following order:
1. terms inside parentheses or brackets from the inner set of symbols to the outer set
2. exponents and roots
3. multiplication and division as they appear from left to right
4. addition and subtraction as they appear from left to right
As shown in the list above, addition and subtraction is also performed whichever comes first.<|endoftext|>
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# Exponential function
(Redirected from )
The natural exponential function y = ex
In mathematics, an exponential function is a function of the form
${\displaystyle f(x)=b^{x}\,}$
in which the input variable x occurs as an exponent. A function of the form ${\displaystyle f(x)=b^{x+c}}$, where ${\displaystyle c}$ is a constant, is also considered an exponential function and can be rewritten as ${\displaystyle f(x)=ab^{x}}$, with ${\displaystyle a=b^{c}}$.
As functions of a real variable, exponential functions are uniquely characterized by the fact that the growth rate of such a function (i.e., its derivative) is directly proportional to the value of the function. The constant of proportionality of this relationship is the natural logarithm of the base ${\displaystyle b}$:
${\displaystyle {\frac {d}{dx}}{\left(b^{x}\right)}=b^{x}{\log _{e}}{(b)}}$.
The constant e ≈ 2.71828... is the unique base for which the constant of proportionality is 1, so that the function's derivative is itself:
${\displaystyle {\frac {d}{dx}}{\left(e^{x}\right)}=e^{x}{\log _{e}}{(e)}=e^{x}}$.
Since changing the base of the exponential function merely results in the appearance of an additional constant factor, it is computationally convenient to reduce the study of exponential functions in mathematical analysis to the study of this particular function, conventionally called the "natural exponential function",[1][2] or simply, "the exponential function" and denoted by
${\displaystyle x\mapsto e^{x}}$ or ${\displaystyle \exp(\cdot )}$.
The exponential function satisfies the fundamental multiplicative identity
${\displaystyle e^{x+y}=e^{x}e^{y}}$, for all ${\displaystyle x,y\in \mathbb {R} }$.
(In fact, this identity extends to complex-valued exponents.) It can be shown that complete set of continuous, nonzero solutions of the functional equation ${\displaystyle f(x+y)=f(x)f(y)}$ are the exponential functions, ${\displaystyle f:\mathbb {R} \to \mathbb {R} ,\ x\mapsto b^{x}}$, with ${\displaystyle b>0}$.
The argument of the exponential function can be any real or complex number or even an entirely different kind of mathematical object (e.g., a matrix).
Its ubiquitous occurrence in pure and applied mathematics has led mathematician W. Rudin to opine that the exponential function is "the most important function in mathematics".[3] In applied settings, exponential functions model a relationship in which a constant change in the independent variable gives the same proportional change (i.e. percentage increase or decrease) in the dependent variable. Such a situation occurs widely in the natural and social sciences; thus, the exponential function also appears in variety of contexts within physics, chemistry, engineering, mathematical biology, and economics.
Exponential function
Representation ex
Inverse ln x
Derivative ex
Indefinite integral ex + C
The graph of ${\displaystyle y=e^{x}}$ is upward-sloping, and increases faster as ${\displaystyle x}$ increases. The graph always lies above the ${\displaystyle x}$-axis but can get arbitrarily close to it for negative ${\displaystyle x}$; thus, the ${\displaystyle x}$-axis is a horizontal asymptote. The slope of the tangent to the graph at each point is equal to its ${\displaystyle y}$-coordinate at that point, as implied by its derivative function (see above). Its inverse function is the natural logarithm, denoted ${\displaystyle \log }$,[4] ${\displaystyle \ln }$,[5] or ${\displaystyle \log _{e}}$; because of this, some old texts[6] refer to the exponential function as the antilogarithm.
## Formal definition
The exponential function (in blue), and the sum of the first n + 1 terms of the power series on the left (in red).
The exponential function ${\displaystyle \exp :\mathbb {C} \to \mathbb {C} }$ can be characterized in a variety of equivalent ways. Most commonly, it is defined by the following power series:[3]
${\displaystyle \exp(z)=\sum _{k=0}^{\infty }{z^{k} \over k!}=1+z+{z^{2} \over 2}+{z^{3} \over 6}+{z^{4} \over 24}+\cdots }$
Since the radius of convergence of this power series is infinite, this definition is applicable to all complex numbers ${\displaystyle z}$. The constant e can then be defined as ${\textstyle e=\exp(1)=\sum _{k=0}^{\infty }(1/k!)}$.
Less commonly, the real exponential function is defined as the solution y to the equation
${\displaystyle x=\int _{1}^{y}{1 \over t}\mathrm {d} t}$
The exponential function can also be defined as the following limit:[7]
${\displaystyle e^{x}=\lim _{n\rightarrow \infty }\left(1+{\frac {x}{n}}\right)^{n}}$.
## Overview
The red curve is the exponential function. The black horizontal lines show where it crosses the green vertical lines.
The exponential function arises whenever a quantity grows or decays at a rate proportional to its current value. One such situation is continuously compounded interest, and in fact it was this observation that led Jacob Bernoulli in 1683[8] to the number
${\displaystyle \lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n}}$
now known as e. Later, in 1697, Johann Bernoulli studied the calculus of the exponential function.[8]
If a principal amount of 1 earns interest at an annual rate of x compounded monthly, then the interest earned each month is x/12 times the current value, so each month the total value is multiplied by (1 + x/12), and the value at the end of the year is (1 + x/12)12. If instead interest is compounded daily, this becomes (1 + x/365)365. Letting the number of time intervals per year grow without bound leads to the limit definition of the exponential function,
${\displaystyle \exp(x)=\lim _{n\to \infty }\left(1+{\frac {x}{n}}\right)^{n}}$
first given by Euler.[7] This is one of a number of characterizations of the exponential function; others involve series or differential equations.
From any of these definitions it can be shown that the exponential function obeys the basic exponentiation identity,
${\displaystyle \exp(x+y)=\exp(x)\cdot \exp(y)}$
which is why it can be written as ex.
The derivative (rate of change) of the exponential function is the exponential function itself. More generally, a function with a rate of change proportional to the function itself (rather than equal to it) is expressible in terms of the exponential function. This function property leads to exponential growth and exponential decay.
The exponential function extends to an entire function on the complex plane. Euler's formula relates its values at purely imaginary arguments to trigonometric functions. The exponential function also has analogues for which the argument is a matrix, or even an element of a Banach algebra or a Lie algebra.
## Derivatives and differential equations
The derivative of the exponential function is equal to the value of the function. From any point P on the curve (blue), let a tangent line (red), and a vertical line (green) with height h be drawn, forming a right triangle with a base b on the x-axis. Since the slope of the red tangent line (the derivative) at P is equal to the ratio of the triangle's height to the triangle's base (rise over run), and the derivative is equal to the value of the function, h must be equal to the ratio of h to b. Therefore, the base b must always be 1.
The importance of the exponential function in mathematics and the sciences stems mainly from its definition as the unique function which is equal to its derivative and is equal to 1 when x = 0. That is,
${\displaystyle {\frac {d}{dx}}e^{x}=e^{x}\quad {\text{and}}\quad e^{0}=1}$
Functions of the form cex for constant c are the only functions that are equal to their derivative (by the Picard–Lindelöf theorem). Other ways of saying the same thing include:
• The slope of the graph at any point is the height of the function at that point.
• The rate of increase of the function at x is equal to the value of the function at x.
• The function solves the differential equation y′ = y.
• exp is a fixed point of derivative as a functional.
If a variable's growth or decay rate is proportional to its size—as is the case in unlimited population growth (see Malthusian catastrophe), continuously compounded interest, or radioactive decay—then the variable can be written as a constant times an exponential function of time. Explicitly for any real constant k, a function f: RR satisfies f′ = kf if and only if f(x) = cekx for some constant c.
Furthermore, for any differentiable function f(x), we find, by the chain rule:
${\displaystyle {\mathrm {d} \over \mathrm {d} x}e^{f(x)}=f'(x)e^{f(x)}}$
## Continued fractions for ex
A continued fraction for ex can be obtained via an identity of Euler:
${\displaystyle e^{x}=1+{\cfrac {x}{1-{\cfrac {x}{x+2-{\cfrac {2x}{x+3-{\cfrac {3x}{x+4-\ddots }}}}}}}}}$
The following generalized continued fraction for ez converges more quickly:[9]
${\displaystyle e^{z}=1+{\cfrac {2z}{2-z+{\cfrac {z^{2}}{6+{\cfrac {z^{2}}{10+{\cfrac {z^{2}}{14+\ddots }}}}}}}}}$
or, by applying the substitution z = x/y:
${\displaystyle e^{\frac {x}{y}}=1+{\cfrac {2x}{2y-x+{\cfrac {x^{2}}{6y+{\cfrac {x^{2}}{10y+{\cfrac {x^{2}}{14y+\ddots }}}}}}}}}$
with a special case for z = 2:
${\displaystyle e^{2}=1+{\cfrac {4}{0+{\cfrac {2^{2}}{6+{\cfrac {2^{2}}{10+{\cfrac {2^{2}}{14+\ddots \,}}}}}}}}=7+{\cfrac {2}{5+{\cfrac {1}{7+{\cfrac {1}{9+{\cfrac {1}{11+\ddots \,}}}}}}}}}$
This formula also converges, though more slowly, for z > 2. For example:
${\displaystyle e^{3}=1+{\cfrac {6}{-1+{\cfrac {3^{2}}{6+{\cfrac {3^{2}}{10+{\cfrac {3^{2}}{14+\ddots \,}}}}}}}}=13+{\cfrac {54}{7+{\cfrac {9}{14+{\cfrac {9}{18+{\cfrac {9}{22+\ddots \,}}}}}}}}}$
## Complex plane
Exponential function on the complex plane. The transition from dark to light colors shows that the magnitude of the exponential function is increasing to the right. The periodic horizontal bands indicate that the exponential function is periodic in the imaginary part of its argument.
As in the real case, the exponential function can be defined on the complex plane in several equivalent forms. The most common definition of the complex exponential function parallels the power series definition for real arguments, where the real variable is replaced by a complex one:
${\displaystyle \exp z:=\sum _{k=0}^{\infty }{\frac {z^{k}}{k!}}}$
Termwise multiplication of two copies of these power series in the Cauchy sense, permitted by Mertens' theorem, shows that the defining multiplicative property of exponential functions continues to hold for all complex arguments:
${\displaystyle \exp(w+z)=\exp(w)\exp(z)}$ for all ${\displaystyle w,z\in \mathbb {C} }$
The definition of the complex exponential function in turn leads to the appropriate definitions extending the trigonometric functions to complex arguments.
In particular, when ${\displaystyle z=it}$ (${\displaystyle t}$ real), the series definition yields the expansion
${\displaystyle \exp(it)={\Big (}1-{\frac {t^{2}}{2!}}+{\frac {t^{4}}{4!}}-{\frac {t^{6}}{6!}}+\cdots {\Big )}+i{\Big (}t-{\frac {t^{3}}{3!}}+{\frac {t^{5}}{5!}}-{\frac {t^{7}}{7!}}+\cdots {\Big )}}$
In this expansion, the rearrangement of the terms into real and imaginary parts is justified by the absolute convergence of the series. The real and imaginary parts of the above expression in fact correspond to the series expansions of ${\displaystyle \cos t}$ and ${\displaystyle \sin t}$, respectively.
This correspondence provides motivation for defining cosine and sine for all complex arguments in terms of ${\displaystyle \exp(\pm iz)}$ and the equivalent power series:[10]
${\displaystyle \cos z:={\frac {1}{2}}{\Big [}\exp(iz)+\exp(-iz){\Big ]}=\sum _{k=0}^{\infty }(-1)^{k}{\frac {z^{2k}}{(2k)!}}}$ and ${\displaystyle \sin z:={\frac {1}{2i}}{\Big [}\exp(iz)-\exp(-iz){\Big ]}=\sum _{k=0}^{\infty }(-1)^{k}{\frac {z^{2k+1}}{(2k+1)!}}}$ for all ${\displaystyle z\in \mathbb {C} }$
The functions exp, cos, and sin so defined have infinite radii of convergence by the ratio test and are therefore entire functions (i.e., holomorphic on ${\displaystyle \mathbb {C} }$). The range of the exponential function is ${\displaystyle \mathbb {C} \setminus \{0\}}$, while the ranges of the complex sine and cosine functions are both ${\displaystyle \mathbb {C} }$ in its entirety, in accord with Picard's theorem, which asserts that the range of a nonconstant entire function is either all of ${\displaystyle \mathbb {C} }$, or ${\displaystyle \mathbb {C} }$ excluding one lacunary value.
These definitions for the exponential and trigonometric functions lead trivially to Euler's formula:
${\displaystyle \exp(iz)=\cos z+i\sin z}$ for all ${\displaystyle z\in \mathbb {C} }$
We could alternatively define the complex exponential function based on this relationship. If ${\displaystyle z=x+iy}$, where ${\displaystyle x}$ and ${\displaystyle y}$ are both real, then we could define its exponential as
${\displaystyle \exp z=\exp(x+iy):=(\exp x)(\cos y+i\sin y)}$
where exp, cos, and sin on the right-hand side of the definition sign are to be interpreted as functions of a real variable, previously defined by other means.[11]
For ${\displaystyle t\in \mathbb {R} }$, the relationship ${\displaystyle {\overline {\exp(it)}}=\exp(-it)}$ holds, so that ${\displaystyle |\exp(it)|=1}$ for real ${\displaystyle t}$ and ${\displaystyle t\mapsto \exp(it)}$ maps the real line (mod ${\displaystyle 2\pi }$) to the unit circle. Based on the relationship between ${\displaystyle \exp(it)}$ and the unit circle, it is easy to see that, restricted to real arguments, the definitions of sine and cosine given above coincide with their more elementary definitions based on geometric notions.
The complex exponential function is periodic with period ${\displaystyle 2\pi i}$ and ${\displaystyle \exp(z+2\pi ik)=\exp z}$ holds for all ${\displaystyle z\in \mathbb {C} ,k\in \mathbb {Z} }$.
When its domain is extended from the real line to the complex plane, the exponential function retains the following properties:
• ${\displaystyle e^{z+w}=e^{z}e^{w}\,}$
• ${\displaystyle e^{0}=1\,}$
• ${\displaystyle e^{z}\neq 0}$
• ${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} z}}e^{z}=e^{z}}$
• ${\displaystyle \left(e^{z}\right)^{n}=e^{nz},n\in \mathbb {Z} }$
for all ${\displaystyle w,z\in \mathbb {C} }$.
Extending the natural logarithm to complex arguments yields the complex logarithm log z, which is a multivalued function.
We can then define a more general exponentiation:
${\displaystyle z^{w}=e^{w\log z}}$
for all complex numbers z and w. This is also a multivalued function, even when z is real. This distinction is problematic, as the multivalued functions log z and zw are easily confused with their single-valued equivalents when substituting a real number for z. The rule about multiplying exponents for the case of positive real numbers must be modified in a multivalued context:
(ez)w
ezw
, but rather (ez)w
= e(z + 2πin)w
multivalued over integers n
See failure of power and logarithm identities for more about problems with combining powers.
The exponential function maps any line in the complex plane to a logarithmic spiral in the complex plane with the center at the origin. Two special cases might be noted: when the original line is parallel to the real axis, the resulting spiral never closes in on itself; when the original line is parallel to the imaginary axis, the resulting spiral is a circle of some radius.
### Computation of ab where both a and b are complex
Complex exponentiation ab can be defined by converting a to polar coordinates and using the identity (eln(a))b
= ab
:
${\displaystyle a^{b}=\left(re^{\theta i}\right)^{b}=\left(e^{\ln(r)+\theta i}\right)^{b}=e^{\left(\ln(r)+\theta i\right)b}}$
However, when b is not an integer, this function is multivalued, because θ is not unique (see failure of power and logarithm identities).
## Matrices and Banach algebras
The power series definition of the exponential function makes sense for square matrices (for which the function is called the matrix exponential) and more generally in any Banach algebra B. In this setting, e0 = 1, and ex is invertible with inverse ex for any x in B. If xy = yx, then ex + y = exey, but this identity can fail for noncommuting x and y.
Some alternative definitions lead to the same function. For instance, ex can be defined as
${\displaystyle \lim _{n\to \infty }\left(1+{\frac {x}{n}}\right)^{n}.}$
Or ex can be defined as f(1), where f: RB is the solution to the differential equation f ′(t) = xf(t) with initial condition f(0) = 1.
## Lie algebras
Given a Lie group G and its associated Lie algebra ${\displaystyle {\mathfrak {g}}}$, the exponential map is a map ${\displaystyle {\mathfrak {g}}}$ G satisfying similar properties. In fact, since R is the Lie algebra of the Lie group of all positive real numbers under multiplication, the ordinary exponential function for real arguments is a special case of the Lie algebra situation. Similarly, since the Lie group GL(n,R) of invertible n × n matrices has as Lie algebra M(n,R), the space of all n × n matrices, the exponential function for square matrices is a special case of the Lie algebra exponential map.
The identity exp(x + y) = exp(x)exp(y) can fail for Lie algebra elements x and y that do not commute; the Baker–Campbell–Hausdorff formula supplies the necessary correction terms.
## Similar properties of e and the function ez
The function ez is not in C(z) (i.e., is not the quotient of two polynomials with complex coefficients).
For n distinct complex numbers {a1, …, an}, the set {ea1z, …, eanz} is linearly independent over C(z).
The function ez is transcendental over C(z).
## exp and expm1
Some calculators provide a dedicated exp(x) function designed to provide a higher precision than achievable by using ex directly.[12][13]
Based on a proposal by William Kahan and first implemented in the Hewlett-Packard HP-41C calculator in 1979, some scientific calculators, computer algebra systems and programming languages (for example C99[14]) support a special exponential minus 1 function alternatively named E^X-1, expm1(x),[14] expm(x),[12][13] or exp1m(x) to provide more accurate results for values of x near zero compared to using exp(x)-1 directly.[12][13][14] This function is implemented using a different internal algorithm to avoid an intermediate result near 1, thereby allowing both the argument and the result to be near zero.[12][13] Similar inverse functions named lnp1(x),[12][13] ln1p(x) or log1p(x)[14] exist as well.[nb 1]
## Notes
1. ^ For a similar approach to reduce round-off errors of calculations for certain input values see trigonometric functions like versine, vercosine, coversine, covercosine, haversine, havercosine, hacoversine, hacovercosine, exsecant and excosecant.
## References
1. ^ Goldstein, Lay; Schneider, Asmar (2006). Brief calculus and its applications (11th ed.). Prentice–Hall. ISBN 0-13-191965-2.
2. ^ Courant; Robbins (1996). Stewart, ed. What is Mathematics? An Elementary Approach to Ideas and Methods (2nd revised ed.). Oxford University Press. p. 448. ISBN 0-13-191965-2. This natural exponential function is identical with its derivative. This is really the source of all the properties of the exponential function, and the basic reason for its importance in applications…
3. ^ a b Rudin, Walter (1987). Real and complex analysis (PDF) (3rd ed.). New York: McGraw-Hill. p. 1. ISBN 978-0-07-054234-1.
4. ^ In pure mathematics, the notation log x generally refers to the natural logarithm of x or a logarithm in general if base is immaterial.
5. ^ The notation ln x is the ISO standard and is prevalent in the natural sciences and secondary education (US). However, some mathematicians (e.g., Halmos) have criticized this notation and prefer to use log x for the natural logarithm of x.
6. ^ Converse; Durrell (1911). Plane and spherical trigonometry. C. E. Merrill Co. p. 12. Inverse Use of a Table of Logarithms; that is, given a logarithm, to find the number corresponding to it, (called its antilogarithm) ...
7. ^ a b Eli Maor, e: the Story of a Number, p.156.
8. ^ a b John J O'Connor; Edmund F Robertson. "The number e". School of Mathematics and Statistics. University of St Andrews, Scotland. Retrieved 2011-06-13.
9. ^ "A.2.2 The exponential function." L. Lorentzen and H. Waadeland, Continued Fractions, Atlantis Studies in Mathematics, page 268.
10. ^ Rudin, Walter (1976). Principles of Mathematical Analysis (PDF). New York: McGraw-Hill. p. 182. ISBN 9780070542358.
11. ^ Apostol, Tom M. (1974). Mathematical Analysis (2nd ed.). Reading, Mass.: Addison Wesley. p. 19. ISBN 978-0201002881.
12. HP 48G Series – Advanced User's Reference Manual (AUR) (4 ed.). Hewlett-Packard. December 1994 [1993]. HP 00048-90136, 0-88698-01574-2. Retrieved 2015-09-06.
13. HP 50g / 49g+ / 48gII graphing calculator advanced user’s reference manual (AUR) (2 ed.). Hewlett-Packard. 2009-07-14 [2005]. HP F2228-90010. Retrieved 2015-10-10.Searchable PDF
14. ^ a b c d Beebe, Nelson H. F. (2002-07-09). "Computation of expm1 = exp(x)−1" (PDF). 1.00. Salt Lake City, Utah, USA: Department of Mathematics, Center for Scientific Computing, University of Utah. Retrieved 2015-11-02.<|endoftext|>
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In this page we have Class 10 Science Acid base and Salts Practice test Paper . Hope you like them and do not forget to like , social shar
and comment at the end of the page.
VERY SHORT ANSWER QUESTIONS
Question 1)Write the neutralization reaction of acids? Question 2Write the name of the products obtained when zinc metal pieces are dropped into sodium hydroxide bottle. Solution
2NaOH +Zn-> Na2ZnO2 + H2
Products are Sodium zincate and hydrogen
Question 3) What is the nature of baking soda? Question 4)What are olfactory indicators? Solution
olfactory indicators are those indicators that help to identify whether the given solution is acidic or basic by changing their smell instead of colour as other indicators do.
olfactory indicator" implies that this is a compound whose can smell you can detect... vanilla, clove and onion are all olfactory indicators as these do not give their distinct odour in highly alkaline(basic) medium.
Question 5) Write an equation for the action of dilute hydrochloric acid on marble chips. Solution
arble is nothing but calcium carbonate
CaCO3 + HCl -> CaCl2 + H20 + CO2
Question 6) Write the chemical name and formula of washing soda? Solution
Question 7) Write the preparation of sodium hydroxide? Question 8) Name the raw materials required to manufacture bleaching powder. Question 9) Write the chemical formula of plaster of paris and gypsum? Question 10) What happen when excess of CO2 passes from lime water? Question 11) Why does dry HCI gas not change the colour of the dry litmus paper? Question 12) What effect does the concentration of H+ (aq) have on acidic nature of the solution? Question 13) Write the reaction between dilute NaOH solution and dilute HCI acid. Question 14) Why does an aqueous solution of an acid conduct electricity? Question 15)What is chloro- alkali process. Solution
When concentrated solution of sodium chloride is electrolysed it forms Chlorine gas, Sodium hydroxide and Hydrogengas. It is called chlor-alkali process because of the products formed- Chlor for chlorine gas and alkali for Sodium hydroxide. Chlorine is formed at anode and Hydrogen is formed at the cathode and Na(OH)2 is formed near the cathode.
The main products formed are: Chlorine and Sodium Hyroxide
Question 16) Write the name and the chemical formula of the organic acid present in vinegar. Solution
name of the acid - acetic acid (CH3C00H)
Question 17) What are alkalies? Given one example of alkalies? Question 18) Is Water a strong acid or a weak acid? Question 19) What is the common name of the compound CaOCl2? Solution
Calcium Oxy chloride. Comman name is bleaching power
Question 20) Name the acids present in wasp sting. Question 21) what is the pH of a neutral solution? Solution
Question 22) Name two indicators which are widely used in laboratories. Question 23) What are olfactory indicators? Give an example Question 24) What is brine? Question 25) What happens to the temperature of water when few drops of concentrated sulphuric acid is added to it? Question 26) Which compound of sodium is used for softening of hard water?
SHORT ANSWER QUESTIONS
Question 1) Why do HCI, HNO3 etc., show acidic characters in aqueous solution while solutions of compounds like alcohol and glucose do not show acidic character? Question 2) What are hydronium ions? Question 3) What is meant by strong acids and weak acids? Classify the following into strong acids and weak acids :-
HCl, HNO3, H2CO3, H2SO3, CH3COOH Solution
A strong acid dissociates completely (100%)
A weak acid is only partly dissociated (less than 100%)
A strong acid has a pH 1.
pH of weak acid is 3-5
Hydrochloric acid, sulphuric acid.
Acetic acid, Formic acid
All the HCl molecules becomes into hydrogen ions & chloride ions when they are dissolved in water
Some of the acid molecules become ions. most of them stay as acid molecules
Question 4) What happens when a solution of sodium hydrogen carbonate is heated? Write equation of the reaction is involved. Question 5)What happens when base react with non metal oxide? Question 6) What is water of crystallization? Give some examples of salt having water of crystallization? Give some example of salt having water of crystallization? Question 7) Classify each of the following substance as a weak acid, strong acid, weak base, strong base, both a weak acid and a weak base, or neither an acid nor a base :
Question 8) What would happen if a small amount of copper oxide is taken in a beaker and dil. HCI is added to it? Question 9) How does plaster of paris reacts with water? Write down the chemical equation? Question 10) What is dilute? Why care must be taken while mixing concentrated nitric acid or sulphuric acid with water? Question 11) Show with the help of an equation show that metal carbonates liberate carbon dioxide on reaction with dilute acid. Question 12) Write the balanced molecular equations showing the complete neutralizations of the following.
HNO3 by NaOH
Ca(OH)2 by HI
HNO3 by KOH Solution
Question 13) Write the formulas of the acid and the base that formed the following salts.
CH3COONa: CH3COOH and NaOH
CuSO4: H2SO4 and Cu(OH)2 KClO3: HClO3 and KOH
Al2(SO4)3 : Al(OH)3 and H2SO4 NH4Cl: NH4OH and HCL Ba(NO2): Ba(OH)2 and HNO2 NH4NO3: NH3 and HNO3
Question 14) Write two observations you would make when quicklime is added to water. Question 15) What are the uses of Bleaching powder? Solution
(i) It is used for bleaching cotton and linen in the textile industry, for bleaching wood pulp in paper factories and for bleaching washed clothes in laundry
(ii) It is used as an oxidising agent in many chemical industries
(iii) It is used for disinfecting drinking water to make it free of germs
Question 16) State the chemical property in each case on which the following uses of baking soda are based :-
as an antacid
as a constituent of baking powder. Question 17) How is plaster of Paris obtained? What reaction is involved in the setting of a paste of plaster of Paris? Question 18)Acids show their properties only in the presence of water? explain? Question 19)A weak acid is added to a concentrated solution of hydrochloric acid. Does the solution become more or less acidic? Question 20) Write the chemical name and formula of washing soda. What happens when crystals of washing soda are exposed to air? Solution
Write word equations and then balanced equations for the reaction taking place when-
dilute sulphuric acid reacts with zinc granules.
dilute hydrochloric acid reacts with magnesium ribbon.Give two important uses of caustic soda and baking soda.
On which factors does the strength of an acid or a base depend? Explain.
Question 22)Explain the following by giving examples:
a. how metal oxides react with acids?
b.How non- metal oxides react with baes? Question 23) a)Why does acidic solution conduct electricity?
(b)Can basic solution conduct electricity?
(c)Can separation of H+ ions in acids take place when HCI is added to a non- aqueous solution? Solution
Both acidic and basic solutions in water conduct electricity. Acids, when dissolved in water release the H+ and bases when dissolved in water release the OH- ions. These ions are charged species and so act as charge carriers. In other words the conductivity of these solutions is due to the movement of these ions.
LONG ANSWER QUESTIONS Question 1) How common salt prepared from sea water? Solution
Sea water contains a large amount of common salt and the salts of other metals dissolved in it. Near the sea-shore, the sea water is collected in shallow pits and allowed to evaporate in sunshine. In a few days, the water evaporates, leaving behind salt. The salt so obtained is collected and transported to big factories, where it is purified and packed for consumtions
Question 2) What is observed when – (i) dilute sulphuric acid is added to solid sodium carbonate. (ii) hot concentrated sulphuric acid is added to sulphur. (iii) Sulphur doioxide is passed through lime water? Also write chemical equations to represent the chemical reaction taking place in each case. Question 3) A student dropped few pieces of marble in dilute hydrochloric acid, contained in a test tube. The evolved gas was then passed through lime water. What change would be observed in lime water? What will happen if excess of gas is passed through lime water? Write balanced chemical equations for all the changes observed. Question 4) A compound X of sodium forms a white powder. It is a constituent of baking powder and is used in some antacid prescriptions. When heated, X gives out a gas and steam. The gas forms a white precipitate with limewater. Write the chemical formula and name of X and the chemical equation for its decomposition on heating. What is its role in baking powder and in antacids? Solution
Baking powder consists of sodium bicarbonate, tartaric acid and small amount of starch. Hence, the compound X of sodium in question which is a constituent of baking powder and is used in antacids is sodium bicarbonate or sodium hydrogencarbonate, which has the chemical formula NaHCO3. It is commonly known as baking soda.
On heating, it decomposes to give sodium carbonate, water and carbon dioxide.
NaHCO3 -> Na2CO3 + H2O +CO2
This gas(CO2) when passed through lime water turns it milky due to the formation of calcium carbonate, which is insoluble in water.
Ca(OH)2 + CO2 ? CaCO3 + H2O
a milk man adds a very small amount of baking soda to fresh milk:
Why does he shift the pH of the fresh milk from 6 to slightly alkaline?
What do you expect to observe when fresh milk comes to boil?
Why does this milk take a long time to set as a curd?
The colour of litmus paper changes only in the presence of ions like hydrogen (H+) or hydronium (H3O+) ions. HCl can produce these ions only in the form of aqueous solution. Hence dry HCl gas does not change the colour of dry litmus paper
Antacid reacts with acid in the stomach and neutralizes it
Question 8) (a)What is pH scale and what’s its range? How is it related to hydronium ion concentration?
(b)Explain any two roles played by pH levels of various chemicals in living organisms. Solution
The term pH is defined as the negative logarithm of H+ ion concentration of a given solution; the concentration being expressed as moles per litre.
athematically pH = log [H+] pH stands for: Power of hydrogen ion concentration, p for power and H for H+ ion concentration.
Some important benchmark values in the pH scale are: pH = 7 indicates neutral solutions e.g., aqueous solutions. pH > 7 to 14 indicates alkaline solutions and pH < 7 to 0 indicate acidic solutions<|endoftext|>
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### Friday Quiz
These are some homework problems that Tricia Colclaser gave her Honors Pre-Calculus students this week. She said they are tougher than anything they might find in their text books.
"Practicing on harder problems gives you a chance to really put all of your skills to work," the hand out says. Some of the skills that might come in handy are Descartes Rule of Signs, Boundedness, and Synthetic division.
Factor each polynomial. (Students were also asked to sketch graphs -- this part is optional.)
Hint: Find all possible rational zeros and test using synthetic division.
1. f(x) = -x^7 - x^6 - x^5 - x^4 + 12x^3 + 12x^2
2. g(x) = 4x^7 - 16x^6 + 7x^5 + 4x^4 +92x^3 - 56x^2 - 288
3. h(x) = x^6 + 2x^5 + 2x^4 + 2x^3 - x^2 - 4x - 2
By Michael Alison Chandler | March 4, 2009; 3:14 PM ET
Previous: Honors Pre-Calculus | Next: Meet Celebrity Mathematician Danica McKellar
I hadn't heard of synthetic division, so I looked it up after the last post. Turns out I'd been doing essentially the same thing without knowing the name. More importantly, I also stumbled across the rational root test, which I was previously unfamiliar with.
The rational root test says that, given a polynomial with integer coefficients, the *possible* rational roots can be obtained by taking all factors of the constant term, divided by all factors of the coefficient of the highest power term. All that remains is to try them out.
So, with that in mind:
1) -x^2 is a factor, leaving x^5+x^4+x^3+x^2-12x-12. Possible roots are ±(1,2,3,4,6,12/1) = ±1,±2,±3,±4,±6,±12.
Synthetic division with -1 gives no remainder, so x+1 is a factor, leaving x^4+x^2-12.
If we substitute y = x^2, then this becomes y^2+y-12, which is a quadratic, and factors to (y+4)(y-3), with zeros at -4 and 3. So x = ±sqrt(-4) = ±2i and x = ±sqrt(3) are zeros.
So, if we limit to reals, the factorization is -x^2*(x+1)*(x+sqrt(3))*(x-sqrt(3))*(x^2+4).
Alternately, we could write -x^2*(x+1)*(x+sqrt(3))*(x-sqrt(3))*(x+2i)*(x-2i).
2) Ok, this one's a bear, because it's got a ton of possibilities. So I simply found the factors of 288, the factors of 4, put them into Excel, and calculated possible rational roots. The only zero I see is at x = 2. So, that gives (x-2)*(4x^6-8x^5-9x^4-14x^3+64x^2+72x+144). I suspect there are no other real roots.
3) Well, this is more like it. x-1 gives x^5+3x^4+5x^3+7x^2+6x+2. x+1 then gives x^4+2x^3+3x^2+4x+2. x+1 again gives x^3+x^2+2x+2. x+1 a third time gives x^2+2, which gives x = ±i*sqrt(2). So, (x-1)*(x+1)^3*(x^2+2), or (x-1)*(x+1)^3*(x+i*sqrt(2))*(x-i*sqrt(2)).
Posted by: tomsing | March 6, 2009 2:02 PM | Report abuse
The comments to this entry are closed.<|endoftext|>
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# Order (group theory)
(Redirected from Order of a group)
In group theory, a branch of mathematics, the term order is used in three different senses:
• The order of a group is its cardinality, i.e., the number of elements in its set.
• The order of an element a of a group, sometimes also period length or period of a, is the smallest positive integer m such that am = e (where e denotes the identity element of the group, and am denotes the product of m copies of a). If no such m exists, a is said to have infinite order.
• The ordering relation of a partially or totally ordered group, which is not related to the above notions.
This article is about the first two senses of order. They are closely related: the order of an element a is equal to the order of its cyclic subgroupa⟩ = {ak for k an integer}, the subgroup generated by a.
The order of a group G is denoted by ord(G) or |G| and the order of an element a is denoted by ord(a) or |a|. Thus, |a| = |a|.
Lagrange's theorem states that for any subgroup H of G, the order of the subgroup divides the order of the group: |H| is a divisor of |G|. In particular, the order |a| of any element is a divisor of |G|.
## Example
Example. The symmetric group S3 has the following multiplication table.
e s t u v w
e e s t u v w
s s e v w t u
t t u e s w v
u u t w v e s
v v w s e u t
w w v u t s e
This group has six elements, so ord(S3) = 6. By definition, the order of the identity, e, is one, since e1 = e. Each of s, t, and w squares to e, so these group elements have order two: |s| = |t| = |w| = 2. Finally, u and v have order 3, since u3 = vu = e, and v3 = uv = e.
## Order and structure
The order of a group G and the orders of its elements give much information about the structure of the group. Roughly speaking, the more complicated the factorization of |G|, the more complicated the structure of G.
For |G| = 1, the group is trivial. In any group, only the identity element a = e has ord(a) = 1. If every non-identity element in G is equal to its inverse (so that a2 = e), then ord(a) = 2; this implies G is abelian since ${\displaystyle ab=(ab)^{-1}=b^{-1}a^{-1}=ba}$ . The converse is not true; for example, the (additive) cyclic group Z6 of integers modulo 6 is abelian, but the number 2 has order 3:
${\displaystyle 2+2+2=6\equiv 0{\pmod {6}}}$ .
The relationship between the two concepts of order is the following: if we write
${\displaystyle \langle a\rangle =\{a^{k}:k\in \mathbb {Z} \}}$
for the subgroup generated by a, then
${\displaystyle \operatorname {ord} (a)=\operatorname {ord} (\langle a\rangle ).}$
For any integer k, we have
ak = e if and only if ord(a) divides k.
In general, the order of any subgroup of G divides the order of G. More precisely: if H is a subgroup of G, then
ord(G) / ord(H) = [G : H], where [G : H] is called the index of H in G, an integer. This is Lagrange's theorem. (This is, however, only true when G has finite order. If ord(G) = ∞, the quotient ord(G) / ord(H) does not make sense.)
As an immediate consequence of the above, we see that the order of every element of a group divides the order of the group. For example, in the symmetric group shown above, where ord(S3) = 6, the orders of the elements are 1, 2, or 3.
The following partial converse is true for finite groups: if d divides the order of a group G and d is a prime number, then there exists an element of order d in G (this is sometimes called Cauchy's theorem). The statement does not hold for composite orders, e.g. the Klein four-group does not have an element of order four). This can be shown by inductive proof.[1] The consequences of the theorem include: the order of a group G is a power of a prime p if and only if ord(a) is some power of p for every a in G.[2]
If a has infinite order, then all powers of a have infinite order as well. If a has finite order, we have the following formula for the order of the powers of a:
ord(ak) = ord(a) / gcd(ord(a), k)
for every integer k. In particular, a and its inverse a−1 have the same order.
In any group,
${\displaystyle \operatorname {ord} (ab)=\operatorname {ord} (ba)}$
There is no general formula relating the order of a product ab to the orders of a and b. In fact, it is possible that both a and b have finite order while ab has infinite order, or that both a and b have infinite order while ab has finite order. An example of the former is a(x) = 2−x, b(x) = 1−x with ab(x) = x−1 in the group ${\displaystyle Sym(\mathbb {Z} )}$ . An example of the latter is a(x) = x+1, b(x) = x−1 with ab(x) = x. If ab = ba, we can at least say that ord(ab) divides lcm(ord(a), ord(b)). As a consequence, one can prove that in a finite abelian group, if m denotes the maximum of all the orders of the group's elements, then every element's order divides m.
## Counting by order of elements
Suppose G is a finite group of order n, and d is a divisor of n. The number of order-d-elements in G is a multiple of φ(d) (possibly zero), where φ is Euler's totient function, giving the number of positive integers no larger than d and coprime to it. For example, in the case of S3, φ(3) = 2, and we have exactly two elements of order 3. The theorem provides no useful information about elements of order 2, because φ(2) = 1, and is only of limited utility for composite d such as d=6, since φ(6)=2, and there are zero elements of order 6 in S3.
## In relation to homomorphisms
Group homomorphisms tend to reduce the orders of elements: if fG → H is a homomorphism, and a is an element of G of finite order, then ord(f(a)) divides ord(a). If f is injective, then ord(f(a)) = ord(a). This can often be used to prove that there are no (injective) homomorphisms between two concretely given groups. (For example, there can be no nontrivial homomorphism h: S3 → Z5, because every number except zero in Z5 has order 5, which does not divide the orders 1, 2, and 3 of elements in S3.) A further consequence is that conjugate elements have the same order.
## Class equation
An important result about orders is the class equation; it relates the order of a finite group G to the order of its center Z(G) and the sizes of its non-trivial conjugacy classes:
${\displaystyle |G|=|Z(G)|+\sum _{i}d_{i}\;}$
where the di are the sizes of the non-trivial conjugacy classes; these are proper divisors of |G| bigger than one, and they are also equal to the indices of the centralizers in G of the representatives of the non-trivial conjugacy classes. For example, the center of S3 is just the trivial group with the single element e, and the equation reads |S3| = 1+2+3.<|endoftext|>
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## Math 117 - Chapter 4 Study Guide
1. Draw a triangle and identify the type of problem (SAS, SSA, etc). Solve if possible. Round your final answers to one decimal place. (While final answers should be rounded to one decimal place, be careful about rounding intermediate steps). Look at problems 4.1.1-16 and 4.2.1-16. Be aware of the ambiguous case.
2. Find the area of the triangle. Look at problems 4.1.17-22.
3. Draw a triangle and identify the type of problem (SAS, SSA, etc). Solve if possible. Round your final answers to one decimal place. (While final answers should be rounded to one decimal place, be careful about rounding intermediate steps). Look at problems 4.1.1-16 and 4.2.1-16. Be aware of the ambiguous case.
4. Consider the given complex number. Graph the complex number and find its absolute value. Look at problems 4.4.1-8, 13-20
5. Draw a triangle and identify the type of problem (SAS, SSA, etc). Solve if possible. Round your final answers to one decimal place. (While final answers should be rounded to one decimal place, be careful about rounding intermediate steps). Look at problems 4.1.1-16 and 4.2.1-16. Be aware of the ambiguous case.
6. Convert the complex number from standard form into trigonometric form. Look at problems 4.4.13-20
7. Convert the complex number from trigonometric form to standard form. Look at problems 4.4.21-28
8. Find all complex solutions to the equation. Look at problems 4.4.67-72
9. Simplify the following and write the answer in standard form for a complex number. Three parts. Look at problems 4.3.1-44
10. Find the component form of the vector given the initial and terminal points. Look at problems 4.6.1-6
11. Given two vectors, find the angle between the vectors. Look at problems 4.6.63-68.
12. Perform the indicated operation on the complex number in trigonometric form. After performing the operation, write your answer in standard form. Angles have been chosen so the answers have exact values. Three parts. Look at problems 4.4.29-32, 45-46
13. Application of vectors. Look at problems 4.5.25-30.
14. Perform the indicated calculations for the given vectors. Express your answer as a linear combination of the i and j vectors. Look at problems 4.6.45-48
15. You are given the magnitude of two vectors and the angle between them. Find the magnitude of the resultant vector using the law of cosines and then angle the resultant vector forms with the vector u using the law of sines. Look at problems 4.5.17-24
16. Perform the indicated calculations for the given vectors. Look at problems 4.6.9-26
### Notes:
• Some of the problems are directly from the text.
• The test was derived by looking at the chapter review and making problems similar to those problems, so you may want to look at the chapter review first and then if you have problems with those, go back to the problems given above as references.
Points for each problem
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Tot. 8 5 8 5 8 4 4 5 9 4 5 9 5 6 6 9 100<|endoftext|>
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# GMAT Avengers Study Group: Interest Rates:
by on February 4th, 2015
Let’s first talk about Simple Interest and Compound Interest.
Simple Interest – The formula for SI is principal * interest rate * time, where “principal” is the starting amount and “rate” is the interest rate at which the money grows per a given period of time (note: express the rate as a decimal in the formula). Time must be expressed in the same units used for time in the Rate.
Compound Interest – The formula to calculate compound interest:
Final balance = where:
P = the principal (the initial investment).
r = the annual interest rate expressed as a decimal
c = the number of times the interest is compounded each year
n = the number of years the investment collects interest
The important thing is to make sure we know how to calculate these interests and are able to manipulate things accordingly. For example, it may be the case that we have been told to calculate the rate of the interest when everything else would have been given to us. So, it is not like that we will be asked to only calculate SI or CI. All the terms used in the formula are used interchangeably.
IMPORTANT – As long as there is more than one compounding period, then compound interest always earns more than simple interest. We always get more interest, and larger account value overall, when the compounding period decreases; the more compounding periods we have, the more interest we earn.
The articles that we shared on our Facebook Event Page are as follows:
## Practice Problems
Data Sufficiency (Sample answer choices are given below)
• Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
• Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
• BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
• Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.
1) If \$5000 invested for one year at p percent simple annual interest yields \$500,what amount must be invested at k percent simple annual interest for one year to yield the same amount of dollars?
(A) k = 0.8p
(B) k = 8
2) If \$24,000 invested at x percent simple annual interest yields an interest of z dollars at the end of one year. How much is it necessary to invest during one year at y percent simple annual interest in order to get an interest of z dollars?
(A) x = 4
(B) x/y = 2/3
3) If a loan of P dollars, at an interest rate of r percent per year compounded monthly, is payable in n monthly installments of m dollars each, then m is determined by the formula
John and Sue took out loans whose monthly installments were determined by using the formula above. Both loans had the same interest rate and the same number of monthly installments. John’s monthly installment was what percent of Sue’s monthly installment?
(A) The amount of Sue’s loan was 4 times the amount of John’s loan.
(B) Sue’s monthly installment was 4 times John’s monthly installment.
4) On May 1 of last year, Jasmin invested x dollars in a new account at an interest rate of 6 percent per year, compounded monthly. If no other deposits or withdrawals were made in the account and the interest rate did not change, what is the value of x?
(A) As of June 1 of last year, the investment had earned \$200 in interest.
(B) As of July 1 of last year, the investment had earned \$401 in interest.
Problem Solving
1) An investor opened a money market account with a single deposit of \$6000 on Dec. 31, 2001. The interest earned on the account was calculated and reinvested quarterly. The compound interest for the first 3 quarters of 2002 was \$125, \$130, and \$145, respectively. If the investor made no deposits or withdrawals during the year, approximately what annual rate of interest must the account earn for the 4th quarter in order for the total interest earned on the account for the year to be 10 percent of the initial deposit?
(A) 3.1%
(B) 9.3%
(C) 10.0%
(D) 10.5%
(E) 12.5%
2) An investment of d dollars at k percent simple interest yields \$600 interest over a 2-year period. In terms of d, what dollar amount invested at the same rate will yield \$2,400 interest over a 3-year period?
(A) 2d/3
(B) 3d/4
(C) 4d/3
(D) 3d/2
(E) 8d/3
3) A sum of \$16,000 amounts to \$18,400 in two years according to simple interest. What is the rate of interest?
(1) 6%
(2) 6.5%
(3) 7.5%
(4) 8%
(5) 8.5%
4) Sarah invested \$38,700 in an account that paid 6.2% annual interest, compounding monthly. She left the money in this account, collecting interest for a full three-year period. Approximately how much interest did she earn in the last month of this period?
(1) \$239.47
(2) \$714.73
(3) \$2793.80
(4) \$7,888.83
(5) \$15,529.61
5) Marcus deposited \$8,000 to open a new savings account that earned
five percent annual interest, compounded semi-annually. If there were no other transactions in the account, what the amount of money in Marcuss account one year after the account was opened?
(A) \$8,200
(B) \$8,205
(C) \$8,400
(D) \$8,405
(E) \$8,500
6) Jolene entered an 18-month investment contract that guarantees to pay 2 percent interest at the end of 6 months, another 3 percent interest at the end of 12 months, and 4 percent interest at the end of the 18 month contract. If each interest payment is reinvested in the contract, and Jolene invested \$10,000 initially, what will be the total amount of interest paid during the 18-month contract?
(A) \$506.00
(B) \$726.24
(C) \$900.00
(D) \$920.24
(E) \$926.24
7) A sum of money doubles itself at a compound interest in 15 years. In how may years it will become 8 times?
(A) 30 years
(B) 40 years
(C) 45 years
(D) 50 years
(E) 60 years
8) An investment of \$1000 was made in a certain account and earned interest that was compounded annually. The annual interest rate was fixed for the duration of the investment, and after 12 years the \$1000 increased to \$4000 by earning interest. In how many years after the initial investment was made the \$1000 have increased to \$8000 by earning interest at that rate?
(A) 16
(B) 18
(C) 20
(D) 24
(E) 30
9) Louie takes out a three-month loan of \$1000. The lender charges him 10% interest per month compounded monthly. The terms of the loan state that Louie must repay the loan in three equal monthly payments. To the nearest dollar, how much does Louie have to pay each month?
(A) 333
(B) 383
(C) 402
(D) 433
(E) 483
Data Sufficiency
1. D, Each Statement Alone is sufficient to answer the question
2. B, Statement 2 alone is sufficient to answer the question
3. D, Each Statement Alone is sufficient to answer the question
4. D, Each Statement Alone is sufficient to answer the question
Problem Solving
1. E
2. E
3. C
4. A
5. D
6. E
7. C
8. B
9. C
Make sure you check out the event Facebook page to see all the comments that were being exchanged during the live session.
***
EXCITING NEWS!! “The GMAT Avengers Guide” is now available! This eBook is a collection of transcripts, tips, strategies, recommended readings, best practices, and challenge problems to help you beat the GMAT and achieve your dream score. Download your FREE copy NOW!<|endoftext|>
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Hypertension, or high blood pressure, is characterized by the elevation of systolic and/or diastolic arterial blood pressures. It is diagnosed as primary or secondary hypertension. Primary, or essential, hypertension has no known cause. Secondary hypertension has an identifiable cause:21,22
Primary hypertension is of unknown etiology and may be related to genetic and/or environmental factors.23-27 In most populations studied, when environmental and familial risk factors are controlled, blood pressure increases with age.28-32 Hypertension affects over 1 billion people worldwide and approximately one-quarter of American adults. It is estimated that less than 5% have a curable cause.1,33,34 The World Health Organization (WHO) estimates that hypertension is the third leading cause or mortality worldwide.35 Furthermore, the risk of cardiovascular disease doubles for adults age 40-70 years for each rise of 20 mmHg in systolic or 10 mmHg in diastolic.36 Given the prevalence of hypertension, it is critical for oral healthcare providers to understand the current diagnostic categories (Table 1).37,38
|Category||Systolic (mmHg)||Diastolic (mmHg)|
|Optimal blood pressure||< 120||< 80|
|Normal blood pressure||< 130||< 85|
|Prehypertension||130 to 139||85 to 89|
|Grade 1 hypertension (mild)||140 to 159||90 to 99|
|Grade 2 hypertension (moderate)||160 to 179||100 to 109|
|Grade 3 hypertension (severe)||> 180||> 110|
|Isolated systolic hypertension (grade 1)||140 to 159||< 90|
|Isolated systolic hypertension (grade 2)||> 160||< 90|
Your session is about to expire. Do you want to continue logged in?
WARNING! You did not finish creating your certificate. Please click CONTINUE below to return to your previous page to complete the process. Failure to complete ALL the steps will result in a loss of this test score, and you will not receive credit for this course.<|endoftext|>
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Subject: Calculus
# Antiderivative Indefinite Integral
We say that the inverse of addition is subtraction, the inverse for multiplication is division and the inverse of raising to powers and extracting roots are a few of the famous inverse operations. Then, what's the inverse process of finding the derivative? Any guess? Well, the inverse process of finding the derivative is integration or antidifferentiation (as its name suggest). However, not all differentiable functions has its corresponding anti-derivative since there is a class of functions in which antidifferentiation is impossible.
Nevertheless, let me lead you directly to our topic which is anti-derivative of functions. Basically, when a function f(x) is differentiated with respect to x, then its notations may become f'(x). But, when we want to bring f'(x) back to its original function f(x), then all we have to do is take the inverse process of differentiation. And that means we have to integrate or anti-differentiate f"(x) to make it f(x).
## Notation
Integration is conventionally performed by indicating a symbol \int which is an elongated letter S, standing for summa (Latin for "sum" or "total"). Though Newton has a different notation for integration which is a letter placed inside a box, Leibniz's notation (which is the \int ) is the most widely used notation for integration and the one which we shall be using throughout the Calculus section.
So we now have the symbol for integration which means that the expression
\int f(x)dx
is read as " the integral of the function f(x) with respect to x", where f(x) is also called as the integrand and the dx is called the variable of integration.
### Indefinite and Definite Integral
There are also two types of integral which must not be taken for granted especially when dealing with its notations. These are definite and indefinite integrals whose notations are given below.
\int f(x)dx
#### Definite integral
\int_{a}^{b} f(x)dx
a and b are called lower and upper limits respectively.
## Important Concepts and Theorems
Now you know a lot of basic information about integration and I bet you're really good to go. But we're just getting started. Before introducing to you how to perform antidifferentiation or integration, allow me first to share to you some pretty important theorems and concepts behind integration.
### Integral as an area
A function ƒ of a real variable x and an interval [a, b] of the real line, the definite integral
\int_{a}^{b} f(x)dx
is defined informally to be the net area of the region in the xy-plane bounded by the graph of ƒ, the x-axis, and the vertical lines x = a and x = b. Refer to the figure at the right.
### Indefinite integral
A collection of functions which forms the set of all antiderivative of a certain function is called an indefinite integral. This means that the integral of a function can be many and is non-unique.
Since an indefinite integral is a set all anti-derivatives of function, and due to the fact that the derivative of a constant is zero, thus any constant may be added to an antiderivative and still correspond to the same integral. If we write the constant as C, then an indefinite integral when the antiderivative of f is F ,can be written as
\int f(x)dx= F(x)+C
C is called the constant of integration and is true for all indefinite integrals for the reasons stated above.
### Definite integral
Unlike indefinite integrals, definite integrals can easily be determined because of the upper and lower limits appearing in the integral sign and unlike indefinite integrals, definite integral is unique. Also, it has no constant on integration C. Evaluating an indefinite integral is just finding the anti-derivative of a function. Well, that's not the case for indefinite integrals as stated in the theorem below.
### Theorem
Let f be a well-behaved function such that its anti-derivative is F. Then the definite integral of f is given by
\int_{a}^{b} f(x)dx= F(b)-F(a)
where a and b are the endpoints in the interval [a, b].
## Finding the Integral of a Function
Finding the derivative of any function varies from the type of the function. For example, the formula for finding the derivative of a polynomial function does not necessarily apply for trigonometric functions. While there is really no general formula for finding the integral of any given function, we can refer to tables of integrals for integral values. For beginner's sake here are some of the integrals of a few basic functions.
### Basic Functions
\int dx = x + C
\int a dx = ax + C
\int x dx = \frac{1}{2}\cdot x^2 + C
\int ax dx = \frac{a}{2}\cdot x^2 + C
In general, the integral of any function with r-exponent is given by the following formula:
\int x^r dx = \frac{1}{r+1}\cdot x^{r+1} + C
### Exponential and logarithmic functions
\int \frac{dx}{x} = \ln{x}+ C
\int \frac{dx}{ax} = \frac{1}{a}\ln{x}+ C
\int e^{x}dx = e^{x}+ C
\int e^{bx}dx = \frac{1}{b}e^{bx} + C
where b is a constant
\int a^{x}dx = \frac{1}{\ln{a}}\cdot a^{x} + C
where a is a constant
\int a^{bx}dx = \frac{1}{b\cdot \ln{a}}\cdot a^{bx} + C
where a and b are constants
### Ttrigonometric functions
\int \sin{u} du = -\cos{u} + C
\int \cos{u} du = \sin{u} + C
\int \tan{u} du = \ln{|\sec{u}|} + C
\int \cot{u} du = \ln{|\sin{u}|} + C
\int \sec{u} du = \ln{|\sec{u}+\tan{u}|} + C
\int \csc{u} du = \ln{|\csc{u}-\cot{u}|} + C
We can always check the integrals above if these really are the corresponding integrals of the functions given above. Since integral is an inverse of differentiation as mentioned above, we can differentiate the integral and the answer must be the same to the original function. For example;
\int x dx = \frac{1}{2}\cdot x^2 + C
So let's differentiate the integral above,
D_{x}[ \frac{1}{2}\cdot x^2 + C]
= 2\cdot \frac{1}{2}\cdot x + 0]
=x
which is just the original function, isn't it? the same is true for all types of integrals. You can try on your own all the integrals above. For more integrals, just go to Table of Integrals and List of Integrals.
### Example #1
Finding the area under a curve f(x) is determined by integrating f(x). What figure/shape represents the area whose curve is f(x)=c, where c is any constant, and the integration limits ranges from 0 to c.
Square
### Example #2
What would be the area of the region under the curve f(x)=e^x\sin{\log{x}}+x^2\cos{e^x} when a=b?
Zero
### Example #3
Finding the area under a curve f(x) is determined by integrating f(x). What figure/shape represents the area whose curve is f(x)=c, where c is any constant, and the integration limits ranges from c to 3c.
Rectangle
### Example #4
Solve the following indefinite integral.
\int e^{-x}dx
\int \frac{1}{x}dx = \ln{x} + C
### Example #5
Solve the following.
\int_{a}^{b} f(x) dx + f(a)
f(b)
NEXT TOPIC: Sum rule in integration<|endoftext|>
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Practice Test on Linear Inequations | Linear Inequality Questions and Answers
Practice Test on Linear Inequation has different types of questions. Students can test their skills and knowledge on linear inequations problems by solving all the provided questions on this page. The questions are mainly related to inequalities and finding the solution to the given inequation and draw a graph for the obtained solution set. You can easily draw a graph on a numbered line.
1. Write the equality obtained?
(i) On subtracting 1 from each side 3 > 7
(ii) On adding 3 to each side 12 < 5
(iii) On multiplying (-2) to each side 11 < 4
(iv) On multiplying 4 to each side 15 > 2
Solution:
(i) 3 – 1 > 7 – 1
2 > 6
(ii) 12 + 3 < 5 + 3
15 < 8
(iii) 11 x (-2) 4 x (-2)
-22 < -8
22 > 8
(iv) 15 x 4 > 2 x 4
60 > 8
2. Write the word statement for the following?
(i) x ≥ 15
(ii) x < 2
(iii) x ≤ -5
(iv) x > 16
(v) x ≠ 6
Solution:
(i) The variable x is greater than equal to 15. The possible values of x are 15 and more than 15.
(ii) The variable x is less than 2. The possible values of x are less than 2.
(iii) The variable x is less than and equal to -5. The possible values of x are less than -5.
(iv) The variable x is greater than 16. The possible values of x are more than 16.
(v) The variable x is not equal to 6. The possible values of x are all real numbers other than 6.
3. Find the solution set for each of the following inequations. x ∈ N
(i) x + 5 < 12
(ii) x – 6 > 5
(iii) 5x + 10 ≥ 17
(iv) 2x + 3 ≤ 6
Solution:
(i) x + 5 < 12
Subtract 5 from both sides.
x + 5 – 5 < 12 – 5
x < 7
Replacement set = {1, 2, 3, 4, 5 . .}
Solution set S = {1, 2, 3, 4, 5, 6}
(ii) x – 6 > 5
x – 6 + 6 > 5 + 6
x > 11
Replacement set = {1, 2, 3, 4, 5 . .}
Solution set S = {12, 13, 14, 15, . . . }
(iii) 5x + 10 ≥ 17
Subtract 10 from both sides.
5x + 10 – 10 ≥ 17 – 10
5x ≥ 7
Divide 5 by each side.
5x/5 ≥ 7/5
x ≥ 1.4
Replacement set = {1, 2, 3, 4, 5 . .}
Solution set S = {2, 3, 4, 5 . . .}
(iv) 2x + 3 ≤ 6
Subtract 3 from both sides of the inequation
2x + 3 – 3 ≤ 6 – 3
2x ≤ 3
Both sides of the inequation divide by 2.
2x/2 ≤ 3/2
x ≤ 1.5
Replacement set = {1, 2, 3, 4, 5 . .}
Solution set S = {1, 1.5}
4. Find the solution set for each of the following inequations and represent it on the number line.
(i) 3 < x < 10, x ∈ N
(ii) 3x + 2 ≥ 6, x ∈ N
(iii) 3x/2 < 5, x ∈ N
(iv) -4 < 2x/3 + 1 < – 2, x ∈ N
Solution:
(i) 3 < x < 10, x ∈ N
The two cases are 3 < x and x < 10
It can also represent as x > 3 and x < 10
Replacement set = {1, 2, 3, 4, 5 . .}
The solution set for x > 3 is 4, 5, 6, 7 . . . i.e P = {4, 5, 6, 7 . . .}
And the solution set for x < 10 is 1, 2, 3, 4, 5, 6, 7, 8, 9 i.e Q = {1, 2, 3, 4, 5, 6, 7, 8, 9}
Therefore, solution set of the given inequation = P ∩ Q = {4, 5, 6, 7, 8, 9}
Let us represent the solution set graphically.
The solution set is marked on the number line by dots.
(ii) 3x + 2 ≥ 6, x ∈ N
Subtract 2 from both sides
3x + 2 – 2 ≥ 6 – 2
3x ≥ 4
Divide each side by 3
3x/3 ≥ 4/3
x ≥ 1.33
Replacement set = {1, 2, 3, 4, 5 . .}
Solution set S = {2, 3, 4, 5, . . }
Let us represent the solution set graphically.
The solution set is marked on the number line by dots.
(iii) 3x/2 < 5, x ∈ N
Multiply both sides by 2.
3x/2 x 2 < 5 x 2
3x < 10
divide both sides by 3
3x/3 < 10/3
x < 3.33
Replacement set = {1, 2, 3, 4, 5 . .}
Solution Set S = {1, 2}
Let us represent the solution set graphically.
The solution set is marked on the number line by dots.
(iv) -4 < 2x/3 + 1 < – 2, x ∈ N
The two cases are -4 < 2x/3 + 1 and 2x/3 + 1 < – 2
Case I: -4 < 2x/3 + 1
Subtract 1 from both sides
-4 – 1 < 2x/3 + 1 – 1
-5 < 2x/3
Multiply each side by 3
-5 x 3 < 2x/3 x 3
-15 < 2x
Divide each side by 2
-15/2 < 2x/2
-7.5 < x
x > 7.5
Replacement Set = {1, 2, 3, 4, 5 . .}
Solution Set P = {8, 9, 10, 11 . . . }
Case II: 2x/3 + 1 < – 2
Subtract 1 from both sides
2x/3 + 1 – 1 < – 2 – 1
2x/3 < -3
Multiply 3 to both sides
2x/3 x 3 < -3 x 3
2x < -9
Divide both sides by 2
2x/2 < -9/2
x < -4.5
4.5 > x
Replacement set = {1, 2, 3, 4, 5 . .}
Solution set Q = {1, 2, 3}
Therefore, required solution set S = P ∩ Q
S = Null
5. Find the solution set for each of the following and represent the solution set graphically?
(i) x – 6 < 4, x ∈ W
(ii) 6x + 2 ≤ 20, x ∈ W
(iii) 7x + 3 < 5x + 9, x ∈ W
(iv) 3x – 7 > 5x – 1, x ∈ I
Solution:
(i) x – 6 < 4, x ∈ W
x – 6 + 6 < 4 + 6
x < 10
Replacement set = {0, 1, 2, 3, 4, 5, 6, …}
Therefore, solution set S = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
Let us represent the solution set graphically.
The solution set is marked on the number line by dots.
(ii) 6x + 2 ≤ 20, x ∈ W
Subtract 2 from both sides
6x + 2 – 2 ≤ 20 – 2
6x ≤ 18
Divide each side by 6
6x/6 ≤ 18/6
x ≤ 3
Replacement set = {0, 1, 2, 3, 4, 5, 6, …}
Therefore, solution set S = {0, 1, 2, 3}
Let us represent the solution set graphically.
The solution set is marked on the number line by dots.
(iii) 7x + 3 < 5x + 9, x ∈ W
Move variables to one side and constants to other side of inequation
7x – 5x < 9 – 3
2x < 6
Divide each side by 2
2x/2 < 6/2
x < 3
Replacement set = {0, 1, 2, 3, 4, 5, 6, …}
Therefore, solution set S = {0, 1, 2}
Let us represent the solution set graphically.
The solution set is marked on the number line by dots.
(iv) 3x – 7 > 5x – 1, x ∈ I
Move variables to one side and constants to another side of inequation
-7 + 1 > 5x – 3x
-6 > 2x
divide 2 by each side
-6/2 > 2x/2
-3 > x
Replacement set ={ . . . -4, -3, -2, -1, 0, 1, 2, 3, . . .}
Solution set = { -2, -1, 0, 1, 2, . . . }
Let us represent the solution set graphically.
The solution set is marked on the number line by dots.<|endoftext|>
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# How do you verify the identity (cos3beta)/cosbeta=1-4sin^2beta?
Jan 14, 2017
$\cos \left(3 \beta\right) = \cos \left(2 \beta + \beta\right)$.
Use the sum formula $\cos \left(A + B\right) = \cos A \cos B - \sin A \sin B$ to expand:
LHS:
$\frac{\cos 2 \beta \cos \beta - \sin 2 \beta \sin \beta}{\cos} \beta$
Expand using the identities $\cos 2 x = 2 {\cos}^{2} x - 1$ and $\sin 2 x = 2 \sin x \cos x$:
$\frac{\left(2 {\cos}^{2} \beta - 1\right) \cos \beta - \left(2 \sin \beta \cos \beta\right) \sin \beta}{\cos} \beta$
$\frac{2 {\cos}^{3} \beta - \cos \beta - 2 {\sin}^{2} \beta \cos \beta}{\cos} \beta$
Use ${\sin}^{2} x + {\cos}^{2} x = 1$:
$\frac{2 {\cos}^{3} \beta - \cos \beta - 2 \left(1 - {\cos}^{2} \beta\right) \cos \beta}{\cos} \beta$
$\frac{2 {\cos}^{3} \beta - \cos \beta - 2 \left(\cos \beta - {\cos}^{3} \beta\right)}{\cos} \beta$
$\frac{2 {\cos}^{3} \beta - \cos \beta - 2 \cos \beta + 2 {\cos}^{3} \beta}{\cos} \beta$
$\frac{4 {\cos}^{3} \beta - 3 \cos \beta}{\cos} \beta$
$\frac{\cos \beta \left(4 {\cos}^{2} \beta - 3\right)}{\cos} \beta$
$4 {\cos}^{2} \beta - 3$
Switch into sine now using ${\cos}^{2} x = 1 - {\sin}^{2} x$:
$4 \left(1 - {\sin}^{2} \beta\right) - 3$
$4 - 4 {\sin}^{2} \beta - 3$
$1 - 4 {\sin}^{2} \beta$
Since the $L H S$ equals the $R H S$, we are done here.
Hopefully this helps!
Jan 14, 2017
See the Proof in the Explanation Section.
#### Explanation:
We will need $\cos 2 \beta = 1 - 2 {\sin}^{2} \beta$
$\cos 3 \beta = \underline{\cos 3 \beta + \cos \beta} - \cos \beta \ldots \ldots \ldots \ldots \ldots . . \left(1\right)$
Since, $\cos C + \cos D = 2 \cos \left(\frac{C + D}{2}\right) \cos \left(\frac{C - D}{2}\right)$
$\therefore , \text{ from } \left(1\right) , \cos 3 \beta = 2 \cos 2 \beta \cos \beta - \cos \beta ,$
$= \left(\cos \beta\right) \left(2 \cos 2 \beta - 1\right) ,$
$= \left(\cos \beta\right) \left\{2 \left(1 - 2 {\sin}^{2} \beta\right) - 1\right\} , i . e . ,$
$\cos 3 \beta = \left(\cos \beta\right) \left(1 - 4 {\sin}^{2} \beta\right) ,$
$\Rightarrow \frac{\cos 3 \beta}{\cos} \beta = 1 - 4 {\sin}^{2} \beta$
Enjoy Maths.!<|endoftext|>
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# How do you write an equation of a line given point (1,3) and m=-3/4?
Jan 9, 2017
Use the point-slope formula to write the equation of the line. See the full explanation below:
#### Explanation:
Use the point-slope formula to write the equation of the line.
The point-slope formula states: $\left(y - \textcolor{red}{{y}_{1}}\right) = \textcolor{b l u e}{m} \left(x - \textcolor{red}{{x}_{1}}\right)$
Where $\textcolor{b l u e}{m}$ is the slope and $\textcolor{red}{\left(\left({x}_{1} , {y}_{1}\right)\right)}$ is a point the line passes through.
Substituting the point and the slope from the problem gives the result:
$\left(y - \textcolor{red}{3}\right) = \textcolor{b l u e}{- \frac{3}{4}} \left(x - \textcolor{red}{1}\right)$
We can convert this to the more familiar slope=intercept form by solving for $y$:
$y - \textcolor{red}{3} = \textcolor{b l u e}{- \frac{3}{4}} x - \left(\textcolor{b l u e}{- \frac{3}{4}} \times \textcolor{red}{1}\right)$
$y - \textcolor{red}{3} = \textcolor{b l u e}{- \frac{3}{4}} x + \frac{3}{4}$
$y - \textcolor{red}{3} + 3 = \textcolor{b l u e}{- \frac{3}{4}} x + \frac{3}{4} + 3$
$y - 0 = \textcolor{b l u e}{- \frac{3}{4}} x + \frac{3}{4} + \left(\frac{4}{4} \times 3\right)$
$y = \textcolor{b l u e}{- \frac{3}{4}} x + \frac{3}{4} + \frac{12}{4}$
$y = - \frac{3}{4} x + \frac{15}{4}$<|endoftext|>
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Otto Stern, (born Feb. 17, 1888, Sohrau, Ger. [now Zory, Pol.]—died Aug. 17, 1969, Berkeley, Calif., U.S.), German-born scientist and winner of the Nobel Prize for Physics in 1943 for his development of the molecular beam as a tool for studying the characteristics of molecules and for his measurement of the magnetic moment of the proton.
Stern’s early scientific work was theoretical studies of statistical thermodynamics. In 1914 he became a lecturer in theoretical physics at the University of Frankfurt and in 1923 a professor of physical chemistry at the University of Hamburg. Stern and Walther Gerlach performed their historic molecular-beam experiment at Hamburg in the early 1920s. By shooting a beam of silver atoms through a nonuniform magnetic field onto a glass plate, they found that the beam split into two distinct beams instead of broadening into a continuous band. This experiment verified the space quantization theory, which stated that atoms can align themselves in a magnetic field only in a few directions (two for silver), instead of in any direction, as classical physics had suggested. (See also Stern-Gerlach experiment.) In 1933 Stern measured the magnetic moment (strength of a subatomic particle’s magnetic property) of the proton by using a molecular beam and found that it was actually about 21/2 times the theoretical value.
In 1933, when the Nazis rose to power, Stern was compelled to leave Germany. He went to the United States, where he became research professor of physics at the Carnegie Institute of Technology, Pittsburgh. He remained there until his retirement in 1945.
Learn More in these related Britannica articles:
spectroscopy: Angular momentum quantum numbers…demonstrated by two German physicists, Otto Stern and Walther Gerlach.…
atom: Bohr’s shell model…1922 by other German physicists, Otto Stern and Walther Gerlach. Their experiment took advantage of the magnetism associated with angular momentum; an atom with angular momentum has a magnetic moment like a compass needle that is aligned along the same axis. The researchers passed a beam of silver atoms through…
Stern-Gerlach experiment…1920s by the German physicists Otto Stern and Walther Gerlach. In the experiment, a beam of neutral silver atoms was directed through a set of aligned slits, then through a nonuniform (nonhomogeneous) magnetic field (
seeFigure), and onto a cold glass plate. An electrically neutral silver atom is actually an…
Molecular beam, any stream or ray of molecules moving in the same general direction, usually in a vacuum— i.e.,inside an evacuated chamber. In this context the word molecule includes atoms as a special case. Most commonly, the molecules comprising the beam are at a low density; that is, they are…<|endoftext|>
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# What is the conjugate when dealing with radical expressions?
Aug 28, 2016
Here's an example.
Let $a = 2 , b = 3 , c = 4 \mathmr{and} d = 5$.
We will be left with the following:
$2 \sqrt{3} + 4 \sqrt{5}$
The conjugate has as goal to make a difference of squares, which is why we use it to rationalize denominators. We can find the conjugate by switching the middle sign in the binomial expression.
Hence, the conjugate of $2 \sqrt{3} + 4 \sqrt{5}$ is $2 \sqrt{3} - 4 \sqrt{5}$.
Let's try multiplying the two expressions to see what happens.
$\left(2 \sqrt{3} + 4 \sqrt{5}\right) \left(2 \sqrt{3} - 4 \sqrt{5}\right) = 4 \sqrt{9} + 8 \sqrt{15} - 8 \sqrt{15} - 16 \sqrt{25} = 4 \left(3\right) - 16 \left(5\right) = 12 - 80 = - 68$
So, we start with an expression with lots of irrational numbers, and we multiply it by it's conjugate and get a rational number! Math is so cool sometimes!
Here are a few exercises for your practice. Send me a note when you're ready to be given the answers.
Practice exercises:
$1.$ For the following expressions:
•Write the conjugate
•Multiply the expression by its conjugate
•Simplify the expression if necessary
a) $4 \sqrt{6} + 3 \sqrt{11}$
b) $\sqrt{72} - \sqrt{36}$
c) $2 \sqrt{10} + 3 \sqrt{16}$
d) $\sqrt{19} - \sqrt{21}$
Hopefully this helps, and good luck!<|endoftext|>
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1 Introduction Chapter Introduction This chapter will introduce you to the Ancient Greeks. You will learn about early Greek history, society, and government. Section 1: The Rise of City-States Section 2: Greek Society and Economy Section 3: Democracy in Athens Section 4: Oligarchy in Sparta
2 mystory Pericles: Calm in the Face of Danger Why was Pericles family evacuating from Athens? How were most Athenians probably feeling about the evacuation?
3 mystory Pericles: Calm in the Face of Danger What did the Athenian men plan to do about the Persian invaders? What happened to Pericles dog, Ajax, when his family left Athens?
4 mystory Pericles: Calm in the Face of Danger What evidence supports the idea that Persia was a mighty and fearsome enemy? Persia was a mighty and fearsome enemy. Evidence
5 mystory Pericles: Calm in the Face of Danger What lessons do you think Pericles learned from the experience of seeing Athens defeat the mighty Persians? Lessons of Salamis
6 mystory What is power? Who should have it? How did the Greek and Persian ideas of power differ? Persia Athens
7 The Rise of City-States Describe your local government, including important leadership positions. How does your city or town government work?
8 The Rise of City-States Academic Vocabulary eventual adj., final Studying hard will improve your eventual results at school. exclude v., to shut out or keep from participating It is against the law to exclude people because of their race or gender.
9 The Rise of City-States Physical geography helped shape Greek life and culture. Key Ideas The basic political unit of ancient Greece was the city-state.
10 The Rise of City-States Key Ideas Early Greek history was marked by frequent warfare among small city-states.
11 The Rise of City-States polis a city-state Key Terms citizen a member of a city-state who enjoys legal rights
12 The Rise of City-States Key Terms acropolis the high hill in a city-state where public buildings and temples were located politics the art and practice of government aristocracy a hereditary class of rulers
13 The Rise of City-States What are some important features of Greek geography? mountain ranges limited farm land the sea Mediterranean climate
14 The Rise of City-States How did geography influence the ancient Greeks? Greek Geographic Features and Their Influence Mountains Limited Land Sea Climate
15 The Rise of City-States Ancient Greece: Early History 2000 B.C. Minoan civilization spreads to Greece B.C B.C. Advanced Mycenaean kingdoms trade bronze weapons and pottery B.C. 750 B.C. Culture declines during the dark age.
16 The Rise of City-States What was the importance of the Iliad and the Odyssey to the Greeks? The Iliad and the Odyssey
17 The Rise of City-States Ancient Greece: The City-States The polis included the people of the community, the city and surrounding area, and its government. On the acropolis, or high city, the important activities of government and religion took place. Greek settlers established city-states throughout the Mediterranean.
18 The Rise of City-States Diversity Among the City-States Why was the small size of the polis important? What kinds of governments did the city-states have?
19 The Rise of City-States The city-state became one of the most important features of Greek culture. What evidence supports this idea? Evidence of Importance of City-State
20 The Rise of City-States What is power? Who should have it? Trace the rise of the city-state in ancient Greece.
21 Greek Society and Economy Community Groups Think about groups in your community. Might some of those groups also have existed in ancient Greece? Identify Predict
22 Greek Society and Economy Academic Vocabulary obtain v., to gain Traders obtained olive oil from the Greeks. symbolize v., to represent The stars on our national flag symbolize the states.
23 Greek Society and Economy Key Ideas Greek society was divided according to wealth and legal status. Women had clear roles and few rights in the Greek city-states.
24 Greek Society and Economy Key Ideas Geography and limited resources spurred conquest, trade, and colonization.
25 Greek Society and Economy Key Terms tenant farmer a person who paid rent, either in money or crops, to grow crops on somebody else s land metic a resident foreigner slavery the ownership and control of other people as property
26 Greek Society and Economy Ancient Greek Women Free Greek women were noncitizens. Women s status varied from city-state to city-state. Greek women oversaw most household duties.
27 Greek Society and Economy How did the role and status of women differ in Sparta and Athens? Athenian Women Both Spartan Women
28 Greek Society and Economy Land Ownership and Status What part did landownership play in determining a person s standing in Greek society? How did tenant farmers differ from landowners? How did people become slaves in ancient Greece?
29 Greek Society and Economy What status did noncitizens have in Greek society? Noncitizens in Greek Society Women Metics Slaves
30 Greek Society and Economy Effects of Overpopulation and Land Hunger The ancient Greeks had limited farmland and resources to support a growing population. Some Greek city-states conquered their neighbors to acquire land and resources. Some Greeks established colonies on the Mediterranean Sea and the Black Sea.
31 Greek Society and Economy What did the Greeks consider an ideal site for a colony? Ideal Site for a Colony
32 Greek Society and Economy Effects of Mediterranean Trade In what ways did trading affect the cultures of the Mediterranean? What effect did trading have on the social structure of Greece?
33 Greek Society and Economy Main Ideas About Greek Society and Economy
34 Greek Society and Economy What is power? Who should have it? Who held the most power in the Greek family?
35 Democracy in Athens What words do you connect with democracy? Democracy
36 Democracy in Athens Academic Vocabulary maintain v., to keep and support Our government maintains an army for defense. lecturer n., a person who gives an informative talk to students Our lecturer is an expert on life in ancient Greece.
37 Democracy in Athens Key Ideas Greek city-states experimented with many forms of government, including oligarchy and tyranny. In Athens, democracy developed. Citizens participated in lawmaking and the courts.
38 Democracy in Athens Key Ideas Athenian democracy and the responsibilities of citizenship developed gradually over many years.
39 Democracy in Athens Key Terms oligarchy a government in which a small number of people hold political power phalanx a formation of heavily armed foot soldiers who moved together as a unit
40 Democracy in Athens Key Terms tyranny a government run by one strong leader democracy a government run by many people
41 Democracy in Athens Key Terms citizenship membership in a community direct democracy a political system in which citizens participate directly in decision making representative democracy a political system in which citizens elect others to represent them
42 Democracy in Athens How did tyranny differ from oligarchy in ancient Greece? Tyranny Shared Oligarchy
43 Democracy in Athens The Phalanx In what ways was the phalanx different from earlier battle formations? Why do some believe that there is a connection between the phalanx and the way larger numbers of people gained political power?
44 Democracy in Athens Moving Toward Democracy 594 B.C. Solon reforms the courts, extends voting rights to some non-aristocrats, and stops the practice of turning debtors into slaves. 508 B.C. Cleisthenes increases the number of voters from lower classes and gives the assembly more power B.C. Pericles increases citizen participation in government by paying citizens for jury service.
45 Democracy in Athens Citizens of Athens In Athens, citizens had various rights and responsibilities. Pericles suggested that it was not class but ability that should be recognized for leaders.
46 Democracy in Athens How did Athenian democracy work? How Athenian Democracy Worked Assembly Juries Archons Boule Council Subcommittees
47 Democracy in Athens Summarize ideas about Athenian democracy.
48 Democracy in Athens The Power of Athenian Democracy In what way was citizenship in Athens unique in the ancient world? Why do you think Athenian democracy spread to other city-states?
49 Democracy in Athens Evolution of Democracy Why is Athens described as a limited democracy? Why would it be difficult, if not impossible, to create direct democracy in a large country?
50 Democracy in Athens What is power? Who should have it? How did citizens gain power in Athens?
51 Oligarchy in Sparta Importance of the Military How does the military help our country?
52 Oligarchy in Sparta Academic Vocabulary authority n., people in power Most citizens respect authority. innovation n., a new way of doing things Some innovations, like the computer, change everyone s life.
53 Oligarchy in Sparta Key Ideas Sparta developed an oligarchic government based on military conquest. Sparta differed greatly from Athens in terms of education, citizenship, and women s roles.
54 Oligarchy in Sparta Key Terms ephor a Spartan official, elected by the assembly, who was responsible for the government s day-to-day operations and for oversight of the kings and the council helot one of the conquered Messenians who were forced to farm their land for the Spartans
55 Oligarchy in Sparta Key Terms military state a society organized for the purpose of waging war barracks military housing
56 Oligarchy in Sparta Sparta s Government Sparta s government was an oligarchy with two kings. The council of elders was Sparta s main governing body. An assembly of citizens could pass laws with the council s approval. Ephors, elected by the assembly, oversaw the daily operations of the government. They also ensured that the kings and the council operated within the law.
57 Oligarchy in Sparta Spartan Military Might Why did Sparta become a military state? In what way did the helots enable Spartan men to become a warrior class?
58 Oligarchy in Sparta Steps to Election to the Council of Elders Step 1 Male youths undergo years of military training and service. Step 2 To become citizens, Spartan men must gain entry into a men s club. Step 3 When Spartan men become citizens, they obtain membership in the assembly and the right to land worked by helots. Step 4 At age 60, citizens are eligible to join the council of elders.
59 Oligarchy in Sparta Roles and Rights in Sparta Spartan Men Shared Spartan Women
60 Oligarchy in Sparta Comparing Athens and Sparta
61 Oligarchy in Sparta Sparta and Athens: A Stark Contrast Thucydides made observations about the differences between these city-states: The Athenians were addicted to innovation. The Spartans had a genius for keeping what you have got. In time, the differences between Sparta and Athens led to conflict and war.
62 Oligarchy in Sparta Two Extremes in Ancient Greece Why did other Greek city-states both fear and admire Sparta? Why do you think Thucydides said Athenians were addicted to innovation? Why did he say the Spartans had a genius for keeping what you have got?
63 Oligarchy in Sparta What is power? Who should have it?
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The north-south divide has been the butt of jokes in Britain for years, but research has shown the Watford Gap, which separates the country, was in fact established centuries ago when the Vikings invaded Britain.
According to the archaeologist Max Adams, who made the discovery while researching his new book, the Northamptonshire-Warwickshire boundary known as the Watford Gap is a geographic and cultural reality that can be traced back to the Viking age.
Adams was struck by the absence of Scandinavian placenames south-west of Watling Street, the Roman road that became the A5. “There might be one or two names, but I don’t think there are any, and there are certainly hundreds and hundreds north-east. Clearly the Scandinavian settlers stopped at Watling Street,” Adams said.
“I began to notice that all the rivers’ sources stop pretty much on the line of Watling Street. North-east of that line, all the rivers flow into the Irish Sea or the North Sea. South and west of it, they all flow into the Severn or the Thames.”
He added: “Roman engineers constructing the route between London [Londinium] and the important town of Wroxeter [Viroconium], in what is now Shropshire, chose this ancient line, and it became Watling Street. In the Viking period it became the boundary for a treaty between King Alfred and the Viking leader Guthrum. Connecting the West Midlands with the south-east, it runs through a narrow pass between hills, the Watford Gap.
“I’m not sure whether people on the north side of Watling Street immediately feel themselves different or whether that’s more of a southern joke. But clearly it’s a joke with a very old reality attached to it.
“These days, we’re unaware of which way rivers face and where they flow out to. It doesn’t make any odds to us. We just put bridges over them. But, for most of history, such things have mattered. Your natural trading routes are along rivers and all the medieval monastic estates used the rivers as their arteries of power. So clearly the geography of power has always mattered … Geographically, it slaps you in the face as soon as you figure it out.”
He explained that the Anglo-Saxon kings eventually fortified that line and made it a frontier in the early 10th-century reigns of Eadweard the Elder and his sister Æthelflæd: “So, in a sense, they reinforced the reality of that piece of geography and it seems to have been with us ever since.”
“In 1959, when the M1 was first built, the Watford Gap was its end point – the butt of north-south divide jokes ever since,” said Adams. The M1 service station’s unofficial status as the country’s dividing point was celebrated in 2009 with the unveiling of a new road sign, with one arrow pointing north and another pointing south. Previously called the Blue Boar, the service station became famous as an early-morning hangout for The Beatles and the Rolling Stones, among millions of travellers who were fed and watered there. Linguists have since identified it as the boundary between northern and southern English.
But boundaries are certainly blurred, Adams said: “We find it bizarre that, on the news last night, people were talking about Cheshire as the north … Routinely, politicians describe Hadrian’s Wall as if it was the border between England and Scotland. Well, there’s another 60 miles of England beyond Hadrian’s Wall.”
Adams has excavated widely in Britain and abroad, and he will include his research in a forthcoming book, titled Aelfred’s Britain: War and Peace in the Viking Age, to be published by Head of Zeus on 2 November. It is a companion volume to his previous early medieval histories, The King in the North and In the Land of Giants.
In the new book, he notes that, before the second decade of the 10th century was out, new fortresses or burhs were constructed at 19 sites strung out on a broad line between the Thames and the Mersey, unmistakable in their offensive purpose. That line roughly follows Watling Street.
“It has an ancient and continuing geographic distinction, barely noticed by today’s midlanders. Broadly speaking, to the north-east all the rivers flow into the Wash or North Sea on the east side, or the Irish Sea on the west. To the south and west every river drains into either Severn or Thames. This is England’s natural fault line, its continental divide: the watershed that divided and divides north from south (epitomised by the famous Watford Gap, on the A5/M1 north-east of Daventry); and I have no doubt that Scandinavian armies and settlers knew its imperatives.”<|endoftext|>
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If you find any mistakes, please make a comment! Thank you.
## Exhibit all of the ring homomorphisms from the cartesian square of the integers to the integers
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.5
Describe all ring homomorphisms from $\mathbb{Z} \times \mathbb{Z}$ to $\mathbb{Z}$. In each case, describe the kernel and the image.
Solution: Since $\mathbb{Z}^2$ is the free abelian group on two generators (cf. §6.3 #11), every additive group homomorphism $\varphi : \mathbb{Z}^2 \rightarrow \mathbb{Z}$ is determined uniquely by $\varphi(1,0)$ and $\varphi(0,1)$. Since every ring homomorphism is also and additive group homomorphism, if $\varphi : \mathbb{Z}^2 \rightarrow \mathbb{Z}$ is a ring homomorphism, then $\varphi$ is determined uniquely by $\varphi(1,0) = a$ and $\varphi(0,1) = b$.
Note that $$\varphi(1,1) = \varphi((1,0) + (0,1)) = \varphi(1,0) + \varphi(0,1) = a+b,$$ and on the other hand $$\varphi(1,1) = \varphi((1,1)(1,1)) = \varphi(1,1)\varphi(1,1) = (a+b)^2.$$ Thus we have that $a+b$ is (multiplicatively) idempotent in $\mathbb{Z}$. Thus $a+b \in \{0,1\}$.
Note that for all $(x,y) \in \mathbb{Z}^2$, we have $\varphi(x,y) = xa + yb$ and $$\varphi(x,y) = \varphi((1,1)(x,y)) = (a+b)(xa + yb);$$ thus $$\varphi(x,y) = xa+yb = (a+b)(xa+yb)$$ for all $x,y \in \mathbb{Z}$.
If $a+b=0$, then for all $(x,y) \in \mathbb{Z}^2$, we have $\varphi(x,y) = 0$. Thus $\varphi = 0$.
Suppose now that $a+b = 1$. Since $\varphi$ is a ring homomorphism, for all $(x,y), (z,w) \in \mathbb{Z}^2$, we have $$\varphi((x,y)(z,w)) = \varphi(x,y)\varphi(z,w) = (ax+by)(az+bw)$$ on one hand and $$\varphi((x,y)(z,w)) = \varphi(xz,yw) = axz+byw$$ on the other. Thus $$axz+byw = a^2xz + (xw+yz)ab + b^2yw$$ for all integers $x,y,z,w$.
Choosing $y = w = 0$ and $x,z \neq 0$, we see that $axz = a^2xz$, so that $a^2 = a$. Since $a$ is an integer, we have $a \in \{0,1\}$. Thus in fact $(a,b) \in \{(1,0), (0,1) \}$.
Thus there are precisely three ring homomorphisms $\varphi_{(a,b)} : \mathbb{Z}^2 \rightarrow \mathbb{Z}$, where $\varphi_{(a,b)}(1,0) = a$ and $\varphi_{(a,b)}(0,1) = b$, given by $(a,b) \in \{(0,0), (1,0), (0,1)\}$.
(1) If $\varphi = \varphi_{(0,0)}$, then clearly $\varphi = 0$. Thus $\mathsf{ker}\ \varphi = \mathbb{Z}^2$ and $\mathsf{im}\ \varphi = 0$.
(2) Consider $\varphi = \varphi_{(1,0)}$. It is clear that if $(x,y) \in 0 \times \mathbb{Z}$, then $\varphi(x,y) = 0$. Suppose now that $\varphi(x,y) = x = 0$; then $(x,y) \in 0 \times \mathbb{Z}$. Thus $\mathsf{ker}\ \varphi = 0 \times \mathbb{Z}$. Since $\varphi(1,0) = 1$, $\mathsf{im}\ \varphi = \mathbb{Z}$.
(3) Consider $\varphi = \varphi_{(0,1)}$. By an argument similar to the previous one, $\mathsf{ker}\ \varphi = \mathbb{Z} \times 0$ and $\mathsf{im}\ \varphi = \mathbb{Z}$.<|endoftext|>
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From hurricanes to solar storms, weather in space has the potential to cause havoc on Earth so the European Space Agency (Esa) is planning a space-weather satellite to act like an interstellar forecaster.
In 2023, Esa will send a probe to a gravitationally stable point in space, known as Langrage 5 (L5), where it will study streams of charged particles heading towards Earth. Other probes currently face the Sun, at Larange 1 (L1), so the new positioning would offer an alternative view of the star’s surface and let scientists measure the speed of solar eruptions more precisely. It will be the first Esa mission aimed at forecasting space weather.
Solar storms, or coronal mass ejections (CME), can have an adverse effect on satellites across the Earth. A recent study, co-authored by researchers from the Cambridge Centre for Risk Studies at University of Cambridge, predicted solar storms could cause massive blackouts across the US and cost up to $41.5 billion a day due to the economic costs of disruption.<|endoftext|>
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Dinosaurs from the Late Jurassic period were fussier eaters than their ancestors which led to a more peaceful co-existence, according to a study by the University of Bristol.
Research was carried out into a community of different species of sauropods, such as Diplodocus and Camarasaurus, from the Late Jurassic Morrison Formation, an area of sedimentary rock in the western US, where over 10 species are known to have lived.
The prehistoric lands were dominated by sauropods between 210 and 65 million years ago. They were the biggest land animals at the time, weighing in at around 80 tonnes.
The size of these animals would mean that vast amounts of food would need to be consumed on a daily basis and it was a mystery how multiple types of sauropods were able to live alongside each other and still have enough food to go around.
This is especially because the harsh, semi-arid environment of the Morrison Formation would have made it difficult for plants to grow.
Another question raised was how such big dinosaurs were able to ingest enough food into their bodies to sustain themselves, with such small heads.
The scientists digitally reconstructed the skulls of a Camarasaurus and a Diplodocus through CT scans, along with the jaw and neck muscles of both species.
A computer model of a Camarasaurus skull was created using Finite Element Analysis (FEA), an engineering technique used to uncover stress and strain distribution.
The model was then compared to an existing model of a Diplodocus skull and biomechanical measurements from other sauropod species.
Scientists found that the differences in the animals' skulls and muscles meant that they were built to eat different types of vegetation.
"Our results show that although neither could chew, the skulls of both dinosaurs were sophisticated cropping tools," said David Button, a PhD student. "Camarasaurus had a robust skull and strong bite, which would have allowed it to feed on tough leaves and branches. Meanwhile, the weaker bite and more delicate skull of Diplodocus would have restricted it to softer foods like ferns."
The research has given scientists a better understanding of the evolution of sauropod feeding mechanisms and how they were able to eat enough food to sustain themselves.
"Our study provides insight not only into the ecology of dinosaurs but more generally into the mechanisms supporting species-richness in other animal communities, both from the fossil record and in the present-day," said co-author, Paul Barrett.<|endoftext|>
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# Factorise $\,a(a - 4)(a^2 - 4a - 1) - 20\,$
I am making an attempt to factorise the above equation.
To do this i am expanding the already factored part of the above equation to obtain ->
$\,f(a) = a^4 - 8a^3 + 15a^2 + 4a - 20\,$
Now i do some random guesses to find that f(2) = 0, hence i know that (a-2) will divide f(a) so i perform long division to get f(a) = (a-2)( Something ) and so on... until i get - > $\,f(a) = (a - 2)^2(a + 1)(a - 5)\,$
I feel that i am approaching the problem in an incorrect and long way. Please let me know if i can get the factorised form in a cleaner and faster way.
PS: Another linked question is to factorise $\,f(x) = (x + 1)(x + 3)(x + 5)(x + 7) + 15\,$
$1)$ Call $x=a(a-4)$ then
$$x(x-1)-20=x^2-x-20=(x+4)(x-5)$$
then
$$(a^2-4a+4)(a^2-4a-5)=(a-2)^2(a+1)(a-5)$$
$2)$ For
$$(x+1)(x+3)(x+5)(x+7)+15=[(x+3)(x+5)][(x+1)(x+7)]+15=(x^2+8x+15)(x^2+8x+7)+15$$
Call $y=x^2+8x+7$ then:
$$(y+8)y+15=y^2+8y+15=(y+3)(y+5)$$
• perfect! very clear Commented Jan 26, 2017 at 17:38
• and for f(x)=(x+1)(x+3)(x+5)(x+7)+15 i can take x^2 + 8x = t ... Commented Jan 26, 2017 at 17:39
• Yes that's correct. Commented Jan 26, 2017 at 17:40
Another linked question is to factorise $\,f(x) = (x + 1)(x + 3)(x + 5)(x + 7) + 15\,$
Let $z=x+4$ then:
\begin{align} (z-3)(z-1)(z+1)(z+3)+15 & = (z^2-1)(z^2-9)+15=z^4-10z^2+24 \\ & = (z^2-4)(z^2-6) = (x^2+8x+12)(x^2+8x+10) \\ & = (x+2)(x+6)(x^2+8x+10) \end{align}<|endoftext|>
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As global temperatures rise, warmer oceans are expected to fuel stronger hurricanes, with disastrous consequences.
SEP 10, 2017
A third of the way into the 2017 Atlantic hurricane season, NOAA looked at the ocean and air temperatures and issued an ominous new forecast: the region would likely experience “an above normal hurricane season” that “could be extremely active,” with more named storms than previously expected—14 to 19 this season—and two to five major hurricanes.
By the halfway point of the season, Hurricane Harvey’s destruction stretched along the Texas coast, and Hurricane Irma’s storm surge had turned coastal Florida streets in saltwater rivers after causing massive destruction in the Caribbean. On Irma’s heels, Hurricane Jose was swirling in the Atlantic, Katia struck Mexico’s eastern coast, and the Maria was about to devastate Puerto Rico and the Virgin Islands.
As global temperatures continue to rise, climate scientists have said this is what we should expect—more huge storms, with drastic impacts.
Though scientists are still wrestling with some of the specifics of howclimate change is impacting hurricanes, a lot is known, including the fact that hurricane seasons like this one could be the new norm.
What’s So Extraordinary about These Storms?
Records are tumbling in quick succession this year. Irma, among the strongest Atlantic hurricanes on record, barreled over the islands of the Caribbean as a Category 5 storm this week en route to Florida, while Houston, Texas, was still draining from Harvey’s five-day deluge that broke the continental U.S. rainfall total for a single event.
Major storms are falling outside their normal range (Ophelia became the easternmost on record when it struck Ireland), and at strange times of the year (Tropical Storm Arlene hit in April of this year—one of only two named tropical storms in April, and the northernmost on record for that time of year).
As climate change progresses, scientists aren’t projecting an increase in total storms, but they are expecting a jump in the number of major storms—just like we’re seeing now.
What Does Climate Change Have to Do with It?
If Hurricane Harvey had happened at the end of the 20th century, that amount of rain falling in Houston in a single storm would have been rare—a 1-in-2,000-year event, said Kerry Emanuel, an MIT professor of atmospheric sciences. But as temperatures continue to rise, those rare events are becoming increasingly less rare, he said.
There are myriad reasons why individual storms develop as they do, including a combination of natural and manmade causes. That can make it hard to assess what role climate may have played in an individual storm (though the science behind attribution studies is getting better all the time). What scientists who study hurricanes are confident in, though, is the underlying physics that show that warmer temperatures are among the factors changing the way that storms form.
According to the 2014 National Climate Assessment, the intensity, frequency and duration of North Atlantic hurricanes have increased since the early 1980s. The frequency of the strongest storms—category 4 and 5 hurricanes—has increased too.
How Do Warmer Oceans Feed Hurricanes?
NOAA releases its annual Atlantic hurricane outlook each spring, in advance of the hurricane season that starts on June 1. This year, the agency had toupdate that outlook in August with an expectation of even more storms, due in part to warmer surface water temperatures.
Surface temperatures in the eastern half of the tropical Atlantic Ocean were between 0.5°C and 1°C above average this summer. The NOAA maps below show how sea surface temperatures (SST) change.
Those higher temperatures (as well as higher temperatures in the atmosphere) feed the storms, helping them strengthen. One study based on two decades of data found that hurricanes intensify significantly faster now than they used to. The researchers found that storms reach Category 3 wind speeds nine hours faster than they did in the 1980s.
Why So Much Rainfall in Houston?
Those higher temperatures don’t just result in more intense wind speeds. Warmer air also retains more water vapor, which can result in dramatic rainfall like what happened during Hurricane Harvey.
In the case of Harvey, the rain volume was exacerbated by the fact that the storm stalled over the Houston area, bringing days of relentless downpours. The storm was surrounded by two high pressure systems, which essentially locked the storm in place. “Meteorologically, Southeast Texas, at the time, was pretty much a giant stop sign,” Weather.com meteorologist Jonathan Belles told the Houston Chronicle. By the time the storm moved out, the National Weather Service had recorded 51.88 inches of rain near Mont Belvieu, Texas—a record for rainfall from a single storm in the continental U.S.
This stalling is a frequent feature of extreme events, but at this point, scientists have not found a conclusive link to climate change. There may be a connection, though, according to climate scientist Michael Mann. “More tenuous, but possibly relevant still, is the fact that very persistent, nearly ‘stationary’ summer weather patterns of this sort, where weather anomalies (both high pressure dry hot regions and low-pressure stormy/rainy regions) stay locked in place for many days at a time, appears to be favored by human-caused climate change,” he wrote in a Facebook post late last month.
In a study published online in the journal Nature in March, Mann and coauthors wrote that amplified warming in the Arctic driven by anthropogenic climate change may be leading to an increase in extreme weather events that linger in one place for extended periods of time.
What Are the Biggest Threats from Irma?
The big fear with Irma is the wind. The National Hurricane Center was reporting sustained winds of 185 mph on Wednesday and gusts even higher. Those wind speeds are similar to Hurricane Wilma, the 2005 storm that resulted in at least 62 deaths and an estimated $29.4 billion in damage, of which $21 billion occurred in the United States.
Another concern is the storm surge that can accompany hurricanes. Riding on top of sea levels that have risen due to climate change, Irma’s surge could be particularly dangerous. The National Hurricane Center warned Wednesday of storm surges as high as 20 feet above normal tide levels in the Turks and Caicos Islands and the Southeastern Bahamas.
What About that ‘Hurricane Drought’ Claim?
On Aug. 25, as Hurricane Harvey gained strength and headed for the Texas coast, the conservative Heartland Institute put out a press release decrying any efforts that scientists and the media might make to explain the climate influences on the storm. Bette Grande, a research fellow with Heartland, said: “Though it has been nearly 12 years since a major hurricane has hit the United States—Harvey will be creatively spun to ‘prove’ there are dire effects linked to man-created climate change.”
She was referring to the concept of a so-called “hurricane drought” thatclimate deniers have been circulating—which they say debunks the work of climate scientists.
While no “major hurricanes” made landfall in the United States between 2005 and this year, those weren’t weak tropical storm years—the biggest storms just didn’t hit the U.S. In 2013, Typhoon Haiyan devastated the Philippines with the highest wind speeds ever seen—until Hurricane Patricia broke that record two years later off Mexico’s Pacific Coast, and several other cyclones wreaked havoc elsewhere around the world in the intervening years.
In a 2015 study published in the journal Geophysical Research Letters, two NASA scientists concluded that the lack of major storms in the United States during that period was merely “a matter of luck.”
People in some parts of the United States might also disagree with the concept of a “hurricane drought” during that period.
“Tell the people of coastal Texas that Ike was not a major hurricane,” said Emanuel, the MIT scientist. “Well, Ike was technically just under the ranking of major hurricane, and it completely destroyed a huge part of coastal Texas. Now, tell the people of New York that Sandy wasn’t a major hurricane.”
“There were plenty of hurricanes in that stretch of 12 years,” he said. “They just didn’t happen to make landfall as strong storms in the United States.”<|endoftext|>
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Leafy Spurge Characteristics
In Saskatchewan, leafy spurge (Euphorbia esula L.) is a noxious weed originally from Eastern Europe that infests pasture and native prairie in a diagonal belt from North Battleford to Estevan.
- it's a perennial weed that reproduces by both seed and vegetative root buds.
- the plant grows to a height of one metre
- has long, thin, dark green leaves and can be identified from a distance by distinctive yellow-green flowers.
Leafy spurge is very competitive and easily out-competes many forage and native plant species. The juice of the plant is a white, milky latex that may cause mouth and throat blistering in cattle and contact dermatitis in people.
Cattle avoid spurge-infested areas, greatly reducing the livestock carrying capacity of infested range and pastureland. Chemical control of leafy spurge in pastures is often time consuming and expensive whereas biological control of leafy spurge has been a biocontrol success in North America.
Chemical Control Options
For chemical control options for leafy spurge in pasture and hayfields see the current addition of the Guide to Crop Protection.
The most successful biological controls of leafy spurge are beetles from the flea beetle genera Aphothona. The beetles have been used as biocontrol of leafy spurge since they were introduced into Canada in the 1980s. There are five beetle species that have been released in Canada for control of leafy spurge: Aphthona cyparissiae, A. flava, A. nigriscutis, A. czwalinae and A. lacertosa. The first three species have brown or gold bodies while the last two are black-bodied. Two species have been shown to have had a greater impact on controlling leafy spurge than the others: the black dot spurge beetle, A. nigriscutis, has been successful on drier sandy sites and the black spurge beetle, A. lacertose (Figure 3), has been more successful in moister locations.
The Life Cycle of Beetles that Control Leafy Spurge
All five beetle species have similar life cycles, producing one generation per year. The adult beetles emerge in late June or early July and feed near the top of the shoot and at the leaf edges. The beetles mate and the females lay their eggs in groups of 20 to 30 below the soil surface near a spurge root. Females lay through the summer, producing up to 300 eggs per season, resulting in about 150 offspring in a growing season.
The eggs hatch in about three weeks and the larvae burrow through the soil until they encounter a small root, which they mine. As the larvae grow in size they feed on the larger perennial roots of the purge. Development to the third and final instar takes about two months. Feeding then ceases and the larvae construct an overwintering cell in the soil and become dormant. In the spring the larvae resume feeding for about three weeks, pupate in the soil and then emerge as adults. Since the Aphthona beetles spend their entire larval stage underground feeding on the leafy spurge roots they are sensitive to soil types and conditions.
The Impact of Biocontrol
Although the adult beetles feed on the spurge leaves, it is the larvae that are primarily responsible for the biological control of the plant. The larvae damage the roots and root hairs through feeding, and the feeding wounds provide an entry for various disease-causing organisms. The plant's nutrient reserves become depleted, the plant is no longer able to flower and eventually will wither and die.
The success of a beetle colony and its impact on the site can be gauged using a number of criteria. Optimal sites, where there is good beetle establishment, will have the following characteristics in common:
A count of over 25 beetles per five sweeps after one year (a sweep net is used to collect beetles to determine beetle density);
- A noticeable reduction of flowering spurge after two years;
- A temporary increase in the number of nonflowering stems, which may persist for four years;
- A reduction in the dry weight of spurge and an increase in the dry weight of grasses and other herbaceous species;
- The presence of beetles in a five-pace radius from the release point after one year. It is possible to monitor this progress for four to five years, after which the colony becomes so large that it is difficult to establish boundaries.
Beetle colonies that are optimal should be ready for a harvest of beetles to be taken to other leafy spurge sites after three full years. The target for the biocontrol of spurge is to reduce spurge to a maximum of five per cent cover. Field work has demonstrated that this target can usually be achieved, in the release area, within five years. In one case, a release of these insects on leafy spurge has resulted in a 99 per cent reduction in spurge stand density in one area and a corresponding 30-fold increase in grass biomass after four years.<|endoftext|>
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# Finding Points on a Coordinate Plane
## Finding Points on a Coordinate Plane
Start Quiz
The coordinate plane is formed by two intersecting number lines: the horizontal x-axis and the vertical y-axis. The point where they intersect is called the origin, and is marked as (0, 0). The numbers that identify a point on a coordinate plane are called an ordered pair, and they always are written in the same order: first is the location along the x-axis, and second is the location along the y-axis.
One way to think about the coordinate plane is to break it up into four parts. The two axes divide the coordinate plane into four quadrants:
Quadrant I is in the upper right region, where both x and y values are positive. For example, the point (2, 3) lies in Quadrant I in this coordinate plane:
Quadrant II is in the upper left region, where x values are negative and y values are positive. For example, the point (-3, 4) is in Quadrant II here:
Quadrant III contains points with negative x and y values and is the lower left region. The point (-5, -2) lies in Quadrant III:
Quadrant IV contains points with a positive x value but negative y value and is the lower right region. For instance, (4, -6) is in Quadrant IV:
To plot a point and find its coordinates:
• First identify the x and y values. Let's use the point (-2,5) as an example.
• Look along the x-axis and go left 2 units from the origin.
• Then go up 5 units from the origin along the y-axis.
• Where those values intersect is the point (-2,5):<|endoftext|>
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Not only that, but it's perfectly possible to design planes with airfoils that are symmetrical looking straight down the wing and they still produce lift.
When pilots reach this decision point, we need to be able to see the runway or its associated lights in order to continue the approach, if not, then you must prepare to ascend again and perform a go-around. Let's take a closer look at how it works! Based on Aerodynamics , a public domain War Department training film from 1941.
Share this page Save this page for later or share it by bookmarking with: Think back to our previous discussion of pressure: The downwash isn't so obvious, but it's just as important as it is with a chopper.
But small wings can also produce a great deal of lift if they move fast enough. Fuel tanks: The difference in speed observed in actual wind tunnel experiments is much bigger than you'd predict from the simple equal transit theory. Just like a cyclist leaning into a bend, a plane "leans" into a curve.
A more detailed explanation of why the traditional Bernoulli explanation of lift is wrong, and an alternative account of how wings really work.
The way to make a paper plane steer is to get one wing to generate more lift than the other—and you can do this in all kinds of different ways!
Imagine two air molecules arriving at the front of the wing and separating, so one shoots up over the top and the other whistles straight under the bottom. In other words, the upside-down pilot creates a particular angle of attack that generates just enough low pressure above the wing to keep the plane in the air. Although the Wrights were brilliant experimental scientists, it's important to remember that they lacked our modern knowledge of aerodynamics and a full understanding of exactly how wings work.
More parts of a plane Photo: Their "aeroplanes" were simply pieces of cloth stretched over a wooden framework; they didn't have an airfoil aerofoil profile. Thanks to their successful experiments with powered flight, the airplane is rightfully recognized as one of the greatest inventions of all time. That's intuitively obvious. The Wright brothers had to fly their pioneering Kitty Hawk plane entirely by sight.
Approaching the terminal, the pilots look for the flight's assigned gate and watch for the ramp team leader to start waving illuminated, bright orange batons.
That'll put the plane in the right spot for the passenger boarding bridge.
How airfoil wings generate lift 2: If you're cycling around a curve at speed, some of your centripetal force comes from the tires and some comes from leaning into the bend. McGraw Hill, 2016. The force of the hot exhaust gas shooting backward from the jet engine pushes the plane forward.
PAPIs are sets of red and white lights that show pilots our position with respect to the ideal vertical path that slopes from the sky down to the runway. CNN — Sitting in the terminal building waiting to be called for our flight is a regular occurrence for most of us -- but what's really going on out there on the ramp while we're inside staring at our phones?<|endoftext|>
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The oceans are not a silent world, but dynamic, living symphonies of sound. In water, sound travels five times faster, and many times farther than it does in air.
The Sonic Sea
Whales, dolphins, porpoises, and other marine mammals have evolved to take advantage of this perfect sonic medium. Just as we rely on sight to survive, they depend on sound to hunt for food, find mates, and detect predators.
Over the last fifty years, our increasing ocean presence has drastically transformed the acoustic environment of these majestic creatures. Undersea noise pollution is invisible but it is damaging the web of ocean life.
Three Major Causes of Ocean Noise
The leading contributors to ocean noise come from commercial, industrial, and military sources: Shipping, Seismic, and Sonar.
At any given time, there are up to sixty thousand commercial ships traversing our seas worldwide. Cavitation from propellers and the rumble of engines reverberate through every corner of the ocean.
The incessant and increasing cacophony masks whales’ ability to hear and be heard, hindering their ability to prosper and ultimately to survive.
To detect oil and gas deposits beneath the ocean floor, the petrochemical industry uses seismic airguns, the modern form of exploratory dynamite. Ships tow arrays of these guns, discharging extremely intense pulses of sound toward the sea floor.
During seismic surveys, acoustic explosions continue for days or weeks on end. The blasts disrupt critical behavior and communication among whales and can have massive impacts on fish populations.
Sonar is the principal submarine detection system used by the U.S. Navy and other navies of the world. To detect targets, naval warships generate extremely loud waves of sound that sweep the ocean.
Military sonar acts as an enormous predator. When exposed, some whales go silent, stop foraging, and abandon their habitat. Repeated exposure can harm entire populations of animals, and has led to mass whale strandings from the Canary Islands and the Caribbean to Japan.<|endoftext|>
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Home > Pre-Algebra > Ratios > Determine if Ratios are Equal
## Determine if Ratios are Equal
#### Introduction
Ratios are a useful way to show how two values relate to one another. For example, if you are about to cycle up a hill with a ratio of 1:5, for every yard five yards you travel horizontally, you will have gained a yard in height. A soccer team with a 2:5 win ratio will win two games out of every five played. By showing these ratios, we can easily extrapolate a lot of information (for example, if the team played 10 games or 15 games, we could quickly determine how many they won).
If you get two sets of ratios, you can compare them and determine if they are equal. For example, if another soccer team in the same division had a win ratio of 4:9, we can work out if they win more than the team with the 2:5 ratio.
## Lesson
To find if ratios are equal, we need to find a common number. to do so, we need to find a common factor of both.
Using the examples in the introduction above, we have two soccer teams:
Team A has a win ratio of 2:5 (i.e. it wins 2 games out of every 5 that it plays)
Team B has a win ratio of 4:9 (i.e. it wins 4 games out of every 9 that it plays)
It's easier if we compare the second number in the ratio since that's the number of games played. We need to find a common factor of 5 and 9.
To find a common factor we can compare the multiples in the table below
59
x159
x21018
x31527
x42036
x52545
x63054
x73563
x84072
x94581
Therefore, 45 is the lowest common factor.
Now we need to set Team A’s ratio to 45. To do that, we need to multiply both parts by 9.
$2(9) : 5(9)$
$= 18 : 45$
So, if Team A were to play 45 games, it would win 18.
For Team B, we need to multiply both sides of the ratio by 5.
$4(5) : 9(5)$
$= 20 : 45$
So, if Team B were to play 45 games, it would win 20.
Therefore, Team B has a slightly better win ratio. By setting the terms, we can directly compare if they are equal or not.
## Examples
$1:3$ and $2:5$
To express the ratio '$1$ to $3$' as a fraction, place $1$ over $3$ and reduce
To express the ratio '$2$ to $5$' as a fraction, place $2$ over $5$ and reduce
$\dfrac{1}{3}$ is not equal to $\dfrac{2}{5}$
$2:4$ and $4:8$
To express the ratio '$2$ to $4$' as a fraction, place $2$ over $4$ and reduce
To express the ratio '$4$ to $8$' as a fraction, place $4$ over $8$ and reduce
$\dfrac{2}{4}$ can be reduced, since $2$ is a factor of both $2$ and $4$:
$\dfrac{2}{4} \div \dfrac{2}{2} = \dfrac{1}{2}$
The fraction is now in lowest terms
$\dfrac{4}{8}$ can be reduced, since $4$ is a factor of both $4$ and $8$:
$\dfrac{4}{8} \div \dfrac{4}{4} = \dfrac{1}{2}$
The fraction is now in lowest terms
$\dfrac{2}{4}$ is equal to $\dfrac{4}{8}$
$12:18$ and $9:15$
To express the ratio '$12$ to $18$' as a fraction, place $12$ over $18$ and reduce
To express the ratio '$9$ to $15$' as a fraction, place $9$ over $15$ and reduce
$\dfrac{12}{18}$ can be reduced, since $6$ is a factor of both $12$ and $18$:
$\dfrac{12}{18} \div \dfrac{6}{6} = \dfrac{2}{3}$
The fraction is now in lowest terms
$\dfrac{9}{15}$ can be reduced, since $3$ is a factor of both $9$ and $15$:
$\dfrac{9}{15} \div \dfrac{3}{3} = \dfrac{3}{5}$
The fraction is now in lowest terms
$\dfrac{12}{18}$ is not equal to $\dfrac{9}{15}$<|endoftext|>
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The past perfect can be challenging for some English learners. It can be tough to figure out exactly when or how to use it. But it's really not very difficult. Basically, the past perfect tense occurs before the past tense and is usually used when you tell a story that happened in the past. When you're telling a story, the past tense is usually the base tense or the tense where the story occurs, and everything that happens before that is told using the past perfect.
As always, we should master its structure. The past perfect is formed with had + past participle. For example, the past perfect of fly is had flown; the past perfect of go is had gone. Mastering the structure of the past perfect is the first and important step in learning how to use it.
Once you've mastered the structure of this tense, you can start using it when you're talking about the past. Remember, anything that happened before the past tense should be told in the past perfect.
Here are some examples:
1. I had already eaten when I got to the party. ("Had eaten" happened before "got".)
2. They wanted to watch the new Will Smith comedy, but their friend had just watched it the day before. ("Had watched" happened before "wanted".)
3. He almost had a heart attack when he realized he had forgotten his passport at home.
4. His family traveled a lot. By the time he was 12, he had been to most of Europe and Asia.
5. He had gone to the doctor before he went to the post office.
When we use the past perfect, we make it easy for people to understand our stories because we're making the order of events clear. Our listeners can easily tell what happened first and what happened next. Sometimes, however, you can simply use the past tense instead of the past perfect if it's already clear which event happened first and which event happened next. We can do this when the words "before" or "after" are in the sentence.
Sentence number 5 above is an example. Because it has the word "before", it's already clear that the person went to the doctor first and then he went to the post office. Because of this we can also say, He went to the doctor before he went to the post office.
Similarly, the following two sentences are both correct and mean the same.
1. They went to the museum after they had eaten lunch.
2. They went to the museum after they ate lunch.
Alright, folks. I hope this lesson explained the past perfect well for you. If it had, you should bookmark it for future reference and recommend it to friends who could use the help.
Thanks for practicing your English with us. My name is Joe. Are you following me on Twitter yet? You'll find me there @joeyu2nd. You can also LIKE the small guide site on Facebook for more quick, small lessons. Catch you later.<|endoftext|>
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What is Disposable Income
Disposable income, also known as disposable personal income (DPI), is the amount of money that households have available for spending and saving after income taxes have been accounted for. Disposable personal income is often monitored as one of the many key economic indicators used to gauge the overall state of the economy.
BREAKING DOWN Disposable Income
Disposable income is an important measure of household financial resources. For example, consider a family with a household income of $100,000, and the family has an effective income tax rate of 25%. This household's disposable income would then be $75,000 ($100,000 - $25,000). Economists use DPI as a starting point to gauge households' rates of savings and spending.
Statistical Uses of Disposable Income
Many useful statistical measures and economic indicators derive from disposable income. For example, economists use disposable income as a starting point to calculate metrics such as discretionary income, personal savings rates, marginal propensity to consume (MPC) and marginal propensity to save (MPS).
Disposable income minus all payments for necessities, such as mortgage, health insurance, food and transportation, equals discretionary income. This portion of disposable income can be spent on what the income earner chooses or, alternatively, it can be saved. The personal savings rate is the percentage of disposable income that goes into savings for retirement or use at a later date. Marginal propensity to consume represents the percentage of each additional dollar of disposable income that gets spent, while marginal propensity to save denotes the percentage that gets saved.
For several months in 2005, the average personal savings rate dipped into negative territory for the first time since 1933. This means that in 2005, Americans were spending all of their disposable income each month and then tapping into debt for further spending.
Disposable Income for Wage Garnishment
The federal government uses a slightly different method to calculate disposable income for wage garnishment purposes. Sometimes, the government garnishes an income earner's wages for payment of back taxes or delinquent child support. It uses disposable income as a starting point to determine how much to seize from the earner's paycheck. As of 2016, the amount garnished may not exceed 25% of a person's disposable income or the amount by which a person's weekly income exceeds 30 times the federal minimum wage, whichever is less.
In addition to income taxes, the government subtracts health insurance premiums and involuntary retirement plan contributions from gross income when calculating disposable income for wage garnishment purposes. Returning to the above example, if the family described pays $10,000 per year in health insurance premiums and is required to contribute $5,000 to a retirement plan, its disposable income for wage garnishment purposes shrinks from $75,000 to $60,000.<|endoftext|>
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Understanding Skin Cancer And Its types
In the United States, skin cancer is one of the most common cancer and every year thousands of people are diagnosed by the skin cancer. Skin is the largest organ of the body and is made up of cells. It performs various functions including, protection against chemicals, biological assailants and it also plays a role in thermoregulation. The skin is composed of two main layers - epidermis, dermis.
1. Epidermis - The outermost layer of the skin is known as epidermis and it is primarily composed of two types of cells - keratinocytes and dendritic cells. Keratinocytes functions to synthesize keratin, a protein that plays a protective role. According to the keratinocyte morphology, the epidermis is commonly divided into four layers:
- The basal cell layer - It is the primary location of active cells in the epidermis that give rise to the cell of the outer epidermal layer.
- The squamous cell layer - It composed of various cells that differ in shape and structure, depending on the location of the cells.
- The granular cell layer - It is the most superficial layer of the epidermis, containing living cells that hold abundant keratohyalin granules in their cytoplasm and are responsible for the synthesis of proteins that are involved in keratinocytes.
- The horny cell layer - It is rich in proteins and it provides protection to the epidermis and acts as a barrier to prevent water loss and invasion by the foreign substance.
2. Dermis - This layer of the skin is located underneath the epidermis layer and it comprises of a bulk of skin that provides elasticity and tensile strength. It protects the body from mechanical injury, binds water and aids thermal regulation. Also, it interacts with the epidermis layer to maintain the properties of the tissues.
Types of Skin Cancer
Cancer is a serious disease that occurs when the healthy and normal cells of the body functions abnormally and leads to an uncontrolled growth. Skin cancer is the type of cancer that starts in the skin. The three main types of skin cancer - basal cell carcinoma, squamous cell carcinoma and melanoma begins in the epidermis layer.
Basal cell carcinoma is the most common skin cancer in humans and it begins with healthy cells of the outer layer of the skin change and grows out of control. It can grow at any age but the incidence of developing basal cell carcinoma increases after the age of 40. Basal cell carcinoma is characterized by the open sores on the skin that bleeds and remains open to few weeks, reddish patch, itching, and red, white or translucent lump. The most significant factors that play a role in the development of basal cell carcinoma are fair skin type and an exposure to the ultraviolet radiations. Other risk factors include immune suppression, exposure to arsenic, prior treatment with ionizing radiations and certain hereditary disorders. It can occur anywhere in the body but primarily, the areas that are most often exposed to the sun are at higher risk such as face, neck, head, and ears. Also, any damaged area due to burn or scars are also affected by this cancer. The treatment of this skin cancer depends on the size of the size of cancer and the area on which it is located.
Squamous cell carcinoma
Squamous cell carcinoma begins in the upper layer of the epidermis and it is an epithelial maliciousness that usually occurs at the areas that are normally covered with the squamous epithelium such as, skin, lips, mouth, prostate, lungs, vagina, and cervix. Squamous cell carcinoma is characterized by the scaly and crusty areas of the lump that may bleed and become inflamed. It appears to be thickened, red and scaly spot and it may also look like a sore that has not healed. The most leading risk factor for squamous cell carcinoma include sum exposure, advancing age, and UVR sensitive skin. According to the researchers, cumulative chronic UVR exposure is the strongest environmental factor that leads to the development of squamous cell carcinoma. Also, immunosuppressive agents such as chemotherapy, classical immunosuppressives are considered to have an impact on the risk factor of this skin cancer.
Melanoma is cancer that begins in the melanocytes. The melanocyte are the pigment synthesizing cells that are confined in the skin predominantly to the basal layer. These are responsible for the production of the pigment melanin and its transfer to keratinocytes. Melanoma is characterized as a new or existing spot on the body that changes in size, shape, and color over several weeks or months. It can appear in more than one color such as brown, pink, red, and white. Melanoma is not as common as basal cell carcinoma and squamous cell carcinoma but it is considered to be one of the most serious types of skin cancer. This is because it is more likely to spread to other parts of the body such as the liver, lung, bones, and brain. Unprotected exposure to the ultraviolet rays remains the most significant cause of the melanoma and other skin cancers. Increasing age and immune suppression are also said to be the risk factors for the development of melanoma cancer.
Exposure to ultraviolet is one of the common cause of skin cancers, therefore it is recommended to take preventive steps to avoid an exposure to sun for a longer duration. Always use sunblocks that will protect your skin from sunburn and other diseases to some extent.
Share this post with your family and friends to create awareness.<|endoftext|>
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Freedom and Tradition: An Introduction to Classical Liberalism and Conservatism
A Course with Professor Thomas Patrick Burke
The publication in 1776 of Adam Smith’s book An Inquiry into the Nature and Causes of the Wealth of Nationsbrought about a revolution in the economic life of the human race which is still continuing. From the beginnings of recorded history, governments have used their power to influence commerce in ways they thought desirable, often to give special benefits to particular groups of people. The Romans organized their empire so that the people of the city of Rome (who often had decisive influence in the selection of the emperor) had free “bread and circuses” at the expense of those who lived in more remote regions (such as the Egyptians, who were compelled to provide the wheat for the bread at low Roman prices). During the middle ages in much of Europe the “three-field” system was imposed on farmers, whereby all the land of the village was divided up into long, narrow strips distributed over fertile and arid land in the interests of “fairness.” The Elizabethans legislated a maximum wage, to protect employers from excessive competition; and so on. In international trade before Adam Smith the prevailing system was what is known as “mercantilism,” which we will need to explain at greater length.
This system was based on the apparently common-sense assumption that wealth consisted in money, and therefore that the wealth of any particular nation consisted in the amount of money the nation as a whole possessed, and especially its government. It followed from this that the aim of trade was to maximize the nation’s quantity of money. And since money is obtained by selling rather than by buying, the aim of trade was to sell as much as possible and buy as little as possible, i.e. to export as much as possible and import as little as possible. This principle led the nations of Europe, especially Britain and France, which were the furthest developed as nation-states, to create overseas empires whose peoples would be obliged to buy the goods produced by the mother-country at maximum prices, and sell it their own products at minimum prices. (Actually the British Empire grew initially mostly by accident, without any grand plan of the British government, as a result of the need to protect British traders from assault by envious locals. But once an area was brought into the empire, the mercantilist system was imposed through tariffs and other means.) Inevitably this led to armed conflict between these empires, as in the French and Indian War in North America, since a benefit to one was a loss to the other. Adam Smith’s thesis is that this immense system stretching right around the globe was an immense mistake and a recipe for poverty rather than for wealth. The wealth of a nation does not consist in money, he argued, but in the goods and services available to its citizens. These are two very different things. Money, i.e. gold and silver, is a commodity and its price can change depending on its availability. Spain received a great deal of gold from its conquests in South America, but this did not help the living standards of the Spanish people, who remained poor. The aim of trade, then, is to maximize the availability of goods and services. This is accomplished, not by selling, but rather by buying. That nation will be wealthiest which buys as much as it can from the labor of others, while selling as little as possible of its own labor to pay for it. (Smith adopts the view suggested by John Locke, known as the labor theory of value, that the value of any product is created chiefly by the labor that goes into it. This was intended just as part of his theory of private property: property is acquired in the first instance out of nature by the labor of the person who makes it available for human consumption. Many subsequent writers, such as Marx, misunderstood this to mean that there is an objective value in every object of commerce, namely the number of hours of labor invested in it, which ought to determine its price. Smith does not go so far, though he does accept the idea of a “natural value.” Hobbes, by contrast, had stated what is now the universal view of economists, that the market value of any product is given by the demand for it. “The value of all things contracted for is measured by the appetite of the contractors, and therefore the just value is that which they be contented to give.”) Since we must labor to produce exports, but not imports, the purpose of exports is just to pay for imports, and so the less we have to pay for any given quantity of imports, the better. But in the long run the best way to achieve this is not by the government giving special favors to imports, for that will instigate similar measures by other governments in favor of their own imports and against our exports, but by “the simple system of natural liberty” which allows everyone to buy or sell what he likes. Smith begins his work with a discussion of the central role of labor.
“The annual labour of every nation is the fund which originally supplies it with all the necessaries and conveniences of life which it annually consumes, and which consist always either in the immediate produce of that labour, or in what is purchased with that produce from other nations.
According therefore as this produce, or what is purchased with it, bears a greater or smaller proportion to the number of those who are to consume it, the nation will be better or worse supplied with all the necessaries and conveniences for which it has occasion. (Introduction)”
This depends in the first instance on the “skill, dexterity and judgment” with which its labor is generally supplied rather than on the number of people employed.
“Among the savage nations of hunters and fishers, every individual who is able to work, is more or less employed in useful labour, and endeavours to provide, as well as he can, the necessaries and conveniences of life, for himself, or such of his family or tribe as are either too old, or too young, or too infirm to go a hunting and fishing. Such nations, however, are so miserably poor that, from mere want, they are frequently reduced, or, at least, think themselves reduced, to the necessity sometimes of directly destroying, and sometimes of abandoning their infants, their old people, and those afflicted with lingering diseases, to perish with hunger, or to be devoured by wild beasts. Among civilised and thriving nations, on the contrary, though a great number of people do not labour at all, many of whom consume the produce of ten times, frequently of a hundred times more labour than the greater part of those who work; yet the produce of the whole labour of the society is so great that all are often abundantly supplied, and a workman, even of the lowest and poorest order, if he is frugal and industrious, may enjoy a greater share of the necessaries and conveniences of life than it is possible for any savage to acquire. (Introduction).”
This naturally gives rise to the question, what causes some labor to be more productive than other labor? Smith answers that it is the division of labor. It would be difficult to exaggerate the importance of this concept, which has application in many different fields. It is the idea that instead of a single individual making an entire complex product, the process of production is divided up between a number of people who each make a particular portion of it. Smith argues that this specialization is the main factor in productivity. He gives a striking example he has personally witnessed.
“To take an example, therefore, from a very trifling manufacture; but one in which the division of labour has been very often taken notice of, the trade of the pin-maker; a workman not educated to this business (which the division of labour has rendered a distinct trade), nor acquainted with the use of the machinery employed in it (to the invention of which the same division of labour has probably given occasion), could scarce, perhaps, with his utmost industry, make one pin in a day, and certainly could not make twenty. But in the way in which this business is now carried on, not only the whole work is a peculiar trade, but it is divided into a number of branches, of which the greater part are likewise peculiar trades. One man draws out the wire, another straights it, a third cuts it, a fourth points it, a fifth grinds it at the top for receiving the head; to make the head requires two or three distinct operations; to put it on is a peculiar business, to whiten the pins is another; it is even a trade by itself to put them into the paper; and the important business of making a pin is, in this manner, divided into about eighteen distinct operations, which, in some manufactories, are all performed by distinct hands, though in others the same man will sometimes perform two or three of them. I have seen a small manufactory of this kind where ten men only were employed, and where some of them consequently performed two or three distinct operations. But though they were very poor, and therefore but indifferently accommodated with the necessary machinery, they could, when they exerted themselves, make among them about twelve pounds of pins in a day. There are in a pound upwards of four thousand pins of a middling size. Those ten persons, therefore, could make among them upwards of forty-eight thousand pins in a day. Each person, therefore, making a tenth part of forty-eight thousand pins, might be considered as making four thousand eight hundred pins in a day. But if they had all wrought separately and independently, and without any of them having been educated to this peculiar business, they certainly could not each of them have made twenty, perhaps not one pin in a day; that is, certainly, not the two hundred and fortieth, perhaps not the four thousand eight hundredth part of what they are at present capable of performing, in consequence of a proper division and combination of their different operations.”
(In the following century Darwin will make brilliant use of this concept to explain evolution, especially the evolution of the sexes.) This higher productivity results from a number of factors: specialization improves the dexterity of the worker, saves time, and, perhaps especially, leads the specialized workmen to invent labor-saving machines and methods. When the division of labor is employed widely, it produces a vastly higher standard of living throughout the whole society.
“It is the great multiplication of the productions of all the different arts, in consequence of the division of labour, which occasions, in a well-governed society, that universal opulence which extends itself to the lowest ranks of the people.”
If we examine even the simplest product, we will find that an astonishing number of people have contributed to its making.
“Observe the accommodation of the most common artificer or day-labourer in a civilised and thriving country, and you will perceive that the number of people of whose industry a part, though but a small part, has been employed in procuring him this accommodation, exceeds all computation. The woollen coat, for example, which covers the day-labourer, as coarse and rough as it may appear, is the produce of the joint labour of a great multitude of workmen. The shepherd, the sorter of the wool, the wool-comber or carder, the dyer, the scribbler, the spinner, the weaver, the fuller, the dresser, with many others, must all join their different arts in order to complete even this homely production. How many merchants and carriers, besides, must have been employed in transporting the materials from some of those workmen to others who often live in a very distant part of the country! How much commerce and navigation in particular, how many ship-builders, sailors, sail-makers, rope-makers, must have been employed in order to bring together the different drugs made use of by the dyer, which often come from the remotest corners of the world! What a variety of labour, too, is necessary in order to produce the tools of the meanest of those workmen! To say nothing of such complicated machines as the ship of the sailor, the mill of the fuller, or even the loom of the weaver, let us consider only what a variety of labour is requisite in order to form that very simple machine, the shears with which the shepherd clips the wool. The miner, the builder of the furnace for smelting the ore, the seller of the timber, the burner of the charcoal to be made use of in the smelting-house, the brickmaker, the brick-layer, the workmen who attend the furnace, the mill-wright, the forger, the smith, must all of them join their different arts in order to produce them. Were we to examine, in the same manner, all the different parts of his dress and household furniture, the coarse linen shirt which he wears next his skin, the shoes which cover his feet, the bed which he lies on, and all the different parts which compose it, the kitchen-grate at which he prepares his victuals, the coals which he makes use of for that purpose, dug from the bowels of the earth, and brought to him perhaps by a long sea and a long land carriage, all the other utensils of his kitchen, all the furniture of his table, the knives and forks, the earthen or pewter plates upon which he serves up and divides his victuals, the different hands employed in preparing his bread and his beer, the glass window which lets in the heat and the light, and keeps out the wind and the rain, with all the knowledge and art requisite for preparing that beautiful and happy invention, without which these northern parts of the world could scarce have afforded a very comfortable habitation, together with the tools of all the different workmen employed in producing those different conveniences; if we examine, I say, all these things, and consider what a variety of labour is employed about each of them, we shall be sensible that, without the assistance and co-operation of many thousands, the very meanest person in a civilised country could not be provided, even according to what we very falsely imagine the easy and simple manner in which he is commonly accommodated. Compared, indeed, with the more extravagant luxury of the great, his accommodation must no doubt appear extremely simple and easy; and yet it may be true, perhaps, that the accommodation of a European prince does not always so much exceed that of an industrious and frugal peasant as the accommodation of the latter exceeds that of many an African king, the absolute master of the lives and liberties of ten thousand naked savages.”
If this is so, then the next question must be, what gives rise to the division of labor? Smith’s answer to this is memorable. It is not out of any large or coordinated design, and especially not because they were commanded to by government, but simply out of their individual desire to earn a living: out of their self-interest. (Smith’s concept of “self-interest” is far broader, however, than might be assumed by someone who has not read him: it can include sympathy and concern for the well-being of our neighbor, for example.)
“This division of labour, from which so many advantages are derived, is not originally the effect of any human wisdom, which foresees and intends that general opulence to which it gives occasion. It is the necessary, though very slow and gradual consequence of a certain propensity in human nature which has in view no such extensive utility; the propensity to truck, barter and exchange one thing for another.”
This is something distinctive of human beings: no animal does it.
“It is common to all men, and to be found in no other race of animals, which seem to know neither this nor any other species of contracts. …Nobody ever saw a dog make a fair and deliberate exchange of one bone for another with another dog. Nobody ever saw one animal by its gestures and natural cries signify to another, this is mine, that yours; I am willing to give this for that. When an animal wants to obtain something either of a man or of another animal, it has no other means of persuasion but to gain the favour of those whose service it requires. A puppy fawns upon its dam, and a spaniel endeavours by a thousand attractions to engage the attention of its master who is at dinner, when it wants to be fed by him. Man sometimes uses the same arts with his brethren, and when he has no other means of engaging them to act according to his inclinations, endeavours by every servile and fawning attention to obtain their good will. He has not time, however, to do this upon every occasion. In civilised society he stands at all times in need of the cooperation and assistance of great multitudes, while his whole life is scarce sufficient to gain the friendship of a few persons. In almost every other race of animals each individual, when it is grown up to maturity, is entirely independent, and in its natural state has occasion for the assistance of no other living creature.”
The secret of the success of the system of exchanges we call the market is that it does not depend on people’s generosity or kindness, but requires nothing more than an appeal to their self-interest.
“… man has almost constant occasion for the help of his brethren, and it is in vain for him to expect it from their benevolence only. He will be more likely to prevail if he can interest their self-love in his favour, and show them that it is for their own advantage to do for him what he requires of them. Whoever offers to another a bargain of any kind, proposes to do this. Give me that which I want, and you shall have this which you want, is the meaning of every such offer; and it is in this manner that we obtain from one another the far greater part of those good offices which we stand in need of. It is not from the benevolence of the butcher, the brewer, or the baker that we expect our dinner, but from their regard to their own interest. We address ourselves, not to their humanity but to their self-love, and never talk to them of our own necessities but of their advantages. Nobody but a beggar chooses to depend chiefly upon the benevolence of his fellow-citizens. Even a beggar does not depend upon it entirely. The charity of well-disposed people, indeed, supplies him with the whole fund of his subsistence. But though this principle ultimately provides him with all the necessaries of life which he has occasion for, it neither does nor can provide him with them as he has occasion for them. The greater part of his occasional wants are supplied in the same manner as those of other people, by treaty, by barter, and by purchase. With the money which one man gives him he purchases food. The old clothes which another bestows upon him he exchanges for other old clothes which suit him better, or for lodging, or for food, or for money, with which he can buy either food, clothes, or lodging, as he has occasion.”
Of course, this does not mean that the market positively excludes kindness and generosity. They always remain welcome possibilities. Only it does not demand them as a prerequisite for success.
“As it is by treaty, by barter, and by purchase that we obtain from one another the greater part of those mutual good offices which we stand in need of, so it is this same trucking disposition which originally gives occasion to the division of labour. In a tribe of hunters or shepherds a particular person makes bows and arrows, for example, with more readiness and dexterity than any other. He frequently exchanges them for cattle or for venison with his companions; and he finds at last that he can in this manner get more cattle and venison than if he himself went to the field to catch them. From a regard to his own interest, therefore, the making of bows and arrows grows to be his chief business, and he becomes a sort of armourer. Another excels in making the frames and covers of their little huts or movable houses. He is accustomed to be of use in this way to his neighbours, who reward him in the same manner with cattle and with venison, till at last he finds it his interest to dedicate himself entirely to this employment, and to become a sort of house-carpenter. In the same manner a third becomes a smith or a brazier, a fourth a tanner or dresser of hides or skins, the principal part of the nothing of savages. And thus the certainty of being able to exchange all that surplus part of the produce of his own labour, which is over and above his own consumption, for such parts of the produce of other men’s labour as he may have occasion for, encourages every man to apply himself to a particular occupation, and to cultivate and bring to perfection whatever talent or genius he may possess for that particular species of business.”
The Wealth of Nations is filled with perceptive comments on human nature. Here is one of them:
“The difference of natural talents in different men is, in reality, much less than we are aware of; and the very different genius which appears to distinguish men of different professions, when grown up to maturity, is not upon many occasions so much the cause as the effect of the division of labour. The difference between the most dissimilar characters, between a philosopher and a common street porter, for example, seems to arise not so much from nature as from habit, custom, and education. When they came into the world, and for the first six or eight years of their existence, they were perhaps very much alike, and neither their parents nor playfellows could perceive any remarkable difference. About that age, or soon after, they come to be employed in very different occupations. The difference of talents comes then to be taken notice of, and widens by degrees, till at last the vanity of the philosopher is willing to acknowledge scarce any resemblance. But without the disposition to truck, barter, and exchange, every man must have procured to himself every necessary and conveniency of life which he wanted. All must have had the same duties to perform, and the same work to do, and there could have been no such difference of employment as could alone give occasion to any great difference of talents.
As it is this disposition which forms that difference of talents, so remarkable among men of different professions, so it is this same disposition which renders that difference useful. Many tribes of animals acknowledged to be all of the same species derive from nature a much more remarkable distinction of genius, than what, antecedent to custom and education, appears to take place among men. By nature a philosopher is not in genius and disposition half so different from a street porter, as a mastiff is from a greyhound, or a greyhound from a spaniel, or this last from a shepherd’s dog. Those different tribes of animals, however, though all of the same species, are of scarce any use to one another. The strength of the mastiff is not, in the least, supported either by the swiftness of the greyhound, or by the sagacity of the spaniel, or by the docility of the shepherd’s dog. The effects of those different geniuses and talents, for want of the power or disposition to barter and exchange, cannot be brought into a common stock, and do not in the least contribute to the better accommodation and conveniency of the species. Each animal is still obliged to support and defend itself, separately and independently, and derives no sort of advantage from that variety of talents with which nature has distinguished its fellows. Among men, on the contrary, the most dissimilar geniuses are of use to one another; the different produces of their respective talents, by the general disposition to truck, barter, and exchange, being brought, as it were, into a common stock, where every man may purchase whatever part of the produce of other men’s talents he has occasion for.”
Adam Smith’s greatest single discovery is what I have termed “the principle of mutual benefit.” This is the fact that exchanges take place for one reason only: both parties consider that they benefit. It can never happen that, in the absence of force and fraud, only one party to an exchange can expect to benefit. If that were the case the exchange would not take place. It is true that the benefit may not be equal, but a lesser benefit is still a benefit, not a loss. He deals with this mainly in his discussion of the “balance of trade,” which occupies the whole of Book 4.
“Nothing, however, can be more absurd than this whole doctrine of the balance of trade, upon which, not only these restraints, but almost all the other regulations of commerce are founded. When two places trade with one another, this doctrine supposes that, if the balance be even, neither of them either loses or gains; but if it leans in any degree to one side, that one of them loses and the other gains in proportion to its declension from the exact equilibrium. Both suppositions are false. A trade which is forced by means of bounties and monopolies may be and commonly is disadvantageous to the country in whose favour it is meant to be established, as I shall endeavour to show hereafter. But that trade which, without force or constraint, is naturally and regularly carried on between any two places is always advantageous, though not always equally so, to both.”
Since every exchange benefits both parties, even if not equally, those benefits become available for further exchanges. The result is that, without anyone intending to benefit society, nonetheless in a system of free exchanges the entire society benefits, as “by an invisible hand.”
“As every individual, therefore, endeavours as much as he can both to employ his capital in the support of domestic industry, and so to direct that industry that its produce may be of the greatest value; every individual necessarily labours to render the annual value of society as great as he can. He generally, indeed, neither intends to promote the public interest, nor knows how much he is promoting it. By preferring the support of domestic to that of foreign industry, he intends only his own security; and by directing that industry in such a manner as its produce may be of the greatest value, he intends only his own gain, and he is in this, as in many other cases, led by an invisible hand to promote an end which was no part of his intention. Nor is it always the worse for the society that it was no part of it. By pursuing his own interest he frequently promotes that of society more effectually than when he really intends to promote it. I have never known much good done by those who affected to trade for the public good. It is an affectation, indeed, not very common among merchants, and very few words need be employed in dissuading them from it.”
He concludes that all systems of government intervention in the market are self-defeating.
“It is thus that every system which endeavours, either by extraordinary encouragements to draw towards a particular species of industry a greater share of the capital of the society than what would naturally go to it, or, by extraordinary restraints, force from a particular species of industry some share of the capital which would otherwise be employed in it, is in reality subversive of the great purpose which it means to promote. It retards, instead of accelerating, the progress of the society towards real wealth and greatness; and diminishes, instead of increasing, the real value of the annual produce of its land and labour.”
At the end of Book 4 he gives what has become the classic statement of the role of government in a free society:
“All systems either of preference or of restraint, therefore, being thus completely taken away, the obvious and simple system of natural liberty establishes itself of its own accord. Every man, as long as he does not violate the laws of justice, is left perfectly free to pursue his own interest his own way, and to bring both his industry and capital into competition with those of any other man, or order of men. The sovereign is completely discharged from a duty, in the attempting to perform which he must always be exposed to innumerable delusions, and for the proper performance of which no human wisdom or knowledge could ever be sufficient; the duty of superintending the industry of private people, and of directing it towards the employments most suitable to the interest of the society. According to the system of natural liberty, the sovereign has only three duties to attend to; three duties of great importance, indeed, but plain and intelligible to common understandings: first, the duty of protecting the society from violence and invasion of other independent societies; secondly, the duty of protecting, as far as possible, every member of the society from the injustice or oppression of every other member of it, or the duty of establishing an exact administration of justice; and, thirdly, the duty of erecting and maintaining certain public works and certain public institutions which it can never be for the interest of any individual, or small number of individuals, to erect and maintain; because the profit could never repay the expense to any individual or small number of individuals, though it may frequently do much more than repay it to a great society.”<|endoftext|>
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# Rotations - O & M Meets Math
By Susan LoFranco on Apr 24, 2017
• The student will recognize the rotation of shapes.
• Find objects in the environment that are rotations.
• Orientation clockwise and counter clockwise
• Understanding the concept of movement in space 90, 180, 270, 360 degrees - both clockwise and counter clockwise
## Materials
• 4 Quadrant Graph Paper - either bold or raise lined
• Markers, dots, tape to connect points on graph
• 4 or 5 spice bottles
## Procedure
Vocabulary: counterclockwise, clockwise, rotation,
Depending on the time the teacher has with the student, Parts 1, 2 and 3 can be done in separate sessions.
### Part 1:
To prep for this part of the lesson, using the masking tape, mark the floor with a large + sign which will represent the four quadrants of the graph paper. This exercise will ask the student to replicate with their body what will later be done on graph paper. Initially the student will move to a stated quadrant and then turn the degrees related to that quadrant. At the most abstract the student should be able to move into a stated quadrant and know the direction to orient their body before moving first.
• Before beginning exercise on the grid you have made on the floor review with the student turning 90, 180, 270 and 360 degrees while standing in place.
• Whatever direction the student is turning in (clockwise or counterclockwise) the student should describe the degree of rotation in the opposite direction needed to get to the same point.
• After reviewing the rotations in place move to the grid on the floor.
• Review the quadrants reinforcing how quadrants are numbered in counter clockwise order beginning with the top right quadrant (1).
• Have the student move clockwise then counterclockwise 90, 180, 270 and 360 degrees into each of the quadrants made by the grid on the floor.
• While the student is rotating, at each turn ask the student how many degrees they are turning in the opposite direction. 90 degrees on way is 270 degrees in the opposite direction.
• Once the student is confident, continue on to Part 2.
### Part 2:
Student will be using The Rotation Worksheet to rotate a figure around the x,y axis.
• With the student, review the quadrants on the graph paper moving clockwise and counterclockwise.
• With the student, turn the graph paper clockwise then counterclockwise 90, 180, 270, and 360 degrees.
• Before beginning the rotation, student should identify the points on the grid which will be rotated.
• Record the points (using the student's media of choice) for use when rotating.
• When beginning the rotation the student should turn the paper (clockwise/counterclockwise) the appropriate number of degrees in order to draw the figure in each of the quadrants.
• With the student, draw the first rotation (90 degrees clockwise) using the coordinates identifed earlier to plot the points on the grid.
• With the student, connect the points to complete the figure.
• With the student, identify how many degrees counterclockwise the figure is.
• If the student is confident, have them continue rotating the figure around the axis, identifying the counterclockwise position at each rotation.
### Part 3 - Assessment
Student will use their understanding of rotations to move the spice jars to another quadrant on the graph paper. The teacher will need to place the spice jars in one quadrant as in image above or however they might fit into one quadrant.
Question to Student: Imagine that you are in your kitchen. You are moving your spices from one cabinet to another. If you want the spices in the same order and the cabinet you are moving the spices to is 270 degrees clockwise on the grid from where the spice jars already sit what will be the new formation of the spice jars?
## Variations
Attached File(s):<|endoftext|>
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# Binary Number Systems and Codes - PowerPoint PPT Presentation
Binary Number Systems and Codes
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Binary Number Systems and Codes
## Binary Number Systems and Codes
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##### Presentation Transcript
1. Binary Number SystemsandCodes ECEn 224
2. Positional Numbers • What does 5132.13 really mean? • Depends on the number base! • Assuming base 10: 5132.1310 = 5x103 + 1x102 + 3x101 + 2x100 + 1x10-1 + 3x10-2 • Assuming base 6: 5132.136 = 5x63 + 1x62 + 3x61 + 2x60 + 1x6-1 + 3x6-2 • We often use a subscript to indicate the base. ECEn 224
3. Positional Number Examples 527.4610 = (5 x 102) + (2 x 101) + (7 x 100) + (4 x 10-1) + (6 x 10-2) 527.468 = (5 x 82) + (2 x 81) + (7 x 80) + (4 x 8-1) + (6 x 8-2) 527.465 = illegalwhy? 1011.112 = (1 x 23) + (0 x 22) + (1 x 21) + (1 x 20) + (1 x 2-1) + (1 x 2-2) This works for binary as well… ECEn 224
4. Conversion from Binary Convert 101011.112 to base 10: 101011.112 = 1x25 + 0x24 + 1x23 + 0x22 + 1x21 + 1x20 + 1x2-1 + 1x2-2 = 32 + 0 + 8 + 0 + 2 + 1 + ½ + ¼ = 43.7510 ECEn 224
5. - 64 1x26 - 32 1x25 - 16 1x24 - 0 0x23 - 0 0x22 - 2 1x21 - 0 0x20 Convert 11410 to binary: 114 50 18 11410 = 11100102 2 Read this way 2 2 0 0 This method also works for fractional numbers. ECEn 224
6. 2 2 2 2 2 2 2 An Alternate Method 114 57 R 0 28 R 1 11410 = 11100102 14 R 0 Read this way 7 R 0 3 R 1 1 R 1 0 R 1 ECEn 224
7. (0).8 (1).6 (1).2 process starts repeating here (0).4 (0).8 Converting fractions from base 10 to binary: Convert 0.710 to binary 0.7 x 2 (1).4 0.710 = 0.1 0110 0110 …2 x 2 x 2 Read this way x 2 x 2 x 2 ECEn 224
8. Convert 114.710 to binary: 114.7 - 64 1x26 50.7 - 32 1x25 18.7 - 16 1x24 2.7 - 0 0x23 2.7 - 0 0x22 2.7 - 2 1x21 0.7 - 0 0x20 0.7 - 0.5 1x2-1 0.2 - 0.0 0x2-2 0.2 - 0.125 1x2-3 0.075 … We could use the first technique. Read this way 11210 = 1110010.10...2 ECEn 224
9. 2 2 2 2 2 2 2 (0).8 (1).6 (1).2 (0).4 (0).8 Convert 114.710 to binary: 0.7 x 2 114 (1).4 57 R 0 x 2 28 R 1 Or we could combine the second and third techniques. 14 R 0 x 2 7 R 0 Read this way 11210 = 1110010.10110...2 x 2 3 R 1 1 R 1 x 2 0 R 1 x 2 ECEn 224
10. Hexadecimal • Commonly used for binary data • 1 hex digit 4 binary digits (bits) • Need more digits than just 0-9 • Use 0-9, A-F • A-F are for 10-15FA216 = 15x162 + 10x161 + 2x160FA216 = 1111 1010 0010 Each group of 4 bits 1 hex digit ECEn 224
11. Other Notations For Binary and Hex • Binary • 101102 • 10110b • 0b10110 • Hexadecimal • 57316 • 0x573 • 573h • 16#573 ECEn 224
12. Other Codes BCD ASCII Gray ECEn 224
13. Binary Coded Decimal(BCD) Convert 249610 to BCD Code 2 4 9 6 0 0 1 0 0 1 0 0 1 0 0 1 0 1 1 0 Note this is very different from converting to binary which yields: 1 0 0 1 1 1 0 0 0 0 0 02 ECEn 224
14. Binary Coded Decimal(BCD) • Why use BCD? • In some applications it may be easier to work with • Financial institutions must be able to represent base 10 fractions (e.g., 1/10) • 0.110 = 0.00110011001100…2 • Using BCD ensures that numeric results are identical to base 10 results ECEn 224
15. Binary Codes ASCII Code • ASCII American Standard Code for Information Interchange • ASCII is a 7-bit code used to represent letters, symbols, and terminal codes • There are also Extended ASCII codes, represented by 8-bit numbers • Terminal codes include such things as:Tab (TAB) Line feed (LF) Carriage return (CR) Backspace (BS) Escape (ESC) And many more! ECEn 224
16. Binary Codes ASCII Code ECEn 224
17. Binary Codes Extended ASCII Code ECEn 224
18. Binary CodesASCII Code (partial) Convert “help” to ASCII h e l p 1101000 1100101 1101100 1111000 0x68 0x65 0x6C 0x70 ECEn 224
19. Binary CodesGray Code • Only one bit changes with each number increment • Not a weighted code • Useful for interfacing to some physical systems ECEn 224
20. Gray Codes are Not Unique ECEn 224
21. Codes - Summary • Bits are bits… • Modern digital devices represent everything as collections of bits • A computer is one such digital device • You can encode anything with sufficient 1’s and 0’s • Text (ASCII) • Computer programs (C code, assembly code, machine code) • Sound (.wav, .mp3, …) • Pictures (.jpg, .gif, .tiff) ECEn 224<|endoftext|>
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The sight of the American bison must have been tremendous. Vast herds covered the plains, grazing the tall grasses that are now fodder for our domestic cattle. In a very short time, the bison went from herds numbering in the thousands, to near extinction and back.
In the early 1800’s, bison were the king of the plains. They migrated over large tracts of land in search of newly grown grass. Their intensive grazing would completely annihilate an area. The prairie grasses and forbs that evolved from the repeated disturbance gave rise the to resilient plant community that allows the military to intensively use the prairie of Fort Riley.
The bison soon fell to the long range rifles made famous in the late 19th century. “Skinners” all across the plains harvested the bison without limit. The hides were shipped away and the bones ground for bone meal. With fewer and fewer bison to be found, the Native Americans lost their source for food, shelter and clothing. For a period of time the only remaining bison were found on farms and zoos. In the late 1800’s a few hundred head of bison were found wild, roaming Yellowstone Park.
Although the bison would never freely roam the plains again, the species was saved from extinction through careful breeding programs of the few remaining individuals. Fort Riley was part of the later recovery of the species, housing bison from the early 1950’s until the late 1990’s. The bison herd was a popular visitor’s attraction for many years on post. At first, they were held in the corral by the Post Cemetery. When they outgrew that, most were placed in a pasture south of Williston Point Road. When the herd grew to 58, a decision was made to transfer much of the herd to Kansas State University, to be placed on Konza Prairie.
The bison herd was maintained on Fort Riley until recently, when the remainder of the herd was transferred to Konza Prairie. Today, the Konza Prairie has more than 100 head of bison, many of which are descendants of the Fort Riley herd. The bison on Konza Prairie are used to study large herbivore impacts to the tallgrass prairie.
Today, there are an estimated 1 million bison in North America. Many of these reside on livestock farms around the country. Some bison can still be found in a natural environment in National Parks such as Yellowstone in Wyoming and Windcave National Park in South Dakota. Closer to home, the Konza Prairie allows visitors to view the bison as native grazers of the Kansas Flint Hills. Although Fort Riley no longer has a bison herd, a short trip south of Manhattan to the Konza Prairie is well worth it. Once there, it is easy to conjure up a scene of what life was like 150 years ago when Fort Riley was just getting established and vast bison herds still roamed the prairie.
For more information on this or other natural resources topics, please call 239-6211 or visit our website at. www.riley.army.mil/Services/Fort/Environment/NatResources/
HQs, 1st Infantry Division & Fort Riley
Building 580, 1st Division Drive
Attn: G1 (AGCRA Bison Chapter)
Fort Riley, KS 66442
|Position||Name & Rank||Phone|
|President||Alston, Cathy G., MAJ||(785) [email protected]|
|Vice-President||Knight, Damien, CW3||(785) [email protected]|
|Chapter Sergeant Major||Sims, Tracey, SGM||(785) [email protected]|
|VP Awards||Sanchez, Jessica, SGT||(785) [email protected]|
|Treasurer (Primary)||Williams, Daralyn, SFC||(785) [email protected]|
|Treasurer (Alternate)||Gallegos, Tuvalu, SSG||(785) [email protected]|
|VP Plans & Programs (Primary)||Bryan, Nicole, CW2||(785) [email protected]|
|VP Plans & Programs (Alternate)||Lyons, Mylene, CPT||(785) [email protected]|
|VP Membership (Primary)||Morgan, Sharleen (MCE), CPT||(785) [email protected]|
|VP Membership (Alternate)||Harvey, Sheneida, SFC||(785) [email protected]|
|Secretary (Primary)||Longacre, Anna J., SGT||(785) [email protected]|
|Secretary (Alternate)||White, Emily, SSG||(785) [email protected]|
|Adjutant||Rubio, Nelly, SFC||(785) [email protected]|
|Media Manager & Historian||Jones, Zalanthia I., CPL||(785) [email protected]|
|VP Fund Raising||Simpson, Cedric, SFC||(785) [email protected]|
|VP Marketing & Strategic Comms||Mackbee, Erika, SGT||(785) [email protected]|
AG Soldiers, Army HR Civilians, and AGCRA Members – AGCRA would like to hear from you through a short survey to tell us what AGCRA can do better to meet…
Click here to see the latest newsletter provided by the AGCRA Bison Chapter. Please follow and like us:
SFC (Ret) Ken Fidler, Vice-President of Membership on the AGCRA National Executive Council (NEC) announced his retirement from the Association after almost 15 years of dedicated service. SFC (Ret) Fidler…
AG Corps Leaders, Really need you help with getting the newest version of the AG Corps Creed to our Soldiers. Posted on ACT and the link at the bottom is…<|endoftext|>
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Identify the Needs of the Student
If the child is new to you, review the child’s Individualized Educational Plan to develop a sense of his or her capabilities and areas of need. Learn how the disability impacts their education. This will give you a place to start for the first week or so. During this time you can assess the child engaging with what you have prepared and identify other ways to modify your plans.
Consider the materials needed for the child. The need for durability should be considered as well. If the child mouths objects or tears papers you will need to go durable. Ideas for durable working materials include using dry erase boards instead of paper, using heavy plastics instead of paper shapes, laminate books, laminate papers that will be used several times, use markers or pens instead of pencils.
Identify the Skills
What are the child’s capabilities? A child with significant disabilities can be limited in motor skill as well as comprehension. To modify a project you must consider: How will the child perform and how will I measure his performance?
Can you broaden his or her capabilities with assistive technology? Anytime you broaden the child’s capabilities, your job modifying lessons becomes easier, most importantly the child benefits.
Identify the Interest
Typically a child with a significant disability will have a short attention span or a hard to gain attention. Identifying and using their special interest will help to gain and hold his attention.
If they have a special interest in dolls, clothes, or trains incorporate these items into the modifications as much as possible.
A Modified Lesson Plan for Fractions
A lesson plan to teach basic fractions to a student with significant disabilities can utilize templates cut from plastic colored page dividers and heavy board such as cardboard or even wall paneling cut into pieces.
To make the templates, cut circles from different colors of plastics. One circle will be left whole. Another circle will be cut in half, another will be cut in thirds and the last circle will be cut into fourths.
Next, take your heavy board and draw a circle with hash marks to match each of the templates. Draw one circle per board so the child is not distracted or confused by another circle. The child will be encouraged to observe how the halves (fourths, thirds) join together to make up the whole circle. They can then participate by placing the templates onto the drawings.
You can decide if you want to use different colors within the circle to highlight the individual pieces. One thought, the child may confuse the color with the term depending on cognitive level. This is why one color per circle is recommended.
How to adapt this to a special interest? Does the child like clothes? Instead of cutting circles, teach and reinforce these concepts with a dress cut from the template. Cut one dress in half, one in quarters, one in thirds. Perhaps the drawing to lay the dress out will be a person instead of a circle. The same is true for most any other object the child may have an interest in.
If the special interest is a television character, choose an item from the show which can be cut and reassembled in the same manner as the circles.
Take pictures to document progress.<|endoftext|>
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The U.S. poultry industry produces a huge quantity of feather waste annually, more than one million metric tons, reported Xingen Lei and colleagues at Cornell University’s Department of Animal Science in May 2010. Scientists seek ways to avoid having the feathers end up in dumps or landfills, as this wastes the protein-rich keratin structure of the feathers and causes nitrogen to pollute the environment. One possible route to reclaiming this organic refuse involves worm composting.
Feathers contain keratin, a fibrous protein also found in human and pet hair. While slow to decompose either in regular or worm composting, feathers and hair are organic and make lists of acceptable materials for worm composting compiled by sustainable agriculture organizations.
Rajiv K. Sinha and colleagues at the School of Environmental Engineering at Griffith University in Queensland, Australia, note that earthworms feed on organic wastes rich in nitrogen. Worms are used to compost slaughterhouse waste, including feathers, bones and blood, as well as organic waste from kitchens, gardens, farms, sugar mills, municipal garbage collection.
Chicken feathers degrade during regular composting, but their high keratin content slows the process. They contain about 14 percent nitrogen, much of which may be retained by finished compost, according to Steve Kroening, School of Biological Sciences, University of Canterbury.
Hobbyist gardeners report successful worm composting of feather waste in small quantities using litter-dwelling compost worms. The Bhawalkar Earthworm Research Institute (BERI) in India is developing ways for large-scale processing of organic wastes using larger burrowing earthworms such as nightcrawlers. BERI, run by chemical engineer Uday Bhawalkar, notes the worms can break down feathers and bones.
Feathers associated with slaughterhouses may be contaminated with salmonella and E. coli bacteria. Even though worm composting avoids the high heat associated with regular composting as bacterial activity raises the temperature in the middle of the pile and kills off pathogens, the worms themselves appear to be able to reduce pathogen levels as they digest and process organic material. Indian zoologist Arvind Kumar found that worms could remove E. coli and salmonella from spoiled food and reduced pathogens to a greater extent in fact than regular composting. Reductions of 99.9 percent or greater are possible, he writes in “Verms and Vermitechnology.”
Studies of vermicomposting and pathogens also test the worm’s ability to remove pathogens from biosolids present in sewage, such as a study by Bruce R. Eastman of the Orange County Environmental Protection Division in Florida reported in BioCycle magazine. Eastman’s and Kumar’s research suggest that vermicomposting with feather waste may also result in low tested results of pathogens in the finished vermicompost. In fact, inoculating feather waste with useful feather degrading bacteria such as Bacillus licheniformis may speed the breakdown of feather waste generated by processing plants, notes botany professor Jann Ichida and colleages at Ohio Wesleyan University, a discovery with implications for faster vermicomposting.
- Cornell University: Whole Genome Search for Novel Microbial Enzymes to Hydrolyze Poultry Feathers
- "The Environmentalist"; Vermiculture and Waste Management; Rajiv K. Sinha et al.; 2002
- University of Plymouth Research: Composting Food Wastes: Scientific Aspects
- In Context: Worm Revolution
- Waste to Health: Three Decades With Waste-to-Health Bioconversion
- GardenBanter: Adding Feathers to the Compost Heap?
- Washington State University Cooperative Extension: Composting of Poultry Offal Demonstration Project
- PubMed: Bacterial Inoculum Enhances Keratin Degradation and Biofilm Formation in Poultry Compost
- Achieving BioCycle: Pathogen Stabilization Using Vermicomposting
- "Verms & Vermitechnology"; Arvind Kumar; 2005
- Temperatures for Vermicomposting
- Diseases Found in Cow Manure
- Vermiculture Methods
- What Are the Benefits of Earthworms?
- Types of Larvae Found in Compost Bins
- Lawn Treatment for Worms
- Compost Sawdust
- Compost with Paper
- The Difference Between Compost and Manure
- Types of Steam Turbines
- Lactobacillus for Plant Growth
- What Is Anoxygenic Photosynthesis?<|endoftext|>
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Learn about the Types Of Human Rights. The development of human rights has been a continuous quest. The contributions of great thinkers, philosophers, theologians, social scientists and reformers as well as national, regional and world bodies have come to shape what is today known as Human Rights and Liberties.
Human Rights could be understood as those rights obtained in the United Nations Conventions, Bill of Rights, International laws, as well as continental/regional human rights treaties.
There are several types of human rights which are as follows:
The fundamental rights to life (sanctity of life and physical existence), social rights, economic rights, civil/political rights, moral rights, group rights, rights to development, rights of women and children, and so on.
Right to life, is in fact, the most fundamental of all types of human rights. This is because it qualifies to stand as the foundation, or the super-structure on which all other rights are built. Certainly, every man should have a right to his physical existence and all that support human life (both on himself and that of others). Thus, man’s right to physical existence and integrity; liberty, and freedom from torture, cruel, or inhuman treatment, slavery, servitude, and forced labor, are inalienable to him. And these rights to live and live well extend even to children and the unborn child.
It is pre-eminent and primary of all men without discriminations to own and use material goods and services of the world for a decent livelihood. This is the basis for economic rights. Moreover, all must give to labor the place assigned to it as the only legitimate means of achieving material and economic power and privilege. Thus, economic rights are directed towards ensuring that all citizens without discrimination have opportunity for securing adequate means of livelihood, suitable employment, the duty to work according to one’s ability, as well as the right to receive remuneration according to work done.
Man cannot really and truly be himself without authentic self existence. As culture is a way of life of a people, man cannot do without some ‘roots’ in his culture and values. Therefore, cultural and moral rights refer to having the rights to take part I one’s cultural norms, beliefs and values, which should be seriously respected by other human beings irrespective of their cultural differences. Depriving or denying one of his culture, is to uproot and alienate him, thereby making him less a cultural man and less a human. This is one of the most critical of all types of human rights that have caused a lot of problems all over the world.
Rights to development
The General Assembly of the United Nations in December 1986, proclaimed the right to development. The spirit of this important right is that nations as well as individuals must consciously map out programs to galvanize common efforts aimed at socio-cultural, political and economic expansion to gain not only scientific and technological progress, greater productivity, efficient and higher standard of living, but to organize and develop the political community to be stable and friendly, where every individual realizes his full human potential and status. For this reason, the individuals must be exposed to appropriate education geared towards the development of his physical, intellectual, moral and spiritual potentials.
Rights to Women and Children
On the 1th December 1979, the United Nations General Assembly adopted Resolution 34/180 approving the convention on the Elimination of ‘All Forms of Discrimination Against Women’. This convention set out ways and means for individual governments to eliminate discriminations against women, and to guarantee an equitable distribution of rights and obligations between men and women. The issue of women and children, especially girl child was formally brought to the fore. These types of human rights came as a result of the observation that shows that women and their children have shared a heavy burden of human deprivation, discrimination and degradation especially during wars and adverse economic and political unrest.<|endoftext|>
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Preposition is a word or several words that express place, time, reason and other logical relationships between different parts of the sentence.
There are over 100 prepositions in English. The most common single-word prepositions are:
Although most prepositions are single words, some pairs and groups of words operate like single prepositions:
- They were unable to attend because of the bad weather in Ireland.
- Jack’ll be playing in the team in place of me.
- I’ll meet you in the cafe opposite the cinema.
- It was difficult to sleep during the flight.
- Give that to me.
- They were talking about their trip.
Types of prepositions
Prepositions show the relationship between the noun or pronoun and other words in a sentence. For example, they describe the position of something, the time when something happens, the way in which something is done, etc.
Prepositions of movement
Prepositions of movement are used to show the direction somebody or something is moving to, towards, from, out of, etc.
- We walked across the park.
- The cat jumped out of the box.
- I took the picture off the wall.
- The price of food has gone up in the past two years.
- The boy is running away from me.
Prepositions of place
Prepositions of place are used to say where someone or something is.
- The ball is on the box.
- The ball is between the box and the bear.
- The ball is behind the box.
- The ball is in front of the box.
- The ball is under the box.
- The ball is next to the box.
- The ball is in the box.
- The ball is near the box.
Prepositions of time
Prepositions of time tell us when something happens, and for how long. They are usually used with clock times, mealtimes, parts of the day, months, years, and other durations.
– at night
– at 9.00 / 10.30 / 7.45
– at the weekend
- I start work at 9.00 every day.
- He doesn’t usually go out at night.
- She sometimes works at the weekend.
– on Monday
– on Friday afternoon, on Saturday night
– on the 20th of January,
- I’m meeting Tom on Monday.
- I don’t usually work on Friday afternoon.
- My birthday is on November 27th.
– in the morning / the afternoon / the evening
– in July / September / January
– in winter / spring / summer / autumn
- He usually watches TV in the evening.
- They sometimes go on holiday in July.
- We bought this house in 2012.
- It’s always cold here in winter.
– We always exchange presents at Christmas.
– We always exchange presents on Christmas Day.
– He likes going out at New Year.
– He likes going out on New Year’s Eve.
Prepositions and abstract meanings
Common prepositions that show relationships of space often have abstract as well as concrete meanings.
- There were beautiful mountains beyond the hotel.
- Learning Japanese in a year was beyond them all. (beyond = too difficult for)
Some common prepositions such as ‘at’, ‘in’ and ‘on’ can have abstract meanings:
- I think you will both need to discuss the problem in private.
- All three singers were dressed in black.
- Our dog stays on guard all night, even when he’s sleeping!
Adjectives with prepositions
These are some useful combinations of adjectives and prepositions we should remeber:
- I’m interested in cooking.
- He’s very good at playing tennis.
- Carrots are good for you.
- My sister is afraid of spiders.
- Paris is famous for the Eiffel Tower.
- We’re worried about the English test.
Prepositions at the end of a sentence
There’s a popular myth in English that you may not end a sentence with a preposition. However, we sometimes do separate a preposition from the words which follow it. It is common in informal styles.
- She was someone to whom he could talk. (formal)
- She was someone who he could talk to. (informal)
- Which room are they having breakfast in? (informal)
- In which room are they having breakfast? (formal)
Consider the following examples:
- Where did you come from? (NOT:
From where did you come?)
- That is something I cannot agree with.
- How many of you can I depend on?
Without these prepositions the meaning would not be clear.
But if the meaning is clear without the preposition, just do not use it.
Where is your brother at?
- Correct: Where is your brother?
In this video from 7ESL, you’ll find many useful phrases with prepositions to improve your vocabulary:
Read more about prepositions:<|endoftext|>
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# A-level Mathematics/MEI/C1/Co-ordinate Geometry
< A-level Mathematics | MEI | C1
Co-ordinates are a way of describing position. In two dimensions, positions are given in two perpendicular directions, x and y.
## Straight linesEdit
A straight line has a fixed gradient. The gradient of a line and its y intercept are the two main pieces of information that distinguish one line from another.
### Equations of a straight lineEdit
The most common form of a straight line is ${\displaystyle y=mx+c}$ . The m is the gradient of the line, and the c is where the line intercepts the y-axis. When c is 0, the line passes through the origin.
Other forms of the equation are ${\displaystyle x=a}$ , used for vertical lines of infinte gradient, ${\displaystyle y=b}$ , used for horizontal lines with 0 gradient, and ${\displaystyle px+qy+r=0}$ , which is often used for some lines as a neater way of writing the equation.
#### Finding the equation of a straight lineEdit
You may need to find the equation of a straight line, and only given the co-ordinates of one point on the line and the gradient of the line. The single point can be taken as ${\displaystyle ({x_{1}},{y_{1}})}$ , and the co-ordinates and the gradient can be substituted in the formula ${\displaystyle y-{y_{1}}=m(x-{x_{1}})}$ . Then it is simply a case of rearranging the formula into the form ${\displaystyle y=mx+c}$ .
You may only be given two points, ${\displaystyle ({x_{1}},{y_{1}})}$ and ${\displaystyle ({x_{2}},{y_{2}})}$ . In this case, use the formula ${\displaystyle m={\frac {{y_{2}}-{y_{1}}}{{x_{2}}-{x_{1}}}}}$ to find the gradient and then use the method above.
The steepness of a line can be measured by its gradient, which is the increase in the y direction divided by the increase in the x direction. The letter m is used to denote the gradient.
${\displaystyle m={\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}}$
#### Parallel and perpendicular linesEdit
With the gradients of two lines, you can tell if they are parallel, perpendicular, or neither. A pair of lines are parallel if their gradients are equal, ${\displaystyle m_{1}=m_{2}}$ . A pair of lines are perpendicular if the product of their gradients is -1, ${\displaystyle m_{1}\times m_{2}=-1}$
### Distance between two pointsEdit
Using the co-ordinates of two points, it is possible to calculate the distance between them using Pythagoras' theorem.
The distance between any two points A${\displaystyle ({x_{1}},{y_{1}})}$ and B${\displaystyle ({x_{2}},{y_{2}})}$ is given by ${\displaystyle {\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}}}$
### Mid-point of a lineEdit
When the co-ordinate of two points are known, the mid-point is the point halfway between those points. For any two points A${\displaystyle ({x_{1}},{y_{1}})}$ and B${\displaystyle ({x_{2}},{y_{2}})}$ , the co-ordinates of the mid-point of AB can be found by ${\displaystyle \left({\frac {{x_{1}}+{x_{2}}}{2}},{\frac {{y_{1}}+{y_{2}}}{2}}\right)}$ .
### Intersection of linesEdit
Any two lines will meet at a point, as long as they are not parallel. You can find the point of intersection simply by solving the two equations simultaneously. The lines will intersect at one distinct point (if a solution to their equation exists) or will not intersect at all (if they are parallel). A curve can however intersect a line or another curve at multiple points.
## CurvesEdit
To sketch a graph of a curve, all you need to know is the general shape of the curve and other important pieces of information such as the x and y intercepts and the points of any maxima and minima.
### Curves in the form ${\displaystyle y=x^{n}}$ Edit
Here are the graphs for the curves ${\displaystyle y=x^{1}}$ , ${\displaystyle y=x^{2}}$ , ${\displaystyle y=x^{3}}$ and ${\displaystyle y=x^{4}}$ :
(Need to draw those later, just simple b&w curve sketches for each curve)
Notice how the odd powers of ${\displaystyle x}$ all share the same general shape, moving from bottom-left to top-right, and how all the even powers of ${\displaystyle x}$ share the same "bucket" shaped curve.
### Curves in the form ${\displaystyle y={\frac {1}{x^{n}}}}$ Edit
Just like earlier, curves with an even powers of ${\displaystyle x}$ all have the same general shape, and those with odd powers of ${\displaystyle x}$ share another general shape.
(Images here)
All curves in this form do not have a value for ${\displaystyle x=0}$ , because ${\displaystyle {\frac {1}{0}}}$ is undefined. There are asymptotes on both the ${\displaystyle x}$ and ${\displaystyle y}$ axis, where the curve moves towards increasingly slowly but will never actually touch.
### Intersection of lines and curvesEdit
When a line intersects with a curve, it is possible to find the points of intersection by substituting the equation of the line into the equation of the curve. If the line is in the form ${\displaystyle y=mx+c}$ , then you can replace any instances of ${\displaystyle y}$ with ${\displaystyle mx+c}$ , and then expand the equation out and then factorise the resulting quadratic.
### Intersection of curvesEdit
The same method can be used as for a line and a curve. However, it will only work in simple cases. When an algebraic method fails, you will need to resort to a graphical or Numerical Method. In the exam, you will only be required to use algebraic methods.
## The circleEdit
The circle is defined as the path of all the points at a fixed distance from a single point. The single point is the centre of the circle and the fixed distance is it's radius. This definition is the basis of the equation of the circle.
### Equation of the circleEdit
The equation of the circle is ${\displaystyle {x^{2}}+{y^{2}}=r^{2}}$ for a circle center (0,0) and radius r, and ${\displaystyle {(x-a)^{2}}+{(y-b)^{2}}=r^{2}}$ for a circle centre (a,b) and radius r.
So, for example, a circle with the equation ${\displaystyle {(x+2)^{2}}+{(y-3)^{2}}=25}$ would have centre (-2,3) and radius 5.
### Circle geometryEdit
When presented with a problem, it may appear at first that there is not enough information given to you. However, there are some facts that will help you spot right angles in relation to a circle.
• The angle in a semi-circle is a right angle
• The perpendicular from the centre of a circle to a chord bisects the chord
• The tangent to a circle at a point is perpendicular to the radius through that point<|endoftext|>
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Fractions and Decimals
Ratio and Proportion
Indices & Standard Form
Algebra and Graphs
Trig Word Problems
Sine and Cosine Rule
Area of any Triangle
Angles of Elevation/Depression
Vectors, Matrices and Transformations
Collecting and displaying data
What are the Sine and Cosine rule
So far we have been working with right-angled triangles. We discovered that Pythagoras' Theorem and Trigonometry allow us to calculate lengths in right-angled triangles. Trigonometry we also use when we want to calculate angles in right-angled triangles. But what do we do when we are not given a right-angled triangle? How do we calculate lengths and angles in non-right-angled triangles? In those cases we use the Sine and Cosine rule (which are actually still based upon the trigonometric ratios in right-angled triangles). Have a proper look at all of the following videos and make sure you know exactly how to approach this important IGCSE maths topic. Good luck and have fun!
Solving more Example Questions about the Sine Rule
Once you know what the Sine Rule is and how we can use it to calculate lengths and angles in all triangles, it is time to study the following maths example questions during your maths revision. The first video will prove the sine rule to you and show you it is not a magic formula! The next three videos will provide you with example questions in which the sine rule is used to calculate lengths and angles. Make sure to study all videos for every time something new will be added (something to take care of, a tip etc). So pass your IGCSE GCSE Maths exam and look at the following maths activities!
The previous three videos explain step by step to you how to use the Cosine Rule to calculate angles and lengths in triangles. Hopefully you have studies all three maths activities during your maths revision. Do not think you know it all now and study the next two example questions about the Cosine Rule. Once you have studied those too you will understand everything about the Cosine Rule and pass your next IGCSE GCSE Maths exam.
The worksheet below I created for you to practice your understanding of the sine and cosine rule. If you want to be successful and pass your next maths exam, analyse for each question the situation properly and decide which strategy (sine or cosine rule) to use. I hope you can read the numbers properly because for some reason the quality decreased after uploading the document. Scroll all the way down for the answers (not too quickly of course because you first want to try it yourself!). Good luck and have fun!!
Past Exam Question involving Cosine Rule and more
Check the next video during your maths revision. I will solve a past paper question which includes a question about the cosine rule. I will also explain how to use Trigonometry and Pythagoras' Theorem to calculate lengths and how to calculate the volume of prisms.
Past Paper Question about Bearings, Sine and Cosine rule
One of you asked me to help with the following maths question. I will solve this past paper question for you and explain how to use the sine rule, how to use the cosine rule and how to calculate bearings. A great activity to practise during your maths revision!<|endoftext|>
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## Factors Of 40 | With Easy Division and Prime Factorization
In maths, factors of 40 are the numbers that divide the original number 40 and produce the whole number in quotient form. You can find the original number by multiplying any two factors in pairs. Likewise, multiples of 40 are the lengthened versions, such as 40, 80, 120, 160, 200, 240, and so on. Factors…
## Factors Of 32 | With Easy Division and Prime Factorization
Factors Of 32 A factor of 32 is an integer that can divide 32 completely. When x is a factor of 32, then 32 is evenly divisible by x. In the case of 32, it is a composite number, so it will have more than two factors. Factors of 32: 1, 2, 4, 8, 16…
## Factors Of 56 | With Easy Division and Prime Factorization
Factors Of 56 The factors of 56 are the numbers that can be multiplied together to produce the number 56. There are no fractions or decimals associated with the number 56, so its factor pairs are those whole numbers that are either positive or negative. Consider, for example, the factor pair 56, written as (1,…
## Factors Of 44 | With Easy Division and Prime Factorization
Factors Of 44 Mathematics defines factors of 44 as those numbers that, when multiplied in pairs, give the original number 44. These factors can be both positive and negative. Taking 44 as an example, the pair factors can either be (1, 44) or (-1, -44). The result of multiplying the pair of negative numbers, such…
## Factors Of 55 | With Easy Division and Prime Factorization
Factors Of 55 In Maths, Numbers that divide an original number equally and uniformly are known as factors of 55. It has factors of more than two since 55 is a composite number. Using the division method, we can determine these factors. It is necessary to check whether there is any remainder left after division….
## Factors Of 42 | With Easy Division and Prime Factorization
Factors Of 42 A factor of 42 is a pair of numbers that, when multiplied together, give the original number. Similarly, composite numbers such as 48, 60, 70, 420, 36, 45, 30, etc. have prime factorization that can be determined. It is rather simple and easy to discover the factors of a number of 42….
## Factors Of 33 | With Easy Division and Prime Factorization
Factors Of 33 Factors of 33 are the numbers that multiply to produce 33. If we multiply 3 and 6, we get 18, i.e. 3 × 6 = 18, so 3 and 6 are factors of 18. We can find the factors of 33 in the same way. In this article, you will learn how…
## Factors Of 36 | Easy Division | Prime Factorization
Factors Of 36 In mathematics, factors of 36 are the numbers that divide 36 precisely without leaving a remainder. However, 36 can have positive or negative factors, but it cannot be decimal or fraction. 36 can be expressed as (1, 36) or (-1, -36). The original number is the result of multiplying two negative numbers,…
## Factors of 51 | Division and Prime Factorization | Easy Guide
Factors Of 51 The factors of 51 are the numbers that multiply to produce the number 51. Multiply 3 by 4, for example, and we get 12, i.e., 3 × 4 = 12. Here, 3 and 4 are the factors of 12. Also, you can find the factors of 51 in a similar fashion. This…
## Factors Of 34 | With Easy Division and Prime Factorization
Factors Of 34 An integer factor of 34 is a number that divides the original number evenly, resulting in a whole number as a quotient. The number 34 has more than two factors since it is a composite number. The factors and multiples of 34 are different from each other. In comparison, factors of a…
End of content
End of content<|endoftext|>
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How do you solve the system using the elimination method for 3(x-y)=15 and 2x+7=7?
Jul 22, 2015
$\left(x , y\right) = \left(0 , - 5\right)$
$\textcolor{w h i t e}{\text{XXXX}}$by the elimination method (although the "elimination" has already been done)
Explanation:
Given
[1]$\textcolor{w h i t e}{\text{XXXX}}$$3 \left(x - y\right) = 15$
[2]$\textcolor{w h i t e}{\text{XXXX}}$$2 x + 7 = 7$
Since $y$ has already been eliminated in [2]
if we simplify [2] as
[3]$\textcolor{w h i t e}{\text{XXXX}}$$2 x = 0$
$\Rightarrow$
[4]$\textcolor{w h i t e}{\text{XXXX}}$$x = 0$
We can then substitute $x = 0$ back into [1] to get
[5]$\textcolor{w h i t e}{\text{XXXX}}$$3 \left(0 - y\right) = 15$
or, after simplification:
[6]$\textcolor{w h i t e}{\text{XXXX}}$$y = - 5$<|endoftext|>
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Cystic Fibrosis (CF) is a genetic disorder that primarily affects the lungs, but can also affect the pancreas, liver, kidneys, and intestines. CF is one of the most common diseases passed on from parent to child. Because CF is recessive, someone diagnosed with CF would have to receive the gene from each parent. People born with only a single CF gene are called carriers, and do not have the disease. The gene for CF also contains mutations, more than 17,000 in fact. Genetic testing can help to screen for common CF mutations, but not all of them.
There are over 30,000 patients with CF in the United States, and more than 70,000 worldwide. Approximately 1,000 new diagnoses of CF occur each year, and over 75% of patients are diagnosed by the age of 2. Many different factors, such as age of diagnosis, can affect an individual’s health and the course of the disease. At the center of any treatment is combatting the sticky mucous produced by CF patients. There are a number of different treatments available. Advances in CF treatments have made it possible for patients to live healthy, vibrant lives.
What are possible symptoms of Cystic Fibrosis?
Patients may have some or all of the following symptoms to various degrees:
- Poor growth
- Fatty stool
- Clubbing of the fingers and toes
- Infertility (male patients)
- Very salty-tasting skin
- Persistent coughing, at times with phlegm
- Frequent lung infections including sinus infections, pneumonia, or bronchitis
- Wheezing or shortness of breath<|endoftext|>
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When Alexander Graham Bell patented his improved version of Johann Philipp Reis’ early telephone design, few could have foreseen the impact it would have on how we communicate with each other. The telephones we use today are a far cry from the pioneering efforts of Bell and Reiss, and to celebrate the evolution of one of mankind’s greatest inventions we’ve put together a brief guide to the fascinating history of telephony.
1664 – Tin Cans and String
Robert Hooke undertakes first serious experiments using cans and string. The “Tin Can Telephone” or “String Telephone” is the first, and most basic telephone that has ever existed. It was essentially an acoustic, non-electrical speech-transmitting device made of two twin cans. Sound was broadcasted over an extended wire by mechanical vibrations.
1861 – Johann Philipp Reis Invents First Working Telephone
Johann Philipp Reis invents the first working example of a telephone. Named “the Reis Telephone”, was a device that captured sound and converted it into electrical impulses, which were then transmitted via electrical wires to another device that transformed these pulses into recognizable sounds – similar to the originals acoustic source.
1876 – Alexander Graham Bell Improves on Reis’ Design
Alexander Graham Bell improves the Johann Philipp Reis design and patents it. Building on the prior developments made by others, Alexander Graham Bell invented the first device that produced and transmitted clearly distinguishable replication of the human voice. Bell patented his invention, which became “The Telephone”. This was the first ever device that allowed people to have direct verbal contact across large distance. The first words ever to be spoken through a telephone cable were “Mr. Watson, come here, I want to see you.” This was without doubt the biggest breakthrough in the history of the telephone.
1905 – First Dial Telephone Becomes Widely Available
The first dial telephone becomes widely available. The “Rotary dial” enabled people to make direct calls using a combination of digits laid out on a rotary finger wheel that was attached to the face of the telephone. Electrical impulses generated from the spring action dial were encoded into a specific digit – clever stuff.
1937 – First Ringing Telephone Released
The first telephone to have a ringer installed. We start to see things get a little more compact – The Model 302 telephone included the ringer and network circuitry in the same telephone housing. Previously it was in a separate box.
1956 – World’s First Mobile Telephone Invented
The world’s first mobile telephone is invented. The SRA/Ericsson MTA weighed around 88 pounds and was roughly the size of a suitcase. The sheer weight and size of the device, ironically, meant that is was virtually immobile. So it was an immobile, mobile telephone…
1963 – World’s First Push-Button Telephone is Released
The world’s first push button telephone is released. The push-button telephone meant that instead of dialling keys, buttons were used to make a direct call to another telephone device. These devices had dual-tone-multi-frequency signalling, which uses voice-frequency bands over telephone lines – this became know as ‘Touch-Tone” and consisted of a telephone keypad, upgrading on the rotary dial.
1973 – Martin Cooper Invents First Working Cell Phone
Martin Cooper invents the first fully working cell phone. Motorola produced the first handheld mobile phone – prior to this, mobile telephones were only installed in cars and other vehicles. The original prototype model weighed a total of 1.1kgs – that’s about the weight of 2 packs of ground beef. It was 23 cm long, 13cm deep and 4.45cm. After 30 minutes of use, the device had to be re-charged for around 10 hours. Six years later, the first commercial cell phone becomes available to the public. It cost $3,995.
1993 – World’s First Smartphone is Invented
The world’s first smartphone is invented. A smartphone is essentially a mobile phone with an advanced mobile operating system that combines features of personal computer operating system with other features useful for mobile or handheld use. This prototype consisted of a mobile phone, pager, fax machine and PDA.
1993 – Nokia Releases First Text Messaging Phone
Nokia debuts the first mobile phone with text messaging capability. Nokia was the first handset manufacturer whose total GSM phone line in 1993 supported user sending of SMS text messages. Four years later, in 1997, Nokia expanded on this by introducing the first mobile phone device with a full keyboard.
2000 – First Camera Phone is Released
The first camera phone is released. The Samsung SCH-V200 was capable of taking 20 photos at 350,000-pixel resolutions, which is the equivalent of 0.35 megapixels.
2007 – Apple Releases First Touchscreen Smartphone
Apple released the first touchscreen smartphone. The “iPhone” (also referred to as the iPhone 2G), was the breakthrough of touchscreen smartphone technology, which now dominates the global mobile telephone market. Steve Jobs perfected the idea of using a multi-touchscreen to interact with a computer without the need for a physical keyboard – the same as a tablet computer. The first 8GB model cost $599, but Apple later dropped the price to $399. Apple now take a 18.3% market share of the smartphone market; making them the leading smartphone vendor.<|endoftext|>
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Researchers have long sought an efficient way to untangle DNA in order to study its structure -- neatly unraveled and straightened out -- under a microscope. Now, chemists and engineers at KU Leuven, in Belgium, have devised a strikingly simple and effective solution: they inject genetic material into a droplet of water and use a pipet tip to drag it over a glass plate covered with a sticky polymer. The droplet rolls like a ball over the plate, sticking the DNA to the plate surface. The unraveled DNA can then be studied under a microscope. The researchers described the technique in the journal ACS Nano.
There are two ways to decode DNA: DNA sequencing and DNA mapping. In DNA sequencing, short strings of DNA are studied to determine the exact order of nucleotides -- the bases A, C, G and T -- within a DNA molecule. The method allows for highly-detailed genetic analysis, but is time- and resource-intensive.
For applications that call for less detailed analysis, such as determining if a given fragment of DNA belongs to a virus or a bacteria, scientists opt for DNA mapping. This method uses the longest possible DNA fragments to map the DNA's 'big picture' structure.
In this study, researchers describe an improved version of a DNA mapping technique they previously developed called fluorocoding, explains chemist Jochem Deen: "In fluorocoding, the DNA is marked with a coloured dye to make it visible under a fluorescence microscope. It is then inserted into a droplet of water together with a small amount of acid and placed on a glass plate. The DNA-infused water droplet evaporates, leaving behind the outstretched DNA pattern."
"But this deposition technique is complicated and does not always produce the long, straightened pieces of DNA that are ideal for DNA mapping," he continues. It took a multidisciplinary team of chemists and engineers specialised in how liquids behave to figure out how to optimise the technique.
"Our improved technique combines two factors: the natural internal flow dynamics of a water droplet and a polymer called Zeonex that binds particularly well to DNA," explains engineer Wouters Sempels.
The 'rolling droplet' technique is simple, low-cost and effective: "We used a glass platelet covered with a layer of the polymer Zeonex. Instead of letting the DNA-injected water droplet dry on the plate, we used a pipet tip to drag it across the plate. The droplet rolls like a ball over the plate, sticking the DNA to the plate's surface. The strings of DNA 'captured' on the plate in this way are longer and straighter," explains Wouters Sempels.
To test the technique's effectiveness, the researchers applied it to the DNA of a virus whose exact length was already known. The length of the DNA captured using the rolling droplet technique matched the known length of the virus' DNA.
The rolling droplet technique could be easily applied in a clinical setting to quickly identify DNA features, say the researchers. "Our technique requires very little start-up materials and can be carried out quickly. It could be very effective in determining whether a patient is infected with a specific type of virus, for example. In this study, we focused on viral DNA, but the technique can just as easily be used with human or bacterial DNA," says Wouters Sempels.
The technique could eventually also be helpful in cancer research and diagnosis. "After further refining this technique, we could be able to quickly tell the difference between healthy cells and cancer cells," says Wouters Sempels.
Tell us what you think of Chemistry 2011 -- we welcome both positive and negative comments. Have any problems using the site? Questions?<|endoftext|>
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# Applied Combinatorics
## Section8.1Basic Notation and Terminology
With a sequence $$\sigma=\{a_n:n\ge0\}$$ of real numbers, we associate a “function” $$F(x)$$ defined by
\begin{equation*} F(x)=\sum_{n=0}^\infty a_n x^n. \end{equation*}
The word “function” is put in quotes as we do not necessarily care about substituting a value of $$x$$ and obtaining a specific value for $$F(x)\text{.}$$ In other words, we consider $$F(x)$$ as a formal power series and frequently ignore issues of convergence.
It is customary to refer to $$F(x)$$ as the generating function of the sequence $$\sigma\text{.}$$ As we have already remarked, we are not necessarily interested in calculating $$F(x)$$ for specific values of $$x\text{.}$$ However, by convention, we take $$F(0)=a_0\text{.}$$
### Example8.1.
Consider the constant sequence $$\sigma=\{a_n:n\ge0\}$$ with $$a_n=1$$ for every $$n\ge0\text{.}$$ Then the generating function $$F(x)$$ of $$\sigma$$ is given by
\begin{equation*} F(x)=1+x+x^2+x^3+x^4+x^5+x^6+\cdots\text{,} \end{equation*}
which is called the infinite geometric series.
You may remember that this last expression is the Maclaurin series for the function $$F(x)=1/(1-x)$$ and that the series converges when $$|x|\lt 1\text{.}$$ Since we want to think in terms of formal power series, let’s see that we can justify the expression
\begin{equation*} \frac{1}{1-x}=1+x+x^2+x^3+x^4+x^5+x^6+\cdots=\sum_{n=0}^\infty x^n \end{equation*}
without any calculus techniques. Consider the product
\begin{equation*} (1-x)(1+x+x^2+x^3+x^4+x^5+x^6+\cdots) \end{equation*}
and notice that, since we multiply formal power series just like we multiply polynomials (power series are pretty much polynomials that go on forever), we have that this product is
\begin{equation*} (1+x+x^2+x^3+x^4+x^5+x^6+\cdots)-x(1+x+x^2+x^3+x^4+x^5+x^6+\cdots) = 1. \end{equation*}
Now we have that
\begin{equation*} (1-x)(1+x+x^2+x^3+x^4+x^5+x^6+\cdots) = 1, \end{equation*}
or, more usefully, after dividing through by $$1-x\text{,}$$
\begin{equation*} \frac{1}{1-x} = \sum_{n=0}^\infty x^n. \end{equation*}
The method of Example 8.1 can be adapted to address the finite geometric series $$\sum_{j=0}^n x^j\text{.}$$ In that case, we look at
\begin{align*} (1-x) \sum_{j=0}^nx^j \amp= \sum_{j=0}^n x^j - \sum_{j=0}^n x^{j+1}\\ \amp= (1+x+\cdots + x^n) - (x+x^2+\cdots x^n + x^{n+1})\text{.} \end{align*}
Looking carefully, we see that everything cancels in the final expression except $$1-x^{n+1}\text{.}$$ Dividing both sides by $$1-x$$ gives us
$$1+x+\cdots + x^n = \frac{1-x^{n+1}}{1-x}\tag{8.1.1}$$
as the formula for the sum of a finite geometric series.
### Example8.2.
Just like you learned in calculus for Maclaurin series, formal power series can be differentiated and integrated term by term. The rigorous mathematical framework that underlies such operations is not our focus here, so take us at our word that this can be done for formal power series without concern about issues of convergence.
To see this in action, consider differentiating the power series of the previous example. This gives
\begin{equation*} \frac{1}{(1-x)^2}=1+2x+3x^2+4x^3+5x^4+6x^5+7x^6+\dots=\sum_{n=1}^\infty nx^{n-1}. \end{equation*}
Integration of the series represented by $$1/(1+x) = 1/(1-(-x))$$ yields (after a bit of algebraic manipulation)
\begin{equation*} \log(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\frac{x^5}{5}- \frac{x^6}{6}+\dots=\sum_{n=1}^\infty (-1)^{n+1}\frac{x^n}{n}. \end{equation*}
Before you become convinced that we’re only going to concern ourselves with generating functions that actually converge, let’s see that we can talk about the formal power series
\begin{equation*} F(x)=\sum_{n=0}^{\infty} n! x^n, \end{equation*}
even though it has radius of convergence $$0\text{,}$$ i.e., the series $$F(x)$$ converges only for $$x=0\text{,}$$ so that $$F(0)=1\text{.}$$ Nevertheless, it makes sense to speak of the formal power series $$F(x)$$ as the generating function for the sequence $$\{a_n:n\ge0\}\text{,}$$ $$a_0=1$$ and $$a_n$$ is the number of permutations of $$\{1,2,\dots,n\}$$ when $$n\ge1\text{.}$$
For reference, we state the following elementary result, which emphasizes the form of a product of two power series.<|endoftext|>
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# What is exponential form 6th grade math?
Jan 6, 2021
## What is exponential form 6th grade math?
exponential form. • a way of representing repeated multiplications of the same number. by writing the number as a base with the number of repeats. written as a small number to its upper right.
## What is an example of exponential form?
Exponential notation is an alternative method of expressing numbers. Exponential numbers take the form an, where a is multiplied by itself n times. A simple example is 8=23=2×2×2. For example, 5 ×103 is the scientific notation for the number 5000, while 3.25×102is the scientific notation for the number 325.
What is the power of 6 called?
In arithmetic and algebra the sixth power of a number n is the result of multiplying six instances of n together. So: n6 = n × n × n × n × n × n. Sixth powers can be formed by multiplying a number by its fifth power, multiplying the square of a number by its fourth power, by cubing a square, or by squaring a cube.
What is the exponential form of 1000?
We can write 1000 as 10x10x10, but instead of writing 10 three times we can write the number 1000 in an alternative way too.
### What is 9 superscript 7 written in expanded form?
Step-by-step explanation: Here, asked to write 0.917 in expanded form. We can see that a given number has 9 tenths, 1 hundredths and 7 thousandths. Therefore, the expanded form of the given number would be 0.9+0.01+0.0070.
### What is the difference between standard form and exponential form?
If a quantity is written as the product of a power of 10 and a number that is greater than or equal to 1 and less than 10, then the quantity is said to be expressed in standard form (or scientific notation). It is also known as exponential form. Note that we have expressed 65 as a product of 6.5 and a power of 10.
What is the exponential form of 10000?
10³
What is the exponential form of 72?
Answer: We see that after the factor tree for 72 is complete, the prime numbers at the end of the branches are 2, 3, 2, 3, and 2. Thus, we have three 2’s and two 3’s in the prime factorization of 72, so we write this in exponential form as 23 × 32.
## When do you need an exponential form worksheet?
This series of printable worksheets is drafted to assist students of grade 5, grade 6, and grade 7 in writing numbers in exponential form and converting exponential form back into standard form. Teach them the basic exponent rules to solve these worksheets that focus the place value multipliers as powers of 10.
## Which is the best exponents worksheet for grade 6?
Grade 6 Exponents Worksheets 1 Exponents. 2 Negative or zero exponents. 3 Equations with exponents. Explore all of our exponents worksheets, from reading and writing simple exponents to negative… More
What’s the maximum power of an expanded exponential?
The maximum power of the exponent is 5 (hundred thousands). Transform numbers in standard form to expanded exponential form and vice versa with this assorted collection of pdf worksheets. Use the answer keys to validate your answers.
How to transform expanded exponential form to standard form?
Get them to transform expanded exponential form to standard notation and vice versa. Numbers are provided in standard form. Transform them into the expanded exponential form. The exponents will be expressed in powers of ten ranging up to 10 11. Rewrite the standard number notations in the expanded exponential form in place values up to billions.<|endoftext|>
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The arrival of the Empire Windrush 60 years ago represents a defining moment in British history.
As the ship docked at Tilbury on the River Thames in June 1948 its West Indian passengers were greeted by a maelstrom of racism whipped up by the press and right wing politicians.
But for many of them this was not their first time in England.
The majority of passengers on the Windrush were servicemen and women returning to duty from leave. But the country they had left a short time before was now different.
They were no longer part of the “war effort” but were seen as a “threat to the British way of life”.
Over 10,000 West Indians volunteered to defend Britain against the Nazis during the Second World War. Among them was Donald Clarke. The RAF veteran was born in British Guyana. He enlisted during the war and served in the West Indies, then signed up for a further 12 years of service in 1948.
Donald was among the veterans who opened a new exhibition at the Imperial War Museum in London marking the contribution made to the war from the West Indies.
He made the journey to Britain onboard the Windrush in 1948.
“I was surprised by the hostility we experienced as we arrived at Tilbury docks,” he told Socialist Worker. “And for someone who was in the RAF I was shocked, because as colonials we had volunteered to fight for England.”
Many people in the British empire took part in raising money to help the war effort – in addition to the extra taxes, raw materials and food that flowed from the colonies to support the war.
The total donated by the people in the colonies, in collections, loans and personal contributions, topped an amount equivalent to £6.1 billion today – a huge sacrifice in countries with searing levels of poverty where the vast majority of people earned less than the equivalent of £1 a day.
Some 15,000 black merchant seamen helped keep the vital supply routes open. The majority of them hauled coal on the older, slower ships. Over 5,000 perished at sea.
There were 520 workers from the West Indies working in munitions factories in Britain, and 800 forestry workers from British Honduras – now Belize – cutting timber in Scotland.
Probably one of the groups that has been the least recognised is the flying crews. Considered the elite of the armed forces, fighter pilots and bombing crews are always depicted as white and upper class.
There were, however, 400 black flying crews and 6,000 ground staff, serving on all fronts. These faces are missing from the raft of war films that appeared in the decades following the war.
The influx of black soldiers and workers during the war troubled sections of the establishment. Some British officers attempted to ban white women from mixing with the black soldiers.
The British Colonial Office at the time worried “what the future population of the nation would look like” – but kept its reservations secret until after the war was over.
The general racism, however, was harder to hide.
The arrival in Britain of 150,000 black US soldiers added to the moral panics about “racial mixing”.
Donald said, “The black Americans had a different experience from us. They all served together [the US army was racially segregated]. We were in mixed units.
“Many English people saw a black face and thought we came from Africa, or were black Americans.
We felt the British public were not very well informed about us.”
Many West Indians were targeted by white US soldiers from the South who were stationed in Britain. They often reacted with violence to finding “coloured limeys” mixing freely with white people in pubs.
In one case troops from the West Indies guarding prisoners of war in Egypt were attacked by white South African troops who objected to seeing black men carrying weapons.
The official line during the war was praise for the contribution “from the colonies”. But after the war ended the establishment became obsessed by racial mixing. The press ran stories highlighting the growing number of “piccaninnies” – a term of abuse describing children of mixed race.
Donald first met Doris at the end of the war. They were married in 1948 and settled in south London.
As a white woman, Doris was stunned by the levels of hostility and racism they faced.
She told Socialist Worker, “It was very hard for me, as most of my family were opposed to our marriage.
“In the end we remained friends with those who accepted Donald, and just ignored those who had a problem.”
Doris is still angry that the country that owed so much to the sacrifice of black soldiers could turn against them. But she was determined to resist the racist onslaught.
“We were young at the time, and felt it was two of us against the world,” she said.
Donald eventually bought himself out of the RAF and went to work for Royal Mail – one of the few jobs open to black men.
The racism became more pronounced in the 1960s, feeding the growth of the National Front – the forerunner of the Nazi BNP. By the late 1970s these attitudes began to change, Donald said.
“By then people got used to seeing black people, and most English people began to realise that all the things the press accused us of – that we would do ‘bad things’ – didn’t happen.
“Although I was often homesick, I have no regrets about making the journey. I tell young people today to stay committed to their dreams, don’t give up your goals.”
From War to Windrush is on at the Imperial War Museum, London, until 29 March 2009.
Go to » london.iwm.org.uk
Keep Smiling Through: Black Londoners on the Home Front 1939-1945 is on at the Cuming Museum, 151 Walworth Road, London. It runs until 1 November. Phone 020 7525 2332<|endoftext|>
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# Is the Sum Even or Odd?
3 teachers like this lesson
Print Lesson
## Objective
SWBAT determine whether a sum is even or odd.
#### Big Idea
Students solve word problems and determine whether the sum is even or odd using strategies they learned in days prior (drawing a ten frame , counting by two, making equal groups).
## Hook
5 minutes
Introduce a problem of the day:
Mr. Fear’s class has 9 King Cakes. Ms. Vaughan’s class has 7 King Cakes. How many King Cakes do they have altogether?
Is the total number of King Cakes even or odd?
## Introduction to New Material
10 minutes
Turn and Talk: What is the first step I should take when solving this problem?
Students will likely suggest adding the two numbers. I record student responses.
I hand out white boards and have students work independently on the problem. I circulate to determine what trends are happening and where students might be getting stuck (i.e: some students might choose to subtract instead of add).
Turn and Talk: What strategies did you use to solve this problem?
I have two or three students come to the front and share how they solved the problem--some students may have added in their head, others may have used cubes or drawn ones—make sure to spotlight student(s) who have used different strategies.
Turn and Talk: How do you know whether the answer is even or odd?
I push students to use the ten frames from yesterday in their answers as well as multiples of 2 (C group)
Make sure that students are able to describe the two distinct parts of this problem: (1) Finding the sum and (2) Determining whether the sum is even or odd.
## Guided Practice
10 minutes
Now you are going to go back to your desks and work on a guided practice problem
I allow students 5-8 minutes to work. As students work, I circulate to observe student strategies and prompt students when they are struggling.
When students are finished, bring them back to the rug and review the problem as a group. As students share their strategies, check for any misconceptions (either with adding two numbers OR more likely, with determining whether a two digit-number is odd or even).
## Independent Practice
10 minutes
Independent Practice is differentiated based on proficiency with this skill. (I use yesterday's lesson and success on the guided practice problem to determine groupings).
Group A: In need of intervention.
Students in group A will work with the teacher to complete word problems with sums 1-20. Students in this group should be allowed to use cubes and ten frames in order to understand these concepts concretely.
Group B: Right on track!
Students in group B will complete word problems with sums 1-40.
Group C: Extension
Students in group C will complete word problems with sums 1-100.
## Closing
5 minutes
Today we used what we know about story problems and what we know about even and odd numbers to solve some tricky problems. Before we go, answer the following question in your math journal (or on a blank piece of paper).
Hailey has 9 blue pens and 10 black pens. Is the total number of pens ODD or EVEN?
Allow students to work silently for 3-5 minutes. Then have two students share out their answer to the group.
This final problem serves as an exit ticket and allows me to assess my students' understanding of this skill. As students work, I circulate to check for understanding and note any students who are struggling or whose strategies don't match the problem type. I make sure that I work with those students during my morning math routine so that I can remediate and clear up any misunderstandings.<|endoftext|>
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### Consecutive Numbers
An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
### I'm Eight
Find a great variety of ways of asking questions which make 8.
### Calendar Capers
Choose any three by three square of dates on a calendar page. Circle any number on the top row, put a line through the other numbers that are in the same row and column as your circled number. Repeat this for a number of your choice from the second row. You should now have just one number left on the bottom row, circle it. Find the total for the three numbers circled. Compare this total with the number in the centre of the square. What do you find? Can you explain why this happens?
# Jugs of Wine
##### Stage: 3 Challenge Level:
This is a very well-explained solution submitted by Julia (Wymondham High School):
For the jugs holding 9, 7, 4, and 2 litres, this flow diagram shows how the solution can be achieved in three distinct ways, using eight decantings of the wine. In each case, the 9 litres of wine are being poured back and forth to achieve the required result. We order the jugs by size and use a four digit number to represent the volume of wine in each jug. For example 9000 means there are 9 litres in the 9 litre jug, and the 7, 4 and 2 litre jugs are empty. The solution is found when we have 3330, where 3 litres are in each of the 9, 7 and 4 litre jugs; and the 2 litre jug is empty.
To measure out all the integer amounts from 1 to 8 litres using three jugs, one of which is full and holds 8 litres, there are several possible solutions. For example, for jugs with capacities 8, 3 and 2 litres, the following triples give the numbers of litres in each of the jugs at successive steps and all the amounts from 1 litre to 8 litres occur at some stage of the process: (8,0,0) (5,3,0) (5,1,2) (7,1,0) (7,0,1) (6,0,2) (6,2,0) (4,2,2). There are other solutions for capacities of 8, 4 and 3; for 8, 5 and 4; for 8, 5 and 1 etc.
Two other different students from Wymondham High School, David and Rachel, also submitted good solutions. In their answers, they included a very useful table of results which showed at a glance the state of the jugs after any particular pouring.
Much later two students from Flegg High, Luke and Ian, also submitted a successful solution to this problem. They had found their answer after "hours of trouble, and help from Mrs Fenn".<|endoftext|>
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Allergies: An Introduction
Allergies are a type of hypersensitivity that is mediated by IgE antibodies.
Basically, a person is exposed to a substance by either direct contact, injecting, inhaling or
ingesting. If the substance elicits an immune response then it is known as an allergen. There
are many, many different allergens known.
Some of the common allergens are plant pollen from trees, grasses and other plants. Food,
especially nuts, peanuts, seafood and milk. Also, mold spores can cause lots of problems for
many people. Even drugs such as antibiotics can result in an allergic reaction. Stings from
insects such as bees and wasps and the chemicals in poison ivy and poison sumac. Other common
triggers of allergies are cats, dogs, roaches and dust mites.
After exposure to the allergen IgE (immunoglobulin E) antibodies react and lead to the
production of histamine and arachidonate by basophils and mast cells. This results in an
inflammatory response within only seconds or minues after exposure. The response could be
local to a particular part of the body or throughout the body. For most people the symptoms
are mild to severe irritation but some cases could result in anaphylactic shock.
Symptoms of Allergies
Symptoms of local allergic reactions include allergic rhinitis, allergic conjunctivitis, asthma
and bronchoconstriction and skin problems such as eczema, hives and other rashes and contact dermatitis.
The most common test for allergies is a skin test. In this test the skin on either a person's arm
or back is divided into little sections with a marker. Then a little bit of each suspected allergen
is applied to the appropriate section. It is applied by making a small prick in the skin. If the
person is allergic to the substance a red mark or welt will appear.
Allergy Treatment and Medication
Many people get allergy shots, also called immunotherapy, to treat their allergies. Allergy shots are
injections of progressively larger doses of the allergen(s) the person reacts too. These shots either
make the allergy symptoms less bad or sometimes the person loses their hypersensitivity to the allergen.
There are a number of different medications available to treat allergies. It is important to note that
these medications do not cure or make the allergies go away for good, they just reduce or eliminate the
symptoms while the medication is being taken. The following are the classes of drugs available for allergies:
- Short-acting antihistamines: These are usually non-prescription and may cause drowsiness. An example
that doesn't cause drowsiness is Claritin.
- Longer-acting antihistamines: These usually need a prescription but cause less drowsiness. Examples
are Allegra (fexofenadine) and Zyrtec (cetirizine).
- Nasal corticosteroid sprays are also usually prescription only and include fluticasone (Flonase),
mometasone (Nasonex) and triamcinolone (Nasacort).
- Cromolyn sodium is also a nasal spray (Nasalcrom). There are also eye drops available for those
whose allergies bother their eyes too.
- Leukotriene inhibitors: Singulair (montelukast) is an example and is a prescription medication
for asthma and seasonal allergies.
Cause of Allergies
The cause of allergies is still not clear, although genetics may play a role to some degree. One popular
theory is called the hygiene hypothesis. It surmises that when kids grow up in a really clean environment,
without enough natural antigenic challenges, they may end up with allergies. This theory, if true, would
explain why the number of people with allergies keeps increasing over time and why allergies are more common
in more developed countries.<|endoftext|>
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US UKIndia
Every Question Helps You Learn
What number shirt comes next?
# Verbal Reasoning - Number Series 3
This Math quiz is called 'Verbal Reasoning - Number Series 3' and it has been written by teachers to help you if you are studying the subject at elementary school. Playing educational quizzes is an enjoyable way to learn if you are in the 3rd, 4th or 5th grade - aged 8 to 11.
It costs only \$12.50 per month to play this quiz and over 3,500 others that help you with your school work. You can subscribe on the page at Join Us
In this quiz you have to find the number that continues the sequence in the most sensible way. You will be offered a sequence of numbers, and you have to find the missing number each time. Some questions will be straightforward, and you will just have to work out the relationship between the first two numbers, then between the second and third and so on. Some of the numbers in this quiz need you to work with hundreds and even thousands, so keep a pencil and paper handy to do subtraction and multiplications. An example has been done for you. Enjoy Number Series quiz number 3.
Example: What is the next number in the series of numbers below? Replace the question mark with a number and select the correct answer from the four choices available.
2 4 6 8 10 (?)
12
11
14
2
The correct answer is 12 because the numbers are going up by 2 each time. 10 + 2 = 12.
1.
What is the next number in the series of numbers below? Replace the question mark with a number and select the correct answer from the four choices available.
49, 36, 25, 16, (?)
9
8
18
12
The correct answer is 9 because the numbers are the squares of 7, 6, 5 and 4. We are looking for the square of 3 = 9
2.
What is the next number in the series of numbers below? Replace the question mark with a number and select the correct answer from the four choices available.
22, 66, 198, 594, (?)
1827
1728
1782
1872
The correct answer is 1782 because the numbers are three times the number to their left so we are looking for 594 x 3 = 1782
3.
What is the next number in the series of numbers below? Replace the question mark with a number and select the correct answer from the four choices available.
12, 21, 24, 42, (?)
62
360
38
36
The correct answer is 36 because the number pattern is: the first number is reversed (12 becomes 21) then multiplied by 2 (24) then reversed (42) so we are looking for what it would be if multiplied by 3 = 36. This way the first, third and fifth numbers are 12, 24, 36
4.
What is the next number in the series of numbers below? Replace the question mark with a number and select the correct answer from the four choices available.
8, 27, 64, 125, (?)
200
212
216
214
The correct answer is 216 because the numbers are all the cubes of 2, 3, 4 and 5. We are looking for the cube of 6. 6 x 6 x 6 = 216. Aryabhata was an ancient Indian mathematician who first explained how cubed numbers worked. He was alive over 1500 years ago
5.
What is the next number in the series of numbers below? Replace the question mark with a number and select the correct answer from the four choices available.
58, 127, 196, 265, (?)
334
336
338
330
The correct answer is 334 because the numbers are going up by 69 each time. The only way to work this out is to subtract the second number from the first, the third from the second, and so on
6.
What is the next number in the series of numbers below? Replace the question mark with a number and select the correct answer from the four choices available.
1, 5, 2, 6, (?)
4
7
3
9
The correct answer is 3 because the numbers alternate (skip one in the pattern). So the first number is 1, the third number is 2, making the fifth number 3
7.
What is the next number in the series of numbers below? Replace the question mark with a number and select the correct answer from the four choices available.
27, 37, 29, 39, (?)
33
32
31
34
The correct answer is 31 because every other number is going up by 2 each time (27, 29, 31) and the other numbers are also going up by 2 each time (37, 39). Only 31 works
8.
What is the next number in the series of numbers below? Replace the question mark with a number and select the correct answer from the four choices available.
240, 216, 192, 168, (?)
142
144
146
148
The correct answer is 144 because the numbers are going down by 24 each time. 168 - 24 = 144
9.
What is the next number in the series of numbers below? Replace the question mark with a number and select the correct answer from the four choices available.
16, 32, 48, 64, (?)
68
64
80
62
The correct answer is 80 because the numbers are the 16 times tables. 16 x 5 = 80. Sixteen is a darts player's favorite number when trying to hit a 'double'. If they miss and hit the single, they can still try for double 8, then double 4, double 2 and finally double 1. 32 is the highest score on a dartboard that can be halved all the way down to 1
10.
What is the next number in the series of numbers below? Replace the question mark with a number and select the correct answer from the four choices available.
27, 81, 243, 729, (?)
2187
2134
2156
2178
The correct answer is 2187 because the numbers are multiplied by 3 each time. 729 x 3 = 2187. If you are working through these quizzes in order, you have just completed the third of four Number Series quizzes. Well done!
Author: Stephen O'Hara<|endoftext|>
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### December 8th 2018 archive
This week in Pre-Calc, I had some trouble understanding how to solve absolute values in piecewise notation, and how to solve reciprocal questions. So, today I will explain the steps on how to solve these.
To solve for piecewise notation, first, you will put the equation in absoulute value form. Next, you will will find the zero of the equation (isolate for “x” if it is a linear equation, or factor if it is a quadratic equation). After you have done this, you will put the number that you got from solving for “x” on a number line. You will choose two (if there is one “x” value) or three (if there are two “x” values) test points on your number line. Once youo have tested the points (by putting them back into the original equation), and you have determined which points are negative and positive, you can write the equation in piecewise form. To do this, you write “y” and then “{” and multiply in a negative for the negative value, writing whether it is < or > where you “x” value or values are on the number line. And you will leave the positive points as the original equation, and write < or > to represent where on the numberline “x” is positive.
To solve a reciprocal function, first you will put it into a fraction. Next, you will take the original equation (not in a fraction), and find the NVP. To do this, you will find the zero (the equation cannot be equal to zero) of the equation, and do this by isolating “x” (linear) or by factoring (quadratic). Once you have found what “x” cannot equal, this will be your asymptote. After, you go one to the left from your asymptote and up or down (positive, up or negative, down) and go one to the right from the asymptote and one up or down. You will then draw the shapes (two if it is linear, or three, one is a parabola, if it is quadratic).<|endoftext|>
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340 |
Water is one of our most important resources. The Earth’s freshwater is stored in lakes, rivers, and streams, or below ground in aquifers. Water collecting on the ground, or in a stream, river, lake, sea or ocean, is called surface water.
Groundwater is below the soil surface and develops from the seepage or infiltration of water into the ground. As water moves, both on the surface, and under the ground, suspended or dissolved substances such as pesticides can move with it. Because surface and groundwater are interconnected, cross contamination can occur.
This site provides information on the environmental fate of pesticides, how water may become contaminated with pesticides, and how contamination can be prevented. A glossary of important terms concerning water will explain terms that may be unfamiliar to you.
Photos courtesy of USDA NRCS and Ron Gardner
- Pesticide Fate — Learn what happens to pesticides after they are applied.
- The Problem of Runoff — Pesticide movement by runoff is explained
- The Problem of Leaching — Pesticide leaching is explained.
- Prevent Water Contamination — Practices that prevent contamination are explained on this page.
- Using Buffers to Reduce Pesticide Runoff and Water Erosion — This section describes buffers used to reduce runoff and water erosion potential.
- Wells and Contamination — Active and Abandoned wells and pesticide pollution.
- Glossary of Water Terminology — Learn the language of water.
- Water Education Quiz Module — Test your knowledge of surface and groundwater runoff.
- Resources and References — Publications used in the development of this information.
Compiled by Ron Gardner<|endoftext|>
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# Prim’s Algorithm for Minimum Spanning Tree (MST)
## Introduction to Prim’s algorithm:
We have discussed Kruskal’s algorithm for Minimum Spanning Tree. Like Kruskal’s algorithm, Prim’s algorithm is also a Greedy algorithm. This algorithm always starts with a single node and moves through several adjacent nodes, in order to explore all of the connected edges along the way.
The algorithm starts with an empty spanning tree. The idea is to maintain two sets of vertices. The first set contains the vertices already included in the MST, and the other set contains the vertices not yet included. At every step, it considers all the edges that connect the two sets and picks the minimum weight edge from these edges. After picking the edge, it moves the other endpoint of the edge to the set containing MST.
A group of edges that connects two sets of vertices in a graph is called cut in graph theory. So, at every step of Prim’s algorithm, find a cut, pick the minimum weight edge from the cut, and include this vertex in MST Set (the set that contains already included vertices).
## How does Prim’s Algorithm Work?
The working of Prim’s algorithm can be described by using the following steps:
Step 1: Determine an arbitrary vertex as the starting vertex of the MST.
Step 2: Follow steps 3 to 5 till there are vertices that are not included in the MST (known as fringe vertex).
Step 3: Find edges connecting any tree vertex with the fringe vertices.
Step 4: Find the minimum among these edges.
Step 5: Add the chosen edge to the MST if it does not form any cycle.
Step 6: Return the MST and exit
Note: For determining a cycle, we can divide the vertices into two sets [one set contains the vertices included in MST and the other contains the fringe vertices.]
Recommended Practice
## Illustration of Prim’s Algorithm:
Consider the following graph as an example for which we need to find the Minimum Spanning Tree (MST).
Example of a graph
Step 1: Firstly, we select an arbitrary vertex that acts as the starting vertex of the Minimum Spanning Tree. Here we have selected vertex 0 as the starting vertex.
0 is selected as starting vertex
Step 2: All the edges connecting the incomplete MST and other vertices are the edges {0, 1} and {0, 7}. Between these two the edge with minimum weight is {0, 1}. So include the edge and vertex 1 in the MST.
1 is added to the MST
Step 3: The edges connecting the incomplete MST to other vertices are {0, 7}, {1, 7} and {1, 2}. Among these edges the minimum weight is 8 which is of the edges {0, 7} and {1, 2}. Let us here include the edge {0, 7} and the vertex 7 in the MST. [We could have also included edge {1, 2} and vertex 2 in the MST].
7 is added in the MST
Step 4: The edges that connect the incomplete MST with the fringe vertices are {1, 2}, {7, 6} and {7, 8}. Add the edge {7, 6} and the vertex 6 in the MST as it has the least weight (i.e., 1).
6 is added in the MST
Step 5: The connecting edges now are {7, 8}, {1, 2}, {6, 8} and {6, 5}. Include edge {6, 5} and vertex 5 in the MST as the edge has the minimum weight (i.e., 2) among them.
Include vertex 5 in the MST
Step 6: Among the current connecting edges, the edge {5, 2} has the minimum weight. So include that edge and the vertex 2 in the MST.
Include vertex 2 in the MST
Step 7: The connecting edges between the incomplete MST and the other edges are {2, 8}, {2, 3}, {5, 3} and {5, 4}. The edge with minimum weight is edge {2, 8} which has weight 2. So include this edge and the vertex 8 in the MST.
Add vertex 8 in the MST
Step 8: See here that the edges {7, 8} and {2, 3} both have same weight which are minimum. But 7 is already part of MST. So we will consider the edge {2, 3} and include that edge and vertex 3 in the MST.
Include vertex 3 in MST
Step 9: Only the vertex 4 remains to be included. The minimum weighted edge from the incomplete MST to 4 is {3, 4}.
Include vertex 4 in the MST
The final structure of the MST is as follows and the weight of the edges of the MST is (4 + 8 + 1 + 2 + 4 + 2 + 7 + 9) = 37.
The structure of the MST formed using the above method
Note: If we had selected the edge {1, 2} in the third step then the MST would look like the following.
Structure of the alternate MST if we had selected edge {1, 2} in the MST
### How to implement Prim’s Algorithm?
Follow the given steps to utilize the Prim’s Algorithm mentioned above for finding MST of a graph:
• Create a set mstSet that keeps track of vertices already included in MST.
• Assign a key value to all vertices in the input graph. Initialize all key values as INFINITE. Assign the key value as 0 for the first vertex so that it is picked first.
• While mstSet doesn’t include all vertices
• Pick a vertex u that is not there in mstSet and has a minimum key value.
• Include u in the mstSet
• Update the key value of all adjacent vertices of u. To update the key values, iterate through all adjacent vertices.
• For every adjacent vertex v, if the weight of edge u-v is less than the previous key value of v, update the key value as the weight of u-v.
The idea of using key values is to pick the minimum weight edge from the cut. The key values are used only for vertices that are not yet included in MST, the key value for these vertices indicates the minimum weight edges connecting them to the set of vertices included in MST.
Below is the implementation of the approach:
## C++
`// A C++ program for Prim's Minimum` `// Spanning Tree (MST) algorithm. The program is` `// for adjacency matrix representation of the graph` `#include ` `using` `namespace` `std;` `// Number of vertices in the graph` `#define V 5` `// A utility function to find the vertex with` `// minimum key value, from the set of vertices` `// not yet included in MST` `int` `minKey(``int` `key[], ``bool` `mstSet[])` `{` ` ``// Initialize min value` ` ``int` `min = INT_MAX, min_index;` ` ``for` `(``int` `v = 0; v < V; v++)` ` ``if` `(mstSet[v] == ``false` `&& key[v] < min)` ` ``min = key[v], min_index = v;` ` ``return` `min_index;` `}` `// A utility function to print the` `// constructed MST stored in parent[]` `void` `printMST(``int` `parent[], ``int` `graph[V][V])` `{` ` ``cout << ``"Edge \tWeight\n"``;` ` ``for` `(``int` `i = 1; i < V; i++)` ` ``cout << parent[i] << ``" - "` `<< i << ``" \t"` ` ``<< graph[i][parent[i]] << ``" \n"``;` `}` `// Function to construct and print MST for` `// a graph represented using adjacency` `// matrix representation` `void` `primMST(``int` `graph[V][V])` `{` ` ``// Array to store constructed MST` ` ``int` `parent[V];` ` ``// Key values used to pick minimum weight edge in cut` ` ``int` `key[V];` ` ``// To represent set of vertices included in MST` ` ``bool` `mstSet[V];` ` ``// Initialize all keys as INFINITE` ` ``for` `(``int` `i = 0; i < V; i++)` ` ``key[i] = INT_MAX, mstSet[i] = ``false``;` ` ``// Always include first 1st vertex in MST.` ` ``// Make key 0 so that this vertex is picked as first` ` ``// vertex.` ` ``key[0] = 0;` ` ` ` ``// First node is always root of MST` ` ``parent[0] = -1;` ` ``// The MST will have V vertices` ` ``for` `(``int` `count = 0; count < V - 1; count++) {` ` ` ` ``// Pick the minimum key vertex from the` ` ``// set of vertices not yet included in MST` ` ``int` `u = minKey(key, mstSet);` ` ``// Add the picked vertex to the MST Set` ` ``mstSet[u] = ``true``;` ` ``// Update key value and parent index of` ` ``// the adjacent vertices of the picked vertex.` ` ``// Consider only those vertices which are not` ` ``// yet included in MST` ` ``for` `(``int` `v = 0; v < V; v++)` ` ``// graph[u][v] is non zero only for adjacent` ` ``// vertices of m mstSet[v] is false for vertices` ` ``// not yet included in MST Update the key only` ` ``// if graph[u][v] is smaller than key[v]` ` ``if` `(graph[u][v] && mstSet[v] == ``false` ` ``&& graph[u][v] < key[v])` ` ``parent[v] = u, key[v] = graph[u][v];` ` ``}` ` ``// Print the constructed MST` ` ``printMST(parent, graph);` `}` `// Driver's code` `int` `main()` `{` ` ``int` `graph[V][V] = { { 0, 2, 0, 6, 0 },` ` ``{ 2, 0, 3, 8, 5 },` ` ``{ 0, 3, 0, 0, 7 },` ` ``{ 6, 8, 0, 0, 9 },` ` ``{ 0, 5, 7, 9, 0 } };` ` ``// Print the solution` ` ``primMST(graph);` ` ``return` `0;` `}` `// This code is contributed by rathbhupendra`
## C
`// A C program for Prim's Minimum` `// Spanning Tree (MST) algorithm. The program is` `// for adjacency matrix representation of the graph` `#include ` `#include ` `#include ` `// Number of vertices in the graph` `#define V 5` `// A utility function to find the vertex with` `// minimum key value, from the set of vertices` `// not yet included in MST` `int` `minKey(``int` `key[], ``bool` `mstSet[])` `{` ` ``// Initialize min value` ` ``int` `min = INT_MAX, min_index;` ` ``for` `(``int` `v = 0; v < V; v++)` ` ``if` `(mstSet[v] == ``false` `&& key[v] < min)` ` ``min = key[v], min_index = v;` ` ``return` `min_index;` `}` `// A utility function to print the` `// constructed MST stored in parent[]` `int` `printMST(``int` `parent[], ``int` `graph[V][V])` `{` ` ``printf``(``"Edge \tWeight\n"``);` ` ``for` `(``int` `i = 1; i < V; i++)` ` ``printf``(``"%d - %d \t%d \n"``, parent[i], i,` ` ``graph[i][parent[i]]);` `}` `// Function to construct and print MST for` `// a graph represented using adjacency` `// matrix representation` `void` `primMST(``int` `graph[V][V])` `{` ` ``// Array to store constructed MST` ` ``int` `parent[V];` ` ``// Key values used to pick minimum weight edge in cut` ` ``int` `key[V];` ` ``// To represent set of vertices included in MST` ` ``bool` `mstSet[V];` ` ``// Initialize all keys as INFINITE` ` ``for` `(``int` `i = 0; i < V; i++)` ` ``key[i] = INT_MAX, mstSet[i] = ``false``;` ` ``// Always include first 1st vertex in MST.` ` ``// Make key 0 so that this vertex is picked as first` ` ``// vertex.` ` ``key[0] = 0;` ` ` ` ``// First node is always root of MST` ` ``parent[0] = -1;` ` ``// The MST will have V vertices` ` ``for` `(``int` `count = 0; count < V - 1; count++) {` ` ` ` ``// Pick the minimum key vertex from the` ` ``// set of vertices not yet included in MST` ` ``int` `u = minKey(key, mstSet);` ` ``// Add the picked vertex to the MST Set` ` ``mstSet[u] = ``true``;` ` ``// Update key value and parent index of` ` ``// the adjacent vertices of the picked vertex.` ` ``// Consider only those vertices which are not` ` ``// yet included in MST` ` ``for` `(``int` `v = 0; v < V; v++)` ` ``// graph[u][v] is non zero only for adjacent` ` ``// vertices of m mstSet[v] is false for vertices` ` ``// not yet included in MST Update the key only` ` ``// if graph[u][v] is smaller than key[v]` ` ``if` `(graph[u][v] && mstSet[v] == ``false` ` ``&& graph[u][v] < key[v])` ` ``parent[v] = u, key[v] = graph[u][v];` ` ``}` ` ``// print the constructed MST` ` ``printMST(parent, graph);` `}` `// Driver's code` `int` `main()` `{` ` ``int` `graph[V][V] = { { 0, 2, 0, 6, 0 },` ` ``{ 2, 0, 3, 8, 5 },` ` ``{ 0, 3, 0, 0, 7 },` ` ``{ 6, 8, 0, 0, 9 },` ` ``{ 0, 5, 7, 9, 0 } };` ` ``// Print the solution` ` ``primMST(graph);` ` ``return` `0;` `}`
## Java
`// A Java program for Prim's Minimum Spanning Tree (MST)` `// algorithm. The program is for adjacency matrix` `// representation of the graph` `import` `java.io.*;` `import` `java.lang.*;` `import` `java.util.*;` `class` `MST {` ` ``// Number of vertices in the graph` ` ``private` `static` `final` `int` `V = ``5``;` ` ``// A utility function to find the vertex with minimum` ` ``// key value, from the set of vertices not yet included` ` ``// in MST` ` ``int` `minKey(``int` `key[], Boolean mstSet[])` ` ``{` ` ``// Initialize min value` ` ``int` `min = Integer.MAX_VALUE, min_index = -``1``;` ` ``for` `(``int` `v = ``0``; v < V; v++)` ` ``if` `(mstSet[v] == ``false` `&& key[v] < min) {` ` ``min = key[v];` ` ``min_index = v;` ` ``}` ` ``return` `min_index;` ` ``}` ` ``// A utility function to print the constructed MST` ` ``// stored in parent[]` ` ``void` `printMST(``int` `parent[], ``int` `graph[][])` ` ``{` ` ``System.out.println(``"Edge \tWeight"``);` ` ``for` `(``int` `i = ``1``; i < V; i++)` ` ``System.out.println(parent[i] + ``" - "` `+ i + ``"\t"` ` ``+ graph[i][parent[i]]);` ` ``}` ` ``// Function to construct and print MST for a graph` ` ``// represented using adjacency matrix representation` ` ``void` `primMST(``int` `graph[][])` ` ``{` ` ``// Array to store constructed MST` ` ``int` `parent[] = ``new` `int``[V];` ` ``// Key values used to pick minimum weight edge in` ` ``// cut` ` ``int` `key[] = ``new` `int``[V];` ` ``// To represent set of vertices included in MST` ` ``Boolean mstSet[] = ``new` `Boolean[V];` ` ``// Initialize all keys as INFINITE` ` ``for` `(``int` `i = ``0``; i < V; i++) {` ` ``key[i] = Integer.MAX_VALUE;` ` ``mstSet[i] = ``false``;` ` ``}` ` ``// Always include first 1st vertex in MST.` ` ``// Make key 0 so that this vertex is` ` ``// picked as first vertex` ` ``key[``0``] = ``0``;` ` ` ` ``// First node is always root of MST` ` ``parent[``0``] = -``1``;` ` ``// The MST will have V vertices` ` ``for` `(``int` `count = ``0``; count < V - ``1``; count++) {` ` ` ` ``// Pick the minimum key vertex from the set of` ` ``// vertices not yet included in MST` ` ``int` `u = minKey(key, mstSet);` ` ``// Add the picked vertex to the MST Set` ` ``mstSet[u] = ``true``;` ` ``// Update key value and parent index of the` ` ``// adjacent vertices of the picked vertex.` ` ``// Consider only those vertices which are not` ` ``// yet included in MST` ` ``for` `(``int` `v = ``0``; v < V; v++)` ` ``// graph[u][v] is non zero only for adjacent` ` ``// vertices of m mstSet[v] is false for` ` ``// vertices not yet included in MST Update` ` ``// the key only if graph[u][v] is smaller` ` ``// than key[v]` ` ``if` `(graph[u][v] != ``0` `&& mstSet[v] == ``false` ` ``&& graph[u][v] < key[v]) {` ` ``parent[v] = u;` ` ``key[v] = graph[u][v];` ` ``}` ` ``}` ` ``// Print the constructed MST` ` ``printMST(parent, graph);` ` ``}` ` ``public` `static` `void` `main(String[] args)` ` ``{` ` ``MST t = ``new` `MST();` ` ``int` `graph[][] = ``new` `int``[][] { { ``0``, ``2``, ``0``, ``6``, ``0` `},` ` ``{ ``2``, ``0``, ``3``, ``8``, ``5` `},` ` ``{ ``0``, ``3``, ``0``, ``0``, ``7` `},` ` ``{ ``6``, ``8``, ``0``, ``0``, ``9` `},` ` ``{ ``0``, ``5``, ``7``, ``9``, ``0` `} };` ` ``// Print the solution` ` ``t.primMST(graph);` ` ``}` `}` `// This code is contributed by Aakash Hasija`
## Python3
`# A Python3 program for ` `# Prim's Minimum Spanning Tree (MST) algorithm.` `# The program is for adjacency matrix ` `# representation of the graph` `# Library for INT_MAX` `import` `sys` `class` `Graph():` ` ``def` `__init__(``self``, vertices):` ` ``self``.V ``=` `vertices` ` ``self``.graph ``=` `[[``0` `for` `column ``in` `range``(vertices)]` ` ``for` `row ``in` `range``(vertices)]` ` ``# A utility function to print ` ` ``# the constructed MST stored in parent[]` ` ``def` `printMST(``self``, parent):` ` ``print``(``"Edge \tWeight"``)` ` ``for` `i ``in` `range``(``1``, ``self``.V):` ` ``print``(parent[i], ``"-"``, i, ``"\t"``, ``self``.graph[i][parent[i]])` ` ``# A utility function to find the vertex with` ` ``# minimum distance value, from the set of vertices` ` ``# not yet included in shortest path tree` ` ``def` `minKey(``self``, key, mstSet):` ` ``# Initialize min value` ` ``min` `=` `sys.maxsize` ` ``for` `v ``in` `range``(``self``.V):` ` ``if` `key[v] < ``min` `and` `mstSet[v] ``=``=` `False``:` ` ``min` `=` `key[v]` ` ``min_index ``=` `v` ` ``return` `min_index` ` ``# Function to construct and print MST for a graph` ` ``# represented using adjacency matrix representation` ` ``def` `primMST(``self``):` ` ``# Key values used to pick minimum weight edge in cut` ` ``key ``=` `[sys.maxsize] ``*` `self``.V` ` ``parent ``=` `[``None``] ``*` `self``.V ``# Array to store constructed MST` ` ``# Make key 0 so that this vertex is picked as first vertex` ` ``key[``0``] ``=` `0` ` ``mstSet ``=` `[``False``] ``*` `self``.V` ` ``parent[``0``] ``=` `-``1` `# First node is always the root of` ` ``for` `cout ``in` `range``(``self``.V):` ` ``# Pick the minimum distance vertex from` ` ``# the set of vertices not yet processed.` ` ``# u is always equal to src in first iteration` ` ``u ``=` `self``.minKey(key, mstSet)` ` ``# Put the minimum distance vertex in` ` ``# the shortest path tree` ` ``mstSet[u] ``=` `True` ` ``# Update dist value of the adjacent vertices` ` ``# of the picked vertex only if the current` ` ``# distance is greater than new distance and` ` ``# the vertex in not in the shortest path tree` ` ``for` `v ``in` `range``(``self``.V):` ` ``# graph[u][v] is non zero only for adjacent vertices of m` ` ``# mstSet[v] is false for vertices not yet included in MST` ` ``# Update the key only if graph[u][v] is smaller than key[v]` ` ``if` `self``.graph[u][v] > ``0` `and` `mstSet[v] ``=``=` `False` `\` ` ``and` `key[v] > ``self``.graph[u][v]:` ` ``key[v] ``=` `self``.graph[u][v]` ` ``parent[v] ``=` `u` ` ``self``.printMST(parent)` `# Driver's code` `if` `__name__ ``=``=` `'__main__'``:` ` ``g ``=` `Graph(``5``)` ` ``g.graph ``=` `[[``0``, ``2``, ``0``, ``6``, ``0``],` ` ``[``2``, ``0``, ``3``, ``8``, ``5``],` ` ``[``0``, ``3``, ``0``, ``0``, ``7``],` ` ``[``6``, ``8``, ``0``, ``0``, ``9``],` ` ``[``0``, ``5``, ``7``, ``9``, ``0``]]` ` ``g.primMST()` `# Contributed by Divyanshu Mehta`
## C#
`// A C# program for Prim's Minimum` `// Spanning Tree (MST) algorithm.` `// The program is for adjacency` `// matrix representation of the graph` `using` `System;` `class` `MST {` ` ``// Number of vertices in the graph` ` ``static` `int` `V = 5;` ` ``// A utility function to find` ` ``// the vertex with minimum key` ` ``// value, from the set of vertices` ` ``// not yet included in MST` ` ``static` `int` `minKey(``int``[] key, ``bool``[] mstSet)` ` ``{` ` ``// Initialize min value` ` ``int` `min = ``int``.MaxValue, min_index = -1;` ` ``for` `(``int` `v = 0; v < V; v++)` ` ``if` `(mstSet[v] == ``false` `&& key[v] < min) {` ` ``min = key[v];` ` ``min_index = v;` ` ``}` ` ``return` `min_index;` ` ``}` ` ``// A utility function to print` ` ``// the constructed MST stored in parent[]` ` ``static` `void` `printMST(``int``[] parent, ``int``[, ] graph)` ` ``{` ` ``Console.WriteLine(``"Edge \tWeight"``);` ` ``for` `(``int` `i = 1; i < V; i++)` ` ``Console.WriteLine(parent[i] + ``" - "` `+ i + ``"\t"` ` ``+ graph[i, parent[i]]);` ` ``}` ` ``// Function to construct and` ` ``// print MST for a graph represented` ` ``// using adjacency matrix representation` ` ``static` `void` `primMST(``int``[, ] graph)` ` ``{` ` ``// Array to store constructed MST` ` ``int``[] parent = ``new` `int``[V];` ` ``// Key values used to pick` ` ``// minimum weight edge in cut` ` ``int``[] key = ``new` `int``[V];` ` ``// To represent set of vertices` ` ``// included in MST` ` ``bool``[] mstSet = ``new` `bool``[V];` ` ``// Initialize all keys` ` ``// as INFINITE` ` ``for` `(``int` `i = 0; i < V; i++) {` ` ``key[i] = ``int``.MaxValue;` ` ``mstSet[i] = ``false``;` ` ``}` ` ``// Always include first 1st vertex in MST.` ` ``// Make key 0 so that this vertex is` ` ``// picked as first vertex` ` ``// First node is always root of MST` ` ``key[0] = 0;` ` ``parent[0] = -1;` ` ``// The MST will have V vertices` ` ``for` `(``int` `count = 0; count < V - 1; count++) {` ` ``// Pick the minimum key vertex` ` ``// from the set of vertices` ` ``// not yet included in MST` ` ``int` `u = minKey(key, mstSet);` ` ``// Add the picked vertex` ` ``// to the MST Set` ` ``mstSet[u] = ``true``;` ` ``// Update key value and parent` ` ``// index of the adjacent vertices` ` ``// of the picked vertex. Consider` ` ``// only those vertices which are` ` ``// not yet included in MST` ` ``for` `(``int` `v = 0; v < V; v++)` ` ``// graph[u][v] is non zero only` ` ``// for adjacent vertices of m` ` ``// mstSet[v] is false for vertices` ` ``// not yet included in MST Update` ` ``// the key only if graph[u][v] is` ` ``// smaller than key[v]` ` ``if` `(graph[u, v] != 0 && mstSet[v] == ``false` ` ``&& graph[u, v] < key[v]) {` ` ``parent[v] = u;` ` ``key[v] = graph[u, v];` ` ``}` ` ``}` ` ``// Print the constructed MST` ` ``printMST(parent, graph);` ` ``}` ` ``// Driver's Code` ` ``public` `static` `void` `Main()` ` ``{` ` ``int``[, ] graph = ``new` `int``[, ] { { 0, 2, 0, 6, 0 },` ` ``{ 2, 0, 3, 8, 5 },` ` ``{ 0, 3, 0, 0, 7 },` ` ``{ 6, 8, 0, 0, 9 },` ` ``{ 0, 5, 7, 9, 0 } };` ` ``// Print the solution` ` ``primMST(graph);` ` ``}` `}` `// This code is contributed by anuj_67.`
## Javascript
``
Output
```Edge Weight
0 - 1 2
1 - 2 3
0 - 3 6
1 - 4 5
```
Time Complexity: O(V2), If the input graph is represented using an adjacency list, then the time complexity of Prim’s algorithm can be reduced to O(E * logV) with the help of a binary heap. In this implementation, we are always considering the spanning tree to start from the root of the graph
Auxiliary Space: O(V)
### Other Implementations of Prim’s Algorithm:
Given below are some other implementations of Prim’s Algorithm
#### OPTIMIZED APPROACH OF PRIM’S ALGORITHM:
Intuition
2. Then we create a Pair class to store the vertex and its weight .
3. We sort the list on the basis of lowest weight.
4. We create priority queue and push the first vertex and its weight in the queue
5. Then we just traverse through its edges and store the least weight in a variable called ans.
6. At last after all the vertex we return the ans.
Implementation
## C++
`#include` `using` `namespace` `std;` `typedef` `pair<``int``,``int``> pii;` `// Function to find sum of weights of edges of the Minimum Spanning Tree.` `int` `spanningTree(``int` `V, ``int` `E, ``int` `edges[][3])` `{ ` ` ``// Create an adjacency list representation of the graph` ` ``vector> adj[V];` ` ` ` ``// Fill the adjacency list with edges and their weights` ` ``for` `(``int` `i = 0; i < E; i++) {` ` ``int` `u = edges[i][0];` ` ``int` `v = edges[i][1];` ` ``int` `wt = edges[i][2];` ` ``adj[u].push_back({v, wt});` ` ``adj[v].push_back({u, wt});` ` ``}` ` ` ` ``// Create a priority queue to store edges with their weights` ` ``priority_queue, greater> pq;` ` ` ` ``// Create a visited array to keep track of visited vertices` ` ``vector<``bool``> visited(V, ``false``);` ` ` ` ``// Variable to store the result (sum of edge weights)` ` ``int` `res = 0;` ` ` ` ``// Start with vertex 0` ` ``pq.push({0, 0});` ` ` ` ``// Perform Prim's algorithm to find the Minimum Spanning Tree` ` ``while``(!pq.empty()){` ` ``auto` `p = pq.top();` ` ``pq.pop();` ` ` ` ``int` `wt = p.first; ``// Weight of the edge` ` ``int` `u = p.second; ``// Vertex connected to the edge` ` ` ` ``if``(visited[u] == ``true``){` ` ``continue``; ``// Skip if the vertex is already visited` ` ``}` ` ` ` ``res += wt; ``// Add the edge weight to the result` ` ``visited[u] = ``true``; ``// Mark the vertex as visited` ` ` ` ``// Explore the adjacent vertices` ` ``for``(``auto` `v : adj[u]){` ` ``// v[0] represents the vertex and v[1] represents the edge weight` ` ``if``(visited[v[0]] == ``false``){` ` ``pq.push({v[1], v[0]}); ``// Add the adjacent edge to the priority queue` ` ``}` ` ``}` ` ``}` ` ` ` ``return` `res; ``// Return the sum of edge weights of the Minimum Spanning Tree` `}` `int` `main()` `{` ` ``int` `graph[][3] = {{0, 1, 5},` ` ``{1, 2, 3},` ` ``{0, 2, 1}};` ` ``// Function call` ` ``cout << spanningTree(3, 3, graph) << endl;` ` ``return` `0;` `}`
## Java
`// A Java program for Prim's Minimum Spanning Tree (MST)` `// algorithm. The program is for adjacency list` `// representation of the graph` `import` `java.io.*;` `import` `java.util.*;` `// Class to form pair` `class` `Pair ``implements` `Comparable` `{` ` ``int` `v;` ` ``int` `wt;` ` ``Pair(``int` `v,``int` `wt)` ` ``{` ` ``this``.v=v;` ` ``this``.wt=wt;` ` ``}` ` ``public` `int` `compareTo(Pair that)` ` ``{` ` ``return` `this``.wt-that.wt;` ` ``}` `}` `class` `GFG {` `// Function of spanning tree` `static` `int` `spanningTree(``int` `V, ``int` `E, ``int` `edges[][])` ` ``{` ` ``ArrayList> adj=``new` `ArrayList<>();` ` ``for``(``int` `i=``0``;i());` ` ``}` ` ``for``(``int` `i=``0``;i pq = ``new` `PriorityQueue();` ` ``pq.add(``new` `Pair(``0``,``0``));` ` ``int``[] vis=``new` `int``[V];` ` ``int` `s=``0``;` ` ``while``(!pq.isEmpty())` ` ``{` ` ``Pair node=pq.poll();` ` ``int` `v=node.v;` ` ``int` `wt=node.wt;` ` ``if``(vis[v]==``1``) ` ` ``continue``;` ` ` ` ``s+=wt;` ` ``vis[v]=``1``;` ` ``for``(Pair it:adj.get(v))` ` ``{` ` ``if``(vis[it.v]==``0``)` ` ``{` ` ``pq.add(``new` `Pair(it.v,it.wt));` ` ``}` ` ``}` ` ``}` ` ``return` `s;` ` ``}` ` ` ` ``// Driver code` ` ``public` `static` `void` `main (String[] args) {` ` ``int` `graph[][] = ``new` `int``[][] {{``0``,``1``,``5``},` ` ``{``1``,``2``,``3``},` ` ``{``0``,``2``,``1``}};` ` ` ` ``// Function call` ` ``System.out.println(spanningTree(``3``,``3``,graph));` ` ``}` `}`
## Python3
`import` `heapq` `def` `tree(V, E, edges):` ` ``# Create an adjacency list representation of the graph` ` ``adj ``=` `[[] ``for` `_ ``in` `range``(V)]` ` ``# Fill the adjacency list with edges and their weights` ` ``for` `i ``in` `range``(E):` ` ``u, v, wt ``=` `edges[i]` ` ``adj[u].append((v, wt))` ` ``adj[v].append((u, wt))` ` ``# Create a priority queue to store edges with their weights` ` ``pq ``=` `[]` ` ``# Create a visited array to keep track of visited vertices` ` ``visited ``=` `[``False``] ``*` `V` ` ``# Variable to store the result (sum of edge weights)` ` ``res ``=` `0` ` ``# Start with vertex 0` ` ``heapq.heappush(pq, (``0``, ``0``))` ` ``# Perform Prim's algorithm to find the Minimum Spanning Tree` ` ``while` `pq:` ` ``wt, u ``=` `heapq.heappop(pq)` ` ``if` `visited[u]:` ` ``continue` ` ``# Skip if the vertex is already visited` ` ``res ``+``=` `wt ` ` ``# Add the edge weight to the result` ` ``visited[u] ``=` `True` ` ``# Mark the vertex as visited` ` ``# Explore the adjacent vertices` ` ``for` `v, weight ``in` `adj[u]:` ` ``if` `not` `visited[v]:` ` ``heapq.heappush(pq, (weight, v)) ` ` ``# Add the adjacent edge to the priority queue` ` ``return` `res ` ` ``# Return the sum of edge weights of the Minimum Spanning Tree` `if` `__name__ ``=``=` `"__main__"``:` ` ``graph ``=` `[[``0``, ``1``, ``5``],` ` ``[``1``, ``2``, ``3``],` ` ``[``0``, ``2``, ``1``]]` ` ``# Function call` ` ``print``(tree(``3``, ``3``, graph))`
## C#
`using` `System;` `using` `System.Collections.Generic;` `public` `class` `MinimumSpanningTree` `{` ` ``// Function to find sum of weights of edges of the Minimum Spanning Tree.` ` ``public` `static` `int` `SpanningTree(``int` `V, ``int` `E, ``int``[,] edges)` ` ``{` ` ``// Create an adjacency list representation of the graph` ` ``List> adj = ``new` `List>();` ` ``for` `(``int` `i = 0; i < V; i++)` ` ``{` ` ``adj.Add(``new` `List<``int``[]>());` ` ``}` ` ``// Fill the adjacency list with edges and their weights` ` ``for` `(``int` `i = 0; i < E; i++)` ` ``{` ` ``int` `u = edges[i, 0];` ` ``int` `v = edges[i, 1];` ` ``int` `wt = edges[i, 2];` ` ``adj[u].Add(``new` `int``[] { v, wt });` ` ``adj[v].Add(``new` `int``[] { u, wt });` ` ``}` ` ``// Create a priority queue to store edges with their weights` ` ``PriorityQueue<(``int``, ``int``)> pq = ``new` `PriorityQueue<(``int``, ``int``)>();` ` ``// Create a visited array to keep track of visited vertices` ` ``bool``[] visited = ``new` `bool``[V];` ` ``// Variable to store the result (sum of edge weights)` ` ``int` `res = 0;` ` ``// Start with vertex 0` ` ``pq.Enqueue((0, 0));` ` ``// Perform Prim's algorithm to find the Minimum Spanning Tree` ` ``while` `(pq.Count > 0)` ` ``{` ` ``var` `p = pq.Dequeue();` ` ``int` `wt = p.Item1; ``// Weight of the edge` ` ``int` `u = p.Item2; ``// Vertex connected to the edge` ` ``if` `(visited[u])` ` ``{` ` ``continue``; ``// Skip if the vertex is already visited` ` ``}` ` ``res += wt; ``// Add the edge weight to the result` ` ``visited[u] = ``true``; ``// Mark the vertex as visited` ` ``// Explore the adjacent vertices` ` ``foreach` `(``var` `v ``in` `adj[u])` ` ``{` ` ``// v[0] represents the vertex and v[1] represents the edge weight` ` ``if` `(!visited[v[0]])` ` ``{` ` ``pq.Enqueue((v[1], v[0])); ``// Add the adjacent edge to the priority queue` ` ``}` ` ``}` ` ``}` ` ``return` `res; ``// Return the sum of edge weights of the Minimum Spanning Tree` ` ``}` ` ``public` `static` `void` `Main()` ` ``{` ` ``int``[,] graph = { { 0, 1, 5 }, { 1, 2, 3 }, { 0, 2, 1 } };` ` ``// Function call` ` ``Console.WriteLine(SpanningTree(3, 3, graph));` ` ``}` `}` `// PriorityQueue implementation for C#` `public` `class` `PriorityQueue ``where` `T : IComparable` `{` ` ``private` `List heap = ``new` `List();` ` ``public` `int` `Count => heap.Count;` ` ``public` `void` `Enqueue(T item)` ` ``{` ` ``heap.Add(item);` ` ``int` `i = heap.Count - 1;` ` ``while` `(i > 0)` ` ``{` ` ``int` `parent = (i - 1) / 2;` ` ``if` `(heap[parent].CompareTo(heap[i]) <= 0)` ` ``break``;` ` ``Swap(parent, i);` ` ``i = parent;` ` ``}` ` ``}` ` ``public` `T Dequeue()` ` ``{` ` ``int` `lastIndex = heap.Count - 1;` ` ``T frontItem = heap[0];` ` ``heap[0] = heap[lastIndex];` ` ``heap.RemoveAt(lastIndex);` ` ``--lastIndex;` ` ``int` `parent = 0;` ` ``while` `(``true``)` ` ``{` ` ``int` `leftChild = parent * 2 + 1;` ` ``if` `(leftChild > lastIndex)` ` ``break``;` ` ``int` `rightChild = leftChild + 1;` ` ``if` `(rightChild <= lastIndex && heap[leftChild].CompareTo(heap[rightChild]) > 0)` ` ``leftChild = rightChild;` ` ``if` `(heap[parent].CompareTo(heap[leftChild]) <= 0)` ` ``break``;` ` ``Swap(parent, leftChild);` ` ``parent = leftChild;` ` ``}` ` ``return` `frontItem;` ` ``}` ` ``private` `void` `Swap(``int` `i, ``int` `j)` ` ``{` ` ``T temp = heap[i];` ` ``heap[i] = heap[j];` ` ``heap[j] = temp;` ` ``}` `}` `// This code is contributed by shivamgupta0987654321`
## Javascript
`class PriorityQueue {` ` ``constructor() {` ` ``this``.heap = [];` ` ``}` ` ``enqueue(value) {` ` ``this``.heap.push(value);` ` ``let i = ``this``.heap.length - 1;` ` ``while` `(i > 0) {` ` ``let j = Math.floor((i - 1) / 2);` ` ``if` `(``this``.heap[i][0] >= ``this``.heap[j][0]) {` ` ``break``;` ` ``}` ` ``[``this``.heap[i], ``this``.heap[j]] = [``this``.heap[j], ``this``.heap[i]];` ` ``i = j;` ` ``}` ` ``}` ` ``dequeue() {` ` ``if` `(``this``.heap.length === 0) {` ` ``throw` `new` `Error(``"Queue is empty"``);` ` ``}` ` ``let i = ``this``.heap.length - 1;` ` ``const result = ``this``.heap[0];` ` ``this``.heap[0] = ``this``.heap[i];` ` ``this``.heap.pop();` ` ``i--;` ` ``let j = 0;` ` ``while` `(``true``) {` ` ``const left = j * 2 + 1;` ` ``if` `(left > i) {` ` ``break``;` ` ``}` ` ``const right = left + 1;` ` ``let k = left;` ` ``if` `(right <= i && ``this``.heap[right][0] < ``this``.heap[left][0]) {` ` ``k = right;` ` ``}` ` ``if` `(``this``.heap[j][0] <= ``this``.heap[k][0]) {` ` ``break``;` ` ``}` ` ``[``this``.heap[j], ``this``.heap[k]] = [``this``.heap[k], ``this``.heap[j]];` ` ``j = k;` ` ``}` ` ``return` `result;` ` ``}` ` ``get count() {` ` ``return` `this``.heap.length;` ` ``}` `}` `function` `spanningTree(V, E, edges) {` ` ``// Create an adjacency list representation of the graph` ` ``const adj = ``new` `Array(V).fill(``null``).map(() => []);` ` ``// Fill the adjacency list with edges and their weights` ` ``for` `(let i = 0; i < E; i++) {` ` ``const [u, v, wt] = edges[i];` ` ``adj[u].push([v, wt]);` ` ``adj[v].push([u, wt]);` ` ``}` ` ``// Create a priority queue to store edges with their weights` ` ``const pq = ``new` `PriorityQueue();` ` ``// Create a visited array to keep track of visited vertices` ` ``const visited = ``new` `Array(V).fill(``false``);` ` ``// Variable to store the result (sum of edge weights)` ` ``let res = 0;` ` ``// Start with vertex 0` ` ``pq.enqueue([0, 0]);` ` ``// Perform Prim's algorithm to find the Minimum Spanning Tree` ` ``while` `(pq.count > 0) {` ` ``const p = pq.dequeue();` ` ``const wt = p[0]; ``// Weight of the edge` ` ``const u = p[1]; ``// Vertex connected to the edge` ` ``if` `(visited[u]) {` ` ``continue``; ``// Skip if the vertex is already visited` ` ``}` ` ``res += wt; ``// Add the edge weight to the result` ` ``visited[u] = ``true``; ``// Mark the vertex as visited` ` ``// Explore the adjacent vertices` ` ``for` `(const v of adj[u]) {` ` ``// v[0] represents the vertex and v[1] represents the edge weight` ` ``if` `(!visited[v[0]]) {` ` ``pq.enqueue([v[1], v[0]]); ``// Add the adjacent edge to the priority queue` ` ``}` ` ``}` ` ``}` ` ``return` `res; ``// Return the sum of edge weights of the Minimum Spanning Tree` `}` `// Example usage` `const graph = [[0, 1, 5], [1, 2, 3], [0, 2, 1]];` `// Function call` `console.log(spanningTree(3, 3, graph));`
Output
```4
```
Time Complexity: O(E*log(E)) where E is the number of edges
Auxiliary Space: O(V^2) where V is the number of vertex
### Prim’s algorithm for finding the minimum spanning tree (MST):
1. Prim’s algorithm is guaranteed to find the MST in a connected, weighted graph.
2. It has a time complexity of O(E log V) using a binary heap or Fibonacci heap, where E is the number of edges and V is the number of vertices.
3. It is a relatively simple algorithm to understand and implement compared to some other MST algorithms.
1. Like Kruskal’s algorithm, Prim’s algorithm can be slow on dense graphs with many edges, as it requires iterating over all edges at least once.
2. Prim’s algorithm relies on a priority queue, which can take up extra memory and slow down the algorithm on very large graphs.
3. The choice of starting node can affect the MST output, which may not be desirable in some applications.
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# Eureka Math Geometry Module 1 Lesson 2 Answer Key
## Engage NY Eureka Math Geometry Module 1 Lesson 2 Answer Key
### Eureka Math Geometry Module 1 Lesson 2 Exercise Answer Key
Opening Exercise
You need a compass, a straightedge, and another student’s Problem Set.
Directions:
Follow the directions from another student’s Problem Set write-up to construct an equilateral triangle.
→ What kinds of problems did you have as you followed your classmate’s directions?
→ Think about ways to avoid these problems. What criteria or expectations for writing steps in constructions should be included in a rubric for evaluating your writing? List at least three criteria.
### Eureka Math Geometry Module 1 Lesson 2 Exploratory Challenge Answer Key
Exploratory Challenge 1.
You need a compass and a straightedge.
Using the skills you have practiced, construct three equilateral triangles, where the first and second triangles share a common side and the second and third triangles share a common side. Clearly and precisely list the steps needed to accomplish this construction.
Switch your list of steps with a partner, and complete the construction according to your partner’s steps. Revise your drawing and list of steps as needed.
Construct three equilateral triangles here:
1. Draw a segment AB.
2. Draw circle A: center A, radius AB.
3. Draw circle B: center B, radius BA.
4. Label one intersection as C; label the other intersection as D.
5. Draw circle C: center C, radius CA.
6. Label the intersection of circle C with circle A (or the intersection of circle C with circle B) as E.
7. Draw all segments that are congruent to $$\overline{A B}$$ between the labeled points.
There are many ways to address Step 7; students should be careful to avoid making a blanket statement that would allow segment BE or segment CD.
Exploratory Challenge 2.
On a separate piece of paper, use the skills you have developed in this lesson to construct a regular hexagon. Clearly and precisely list the steps needed to accomplish this construction. Compare your results with a partner, and revise your drawing and list of steps as needed.
1. Draw circle K: center K, any radius.
2. Pick a point on the circle; label this point A.
3. Draw circle A: center A, radius AK.
4. Label the intersections of circle A with circle K as B and F.
5. Draw circle B: center B, radius BK.
6. Label the intersection of circle B with circle K as C.
7. Continue to treat the intersection of each new circle with circle K as the center of a new circle until the next circle to be drawn is circle A.
8. Draw $$\overline{A B}$$, $$\overline{B C}$$, $$\overline{C D}$$, $$\overline{D E}$$, $$\overline{E F}$$, $$\overline{F A}$$.
Can you repeat the construction of a hexagon until the entire sheet is covered in hexagons (except the edges, which are partial hexagons)?
Yes, this result resembles wallpaper, tile patterns, etc.
### Eureka Math Geometry Module 1 Lesson 2 Problem Set Answer Key
Why are circles so important to these constructions? Write out a concise explanation of the importance of circles in creating equilateral triangles. Why did Euclid use circles to create his equilateral triangles in Proposition 1? How does construction of a circle ensure that all relevant segments are of equal length?<|endoftext|>
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R: Binomial Distribution
I. Calculating Exact Binomial Probabilities
If X~B(n, p), one may use the following mathematical formula to calculate P(X = k).
$$P(X = k)= {n \choose k} p^k (1-p)^{n-k} \:\:where \:\: {n \choose k} = \frac{n!}{k!(n-k)!}$$
Likewise, if X~B(n, p), one may use the dbinom(x, n, p) function to calculate P(X = k). In other words, the dbinom(x, n, p) function will provide the same answer as using the mathematical formula shown above.
Example 1:
If X~B(10, 0.3), use the following R code to calculate P(X = 3).
# Method 1 - Use dbinom(k, n, p)
> dbinom(3, 10,0.3)
[1] 0.2668279
# Method 2 - Use the mathematical formula...a little more complicated :)
> choose(10,3)*(0.3)^3*(0.7)^7
[1] 0.2668279
Example 2:
On the way to work, a commuter encounters one stoplight. The probability the commuter encounters a red light on the way to work is 0.3. Over the course of 20 days of commuting, find the probability that the commuter encounters exactly 8 red lights on the way to work.
> dbinom(8, 20, 0.3)
[1] 0.1143967
II. Calculating Cumulative Binomial Probabilities
If X~B(n,p), use pbinom(x, n, p) function to calculate P(X ≤ x).
Example 1:
IX~B(10, 0.3), use the following to code to find P(X ≤ 3).
> pbinom(3,10,0.3) # pbinom(x, n, p)
[1] 0.6496107
Example 2: IX~B(10, 0.3), use the following to code to find P(X > 3).
# Method 1 - Use "lower.tail = FALSE"
> pbinom(3, 10, 0.3, lower.tail = FALSE)
[1] 0.3503893
# Method 2 - Subtract pbinom(x, n, p) from 1
> 1 - pbinom(3, 10, 0.3)
[1] 0.3503893
III. Simulating Binomial Random Variables
In statistics, one often finds the need to simulate random scenarios that are binomial. To do this, we need to use the rbinom() function. To use the rbinom() function, you need to define three parameters:
Example 1:
Let's say you wanted to simulate rolling a dice 5 times, and you wished to count the number of 3's you observe. You could simulate this experiment using the following code:
> rbinom(1,5,1/6) ### rbinom(number of experiments, number of observations per experiment, probability of success)
[1] 2
Conclusion: The above code simulated rolling a die five times. The output of 2 means that there were 2 "successes", or 2 observed 3's as rolling a 3 was labeled as a "success".
Example 2:
Let's say you wanted to simulate a class of 30 different students flipped a coin 10 times and you asked each student to report back the number of tails they observed. You could simulate this experiment using the following code:
> rbinom(30,10,0.5) # rbinom(number of experiments, number of observations per experiment, probability of success)
[1] 7 4 7 5 7 3 5 4 7 5 5 5 3 5 6 6 4 4 4 6 7 5 6 8 5 8 5 4 5 6
Conclusion: The above output shows the number of "successes" recorded by each of the 30 students. For example, the first student recorded seeing 7 heads, while the second student recorded seeing 4 heads.
Example 3:
Let's say you wanted to simulate a class of 30 students rolling a dice 10 times, and you wished to count the number of 3's you observe for each student. You could simulate this experiment, then create a table using the table() function to summarize your results using the following code:
> ### rbinom(number of experiments, number of observations per experiment, probability of success)
> s=rbinom(30,10,1/6)
> table(s)
s
0 1 2 3 4 5
7 9 7 4 2 1
Conclusion: The R output creates a simple table showing, for instance, of the 30 students who rolled their dice 10 times, 7 of them observed no "successes" or rolled no 3's, while 9 students observed exactly one "success".<|endoftext|>
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# How to Simplify Polynomials
To simplify polynomials, you need to find ‘like’ terms and combine them. Here you can learn how to simplify polynomials.
## Step by step guide to simplifying polynomials
• Find “like” terms. (they have same variables with same power).
• Add or Subtract “like” terms using order of operation.
### Simplifying Polynomials – Example 1:
Simplify this expression. $$2x(2x-4)=$$
Solution:
Use Distributive Property: $$2x(2x−4)=4x^2−8x$$
### Simplifying Polynomials – Example 2:
Simplify this expression. $$4x^2+6x+2x^2-4x-3=$$
Solution:
First find “like” terms and combine them: $$4x^2+2x^2= 6x^2$$, $$6x-4x= 2x$$
Now simplify: $$4x^2+6x+2x^2-4x-3=6x^2+2x-3$$
### Simplifying Polynomials – Example 3:
Simplify this expression. $$4x(6x-3)=$$
Solution:
Use Distributive Property: $$4x(6x-3)=24x^2-12x$$
### Simplifying Polynomials – Example 4:
Simplify this expression. $$7x^3+2x^4+2x^3-4x^4-8x=$$
Solution:
First find “like” terms and combine them: $$7x^3+2x^3= 10x^3$$, $$2x^4-4x^4= -2x^4$$
Now simplify and write in standard form: $$7x^3+2x^4+2x^3-4x^4-8x=-2x^4+10x^3-8x$$
## Exercises for Simplifying Polynomials
### Simplify each expression.
1. $$\color{blue}{(12x^3 + 28x^2 + 10x^2 + 4) }$$
2. $$\color{blue}{(2x + 12x^2 – 2) – (2x + 1)}$$
3. $$\color{blue}{(2x^3 – 1) + (3x^3 – 2x^3)}$$
4. $$\color{blue}{(x – 5) (x – 3)}$$
5. $$\color{blue}{(3x + 8) (3x – 8)}$$
6. $$\color{blue}{(8x^2 – 3x) – (5x – 5 – 8x^2)}$$
1. $$\color{blue}{12x^3 + 38x^2 + 4}$$
2. $$\color{blue}{12x^2 – 3 }$$
3. $$\color{blue}{3x^3 – 1}$$
4. $$\color{blue}{x^2 – 8x + 15}$$
5. $$\color{blue}{9x^2 – 64}$$
6. $$\color{blue}{16x^2 – 8x + 5}$$
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International Covenant on Civil and Political Rights
The rights are listed in the Universal Declaration of Human Rights. It is one of the most important treaties in international human rights law. The Covenant was adopted in 1966 with the International Covenant on Economic, Social and Cultural Rights. It was put into force in 1976. The United Nations set up the Committee on Civil and Political Rights (CCPR) to carry out the treaty.
- Article 1. Right to self-determination.
- Article 6. Right to life and on Genocide Convention.
- Article 7. Free from torture and any inhuman treatments.
- Article 8. Free from slavery and unfree labour.
- Article 9. Right to personal security and protection from arbitrary arrest and detention. Right to demand remedy if proved innocent.
- Article 10. The right to be treated with humanity and repected dignity while in detention.
- Article 12. Freedom of movement, freedom to leave and to enter one's own country.
- Article 14. The right to a fair trial and to be regarded as innocent until found guilty.
- Article 16. Right to be recognized as a person before law.
- Article 17. Right to privacy and protection from attack of honour.
- Article 18. Freedom of thought, conscience and freedom of religion.
- Article 19. Freedom of expression and freedom of speech but with special duty and responsibility not to harm others.
- Article 20. prohibition of propaganda of war and any inciting of hatred and discrimination.
- Article 21. Freedom of assembly.
- Article 22. Right to association and trade union.
- Article 25. Right to free and fair voting.
- Article 26. Equality before law and protection from all discriminations by law.
- Article 27. Right to minority groups and ethnic group and to use their own language.
This Covenant has two optional protocols. One of them is to allow a citizen whose rights have been violated to claim before the CCPR. The other is to prohibit the death penalty. There are countries which ignore, or do not agree with, the optional protocols.<|endoftext|>
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Brand-New
Dashboard lnterface
ln the Making
We are proud to announce that we are developing a fresh new dashboard interface to improve user experience.
We invite you to preview our new dashboard and have a try. Some features will become unavailable, but they will be added in the future.
Don't hesitate to try it out as it's easy to switch back to the interface you're used to.
Published on Feb 10,2015
chapter 2 Square root EM Read More
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2 Square RootBy studying this lesson you will be able to• approximate the square root of a positive number which is not a perfect square• use the division method to find an approximate value for the square root of a positive number which is not a perfect square. 2.1 The square root of a positive numberYou have earlier learnt some facts about the square of a number, and the square rootof a positive number. Let us briefly recall what you have learnt.The value of 3 « 3 is 9. We denote 3 « 3 in short by 32. This is read as ‘threesquared’. The ‘2’ in 32 denotes the fact that three is multiplied ‘twice’ over.Accordingly, three squared is 9 and this can be written as 32 = 9.The squares of several numbers are given in the following table.Number How the square of the How the square Square of the number is obtained number of the number is 1 1×1 1 2 2×2 denoted 4 3 3×3 9 4 4×4 12 16 5 5×5 22 25 32 42 52Numbers such as 1, 4, 9, 16 are perfect squares.Finding the square root is the inverse of squaring. For example, since 32 = 9, we saythat the square root of 9 is 3. It will be clear to you, according to the first and lastcolumns of the above table that the square root of 1 is 1, the square root of 4 is 2, the square root of 9 is 3, the square root of 16 is 4 and the square root of 25 is 5.14
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The symbol is used to denote the square root. Accordingly we can write etc.It is clear that every number has a square. However, does every positive numberhave a square root? Let us investigate this.According to the above table, the square root of 4 is 2 and the square root of 9is 3. The square root of a number between 4 and 9 is a value between 2 and 3.Accordingly, it is clear that the square root of a number between 4 and 9 is not aninteger. It is a decimal number. In this lesson we will consider how an approximatevalue is found for this. We call such a value an approximation.Let us for example consider how an approximate value is obtained for the squareroot of 5.Consider the following table.Number How the square How the square The square of the of the number is number 2 of the number is written 2.1 4 2.2 found 22 4.41 2.3 2.12 4.84 2.4 2×2 5.29 2.5 2.1 × 2.1 2.22 5.76 2.6 2.2 × 2.2 6.25 2.7 2.3 × 2.3 2.32 6.76 2.4 × 2.4 7.29 2.5 × 2.5 2.42 2.6 × 2.6 2.7 × 2.7 2.52 2.62 2.72From the values in the right hand side column, the two values that are closest to 5are 4.84 and 5.29. They are the squares of 2.2 and 2.3 respectively. According to theabove table, the square roots of 4.84 and 5.29 are respectively 2.2 and 2.3. This canbe written symbolically as and 'Now let us examine which value from 4.84 and 5.29 is closer to 5.The difference between 4.84 and 5 = 5 – 4.84 = 0.16The difference between 5.29 and 5 = 5.29 – 5 = 0.29Accordingly, the value that is closer to 5 is 4.84. Therefore, 2.2 can be taken as anapproximate value for the square root of 5. The value that is obtained for the squareroot of a positive integer which is correct to the first decimal place is called the 15
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“approximation to the first decimal place” of the square root of the given number(or more simply the “first approximation”)Accordingly, the approximation of square root of 5 to the first decimal place is 2.2.When an approximate value is given, the symbol is used. Accordingly, we canwrite 'By providing reasons in a similar manner, we can conclude that the approximationof square root of 6 to the first decimal place is 2.4 and the approximation of squareroot of 7 to the first decimal place is 2.6That is,By considering the following examples let us now learn a specific method of findingthe first approximation of the square root of a positive number which is not a perfectsquare.Example 1Approximate to the first decimal place.• From the perfect square numbers which are less than 17, the one which is closestto it is 16, and from the perfect square numbers which are greater than 17, the onewhich is closest to it is 25. Accordingly, let us write 16 < 17 < 25• When we write the square root of each of these numbers we obtain.Accordingly, the square root of 17 is greater than 4 which is the square root of 16,and less than 5 which is the square root of 25. i.e., lies between 4 and 5.• To find an approximate value close to , let us check whether 17 is closer to16 or to 25. The difference between 16 and 17 is 1. The difference between 17 and 25 is 8.16
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17 is closer to 16 than to 25. is a value close to 4.i.e., one of the values 4.1, 4.2, 4.3 and 4.4 is the first approximation of• Let us now multiply each of these numbers by itself to identify the number which has a product which is closest to 17.When the first two values are squared we obtain = =Since the value of 4.22 exceeds 17, it is unnecessary to find 4.32 and 4.42.16.81 is the closer value to 17 from these two. First approximation of is 4.1 Example 2Find the first approximation of 'Since 152 = 225 and 162 = 256, writeAccordingly, is a value between 15 and 16.Since 245 is closer to 256 than to 225, is closer to 16 than to 15. The first approximation of is one of 15.5, 15.6, 15.7, 15.8 and 15.9.Let us now determine this value. 15'9 x 15'9 = 252'81 15'8 x 15'8 = 249'64 15'7 x 15'7 = 246'49 15'6 x 15'6 = 243'36 From the above values, 246.49 is closest to 245. First approximation of is 15'7'Exercise 2.1 Find the first approximation of each of the following numbers. 17
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2.2 The Division MethodLet us now consider a method of finding the square root of any positive number.This method is called the division method. Let us study this method by consideringseveral examples. Example 1 Find the square root of 1764.Step 1Separate 1764 as shown below, by grouping the digits of 1764 in pairs, startingfrom the units position and proceeding towards the left.17 64Step 2Find the perfect square number which is closest to the leftmost digit or pair of digitsof the separated number, and as indicated below, write its square root above and tothe left of the drawn lines. Step 3 Write down the product 4 x 4 of the number above and to the left of the lines, belowthe number 17 as indicated, and subtract it from 17. Step 4Now carry down the next two digits 64, as indicated below. Step 5Next, write on the left as shown below, the digit 8, which is two times the numberabove the line, leaving space for another number to be written. (i.e., leave space for 18
P:06
the digit in the units position) Step 6The same digit should be written above the line to the right of 4 and in the space leftin the units position on the left. This digit should be selected so that the product ofthis digit and the number obtained on the left when this digit is written in the unitsposition (in this case 82), is equal to 164, or is the closest number less than 164 thatcan be obtained in this manner. ThenWhen finding the square root of a decimal number, separate the digits in pairs onboth sides of the decimal point, starting at the decimal point as shown below. 3'61 3' 61 12'321 12' 32 10 143'456 1 43' 45 60 Example 2Find the value of . 19
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Example 3Find the value of accurate to two decimal places.We must find the value to three decimal places and round off to two decimal places.To find to three decimal places we must write three pairs of zeros after the decimalpoint. 5 2'3 1 6 5 27 37' 00 00 00 5 x 2 = 10 25 10 2 2 37 2 0452 x 2 = 104 104 3 33 00 31 29523 x 2 = 1046 1046 1 1 71 005231 x 2 = 10462 10462 6 1 04 61 66 39 00 62 77 56 3 61 44 Example 4 Find the value of accurate to two decimal places.As above let us find the value to three decimal places and round off to two decimalplaces. For this there must be three pairs of decimal places after the decimal point. 20
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Exercise 2.21. Find the square root of each of the following numbers.(i) 676 (ii) 1024 (iii) 2209 (iv) 2809 (v) 37212. Find the value accurate to one decimal place. 0.0064 0.0001442.3 Using the square roots of numbers to solve problemsExample 1Find the length of a side of a square of area 441 cm2. Area of the square = (side length)2 Length of a side of the square = area of the square Area of the square = 441 cm2 Length of a side of the square = = 21 cm Example 2324 square shaped garden tiles of area 900 cm2 each have been placed in a squareshaped courtyard so that the courtyard is completely covered with the tiles. Find thelength of a side of the courtyard. The number of garden tiles in one row = 324 = 18 The length of a garden tile = 900 cm = 30 cm The length of a side of the courtyard = 18 × 30 cm = 540 cm = 5.4 m 21
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Exercise 2.31. What is the length of a side of a square shaped piece of cardboard of area 1225 cm2?2. What is the length of a side of a square of area the same as that of a rectangle of length 27 cm and breadth 12 cm?3. 196 children participating in a drill display have been placed such that they form an equal number of rows and columns. How many children are there in a row?4. The surface area of a cube is 1350 cm2. Find the length of a side of the cube.5. A rectangular walkway has been made by placing 10 flat square shaped concrete slabs along 200 rows. If the area of the flat surface of a concrete slab is 231.04 cm2, what is the length and the breadth of the walkway? Miscellaneous Exercise1. Find the value accurate to the second decimal place.2. The length and breadth of a rectangular shaped plot of land are respectively 25 m and 12 m. Find to the nearest metre, the least distance that a child standing at one corner of the plot should travel to reach the diagonally opposite corner of the plot.3. If the length of the hypotenuse of an isosceles right angled triangle is 12 cm, find the length of a remaining side. (Give the answer to two decimal places).4. 9, 16, 25, … is a number pattern. Which term of the pattern is the number 729? 22<|endoftext|>
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# Reflections and Symmetry
Reflections and Symmetry. Lesson 5.2. Across the x-axis. Across the y-axis. Flipping the Graph of a Function. Given the function below We wish to manipulate it by reflecting it across one of the axes. Flipping the Graph of a Function. Consider the function
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## Reflections and Symmetry
E N D
### Presentation Transcript
1. Reflections and Symmetry Lesson 5.2
2. Across the x-axis Across the y-axis Flipping the Graph of a Function • Given the function below • We wish to manipulate it by reflecting it across one of the axes
3. Flipping the Graph of a Function • Consider the function • f(x) = 0.1*(x3 - 9x2 + 5) : place it in y1(x) • graphed on the window -10 < x < 10 and -20 < y < 20
4. Flipping the Graph of a Function • specify the following functions on the Y= screen: • y2(x) = y1(-x) dotted style • y3(x) = -y1(x) thick style • Predict which of these will rotate the function • about the x-axis • about the y-axis
5. use -f(x) use f(-x) Flipping the Graph of a Function • Results • To reflect f(x) in the x-axis or rotate about • To reflect f(x) in the y-axis or rotate about Spreadsheet Demo
6. Even and Odd Functions • If f(x) = f(-x) the graph is symmetric across the y-axis • It is also an even function
7. Even and Odd Functions • If f(x) = -f(x) the graph is symmetric across the x-axis • But ... is it a function ??
8. Even and Odd Functions • A graph can be symmetric about a point • Called point symmetry • If f(-x) = -f(x) it is symmetric about the origin • Also an odd function
9. Applications • Consider a frozen yam placed into a hot oven. Think what the graph of the temperature would look like. Sketch the graph of the temperature of the yam. It is frozen at 0 degrees Fahrenheit and the oven is at 300 degrees Fahrenheit. This will be both a flip and a shift of an exponential function
10. Applications • This is the function • f(x) = 300 - 300(0.97)t • It has been flipped about the y-axis • Then it has been shifted up • Which part did the shift? • Which part did theflip?
11. Reflecting in the Line y = x • Given the function below: • For each (x,y) shown, reverse the values to get (y,x) • Plot the (y,x) values and connect the points
12. Reflecting in the Line y = x • Results • Note: it is not a function.
13. Reflecting in the Line y = x • Try it for this graph … will the result be a function or not?
14. Assignment • Lesson 5.2 • Page 209 • Exercises 1 – 31 odd
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Londinium was a settlement established on the current site of the City of London around AD 43. Its bridge over the River Thames turned the city into a road nexus and major port, serving as a major commercial centre in Roman Britain until its abandonment during the 5th century.
Following its foundation in the mid-1st century, early Londinium occupied the relatively small area of 1.4 km2 (0.5 sq mi), roughly equivalent to the size of present-day Hyde Park, with a fortified garrison on one of its hills. In the year 60 or 61, the rebellion of the Iceni under Boudica forced the garrison to abandon the settlement, which was then razed. Following the Iceni's defeat at the Battle of Watling Street, the city was rebuilt as a planned Roman town and recovered within about a decade. During the later decades of the 1st century, Londinium expanded rapidly, becoming Great Britain's largest city. By the turn of the century, Londinium had grown to perhaps 30,000 or 60,000 people, almost certainly replacing Camulodunum (Colchester) as the provincial capital and by the mid-2nd century, Londinium was at its height. Its forum and basilica were one of the largest structures north of the Alps when the Emperor Hadrian visited Londinium in 122. Excavations have discovered evidence of a major fire that destroyed most of the city shortly thereafter, but the city was again rebuilt. By the second half of the 2nd century, Londinium appears to have shrunk in both size and population.
Although Londinium remained important for the rest of the Roman period, no further expansion resulted. Londinium supported a smaller but stable settlement population as archaeologists have found that much of the city after this date was covered in dark earth—the by-product of urban household waste, manure, ceramic tile, and non-farm debris of settlement occupation, which accumulated relatively undisturbed for centuries. Sometime between 190 and 225, the Romans built a defensive wall around the landward side of the city. Along with Hadrian's Wall and the road network, this wall was one of the largest construction projects carried out in Roman Britain. The London Wall survived for another 1,600 years and broadly defined the perimeter of the old City of London.
Part of a series on the
|History of London|
- 1 Name
- 2 Location
- 3 Status
- 4 History
- 5 Demographics
- 6 Excavation
- 7 Displays
- 8 See also
- 9 Notes
- 10 References
- 11 Further reading
- 12 External links
The etymology of the name Londinium is unknown. Following Geoffrey of Monmouth's pseudohistorical History of the Kings of Britain, it was long derived from an eponymous founder named Lud, son of Heli. There is no evidence such a figure ever existed. Instead, the Latin name was probably based on a native Brittonic placename reconstructed as *Londinion. Morphologically, this points to a structure of two suffixes: -in-jo-. However, the Roman Londinium was not the immediate source of English "London" (Old English: Lunden), as i-mutation would have caused the name to have been Lyndon. This suggests an alternative Brittonic form Londonion; alternatively, the local pronunciation in British Latin may have changed the pronunciation of Londinium to Lundeiniu or Lundein, which would also have avoided i-mutation in Old English. The list of the 28 Cities of Britain included in the 9th-century History of the Britons precisely notes London in Old Welsh as Cair Lundem or Lundein.
The site guarded the Romans' bridgehead on the north bank of the Thames and a major road nexus. It centered on Cornhill and the River Walbrook, but expanded west to Ludgate Hill and east to Tower Hill. Just prior to the Roman conquest, the area had been contested by the Catuvellauni based to its west and the Trinovantes based to its east; it bordered the realm of the Cantiaci on the south bank of the Thames.
The Roman city ultimately covered at least the area of the City of London, whose boundaries are largely defined by its former wall. Londinium's waterfront on the Thames ran from around Ludgate Hill in the west to the present site of the Tower in the east, around 1.5 kilometres (0.93 mi). The northern wall reached Bishopsgate and Cripplegate near the Museum of London, a course now marked by the street "London Wall". Cemeteries and suburbs existed outside the city proper. A round temple has been located west of the city, although its dedication remains unclear. Substantial suburbs existed at St Martin-in-the-Fields in Westminster and around the southern end of the Thames bridge in Southwark, where inscriptions suggest a temple of Isis was located.
The status of Londinium is uncertain. It seems to have been founded as a mere vicus and remained as such even after its recovery from Boudica's revolt. Ptolemy lists it as one of the cities of the Cantiacs, but Durovernum (Roman Canterbury) was their tribal capital (civitas). Starting as a small fort guarding the northern end of the new bridge across the River Thames, Londinium grew to become an important port for trade between Britain and the Roman provinces on the continent. The initial lack of private Roman villas (plentiful elsewhere) suggests military or even Imperial ownership. Tacitus wrote that, at the time of the uprising of Boudica, "Londinium... though undistinguished by the name of 'colony', was much frequented by a number of merchants and trading vessels." Depending on the time of its creation, the modesty of Londonium's first forum may have reflected its early elevation to city (municipium) status or may have reflected an administrative concession to a low-ranking but major Romano-British settlement. It had almost certainly been granted colony (colonia) status prior to the complete replanning of the city's street plan attending the erection of the great second forum around the year 120.
By this time, Britain's provincial administration had also almost certainly been moved to Londinium from Camulodunum (Colchester in Essex). The precise date of this change is unknown and no surviving source explicitly states that Londinium was "the capital of Britain" but there are several strong indications of this status: 2nd-century roofing tiles have been found marked by the "Procurator" or "Publican of the Province of Britain at Londinium", the remains of a governor's palace and tombstones belonging to the governor's staff have been discovered, and the city was well defended and armed, with a new military camp erected at the beginning of the 2nd century, despite being far from any frontier. Despite some corruption to the text, the list of bishops for the 314 Council of Arles indicates that either Restitutus or Adelphius came from Londinium. The city seems to have been the seat of the diocesan vicar and one of the provincial governors following the Diocletian Reforms around the year 300; it had been renamed Augusta—a common epithet of provincial capitals—by 368.
Unlike many cities of Roman Britain, Londinium was not placed on the site of a Celtic oppidum. Prior to the arrival of the Roman legions, the area was almost certainly lightly rolling open countryside traversed by numerous streams now underground. Archaeologist Lacey Wallace notes that "Because no LPRIA settlements or significant domestic refuse have been found in London, despite extensive archaeological excavation, arguments for a purely Roman foundation of London are now common and uncontroversial." The city's Latin name now seems to have derived from an originally Brittonic one. However, significant pre-Roman finds in the Thames, especially the Battersea Shield (Chelsea Bridge, perhaps 4th-century BC) and the Wandsworth Shield (perhaps 1st-century BC), both assumed to be votive offerings deposited a couple of miles upstream of Londinium, suggest the general area was busy and significant. It has been suggested that the area was where a number of territories met. There was probably a ford in that part of the river; other Roman and Celtic finds suggest this was perhaps where the opposed crossing Julius Caesar describes in 54 BC took place.
Londinium grew up around the point on the River Thames narrow enough for the construction of a Roman bridge but still deep enough to handle the era's seagoing ships. Its placement on the Tideway permitted easier access for ships sailing upstream against the current. The remains of a massive pier base for such a bridge were found in 1981 close by the modern London Bridge. Some Claudian-era camp ditches have been discovered, but archaeological excavations undertaken since the 1970s by the Department of Urban Archaeology at the Museum of London (now MOLAS) have suggested the early settlement was largely the product of private enterprise. A timber drain by the side of the main Roman road excavated at No 1 Poultry has been dated by dendrochronology to AD 47, which is likely to be the foundation date.
Following its foundation in the mid-1st century, early Roman London occupied a relatively small area, about 350 acres (1.4 km2) or roughly the area of present-day Hyde Park. Archaeologists have uncovered numerous goods imported from across the Roman Empire in this period, suggesting that early Roman London was a highly cosmopolitan community of merchants from across the Empire and that local markets existed for such objects.
Of the fifteen British routes recorded in the 2nd- or 3rd-century Antonine Itinerary, seven ran to or from Londinium. Most of these have been shown to have been initially constructed near the time of the city's foundation around AD 47. The roads are now known by Welsh or Old English names, as their original Roman names have been entirely lost due to the lack of written and inscribed sources. (It was customary elsewhere to name roads after the emperor during whose principate they were completed, but the number and vicinity of routes completed during the time of Claudius would seem to have made this impractical in Britain's case.)
The road from the Kentish ports of Rutupiae (Richborough), Dubris (Dover), and Lemanis (Lympne) via Durovernum (Canterbury) seems to have first crossed the Thames at a natural ford near Westminster before being diverted north to the new bridge at London. The Romans enabled the road to cross the marshy terrain without subsidence by laying down substrates of one to three layers of oak logs. This route, now known as Watling Street, then passed through the town from the bridgehead in a straight line to reconnect with its northern extension towards Viroconium (Wroxeter) and the legionary base at Deva Victrix (Chester). The Great Road ran northeast across Old Ford to Camulodunum (Colchester) and thence northeast along Pye Road to Venta Icenorum (Caistor St Edmund). Ermine Street ran north from the city to Lindum (Lincoln) and Eboracum (York). The Devil's Highway connected Londinium to Calleva (Silchester) and its roads to points west over the bridges near modern Staines. A minor road led southwest to the city's main cemetery and the old routes to the ford at Westminster. Stane Street to Noviomagus (Chichester) did not reach Londinium proper but ran from the bridgehead in the southern suburb at Southwark. These roads varied from 12–20 m (39–66 ft) wide.
After its reconstruction in the AD 60s, the streets within Londinium itself largely adhered to a grid. By analogy with Roman forts, the main east-west street is now generally called the Via Decumana ("Tenth Cohort Way"), while the main north-south street (interrupted by the forum north of its intersection with the Via Decumana) is known as the Via Principalis ("Headquarters Way"). These names would not have been used for the civilian settlement at the time. The main streets were 9–10 m (30–33 ft) wide, while side streets were usually about 5 m (16 ft) wide.
In the year 60 or 61, a little more than ten years after Londinium was founded, the king of the Iceni died. He had possibly been installed by the Romans after the Iceni's failed revolt against P. Ostorius Scapula's disarmament of the allied tribes in AD 47 or may have assisted the Romans against his tribesmen during that revolt. His will had divided his wealth and lands between Rome and his two daughters, but Roman law forbade female inheritance and it had become common practice to treat allied kingdoms as life estates that were annexed upon the ruler's death, as had occurred in Bithynia and Galatia. Roman financiers including Seneca called in all the king's outstanding loans at once and the provincial procurator confiscated the property of both the king and his nobles. Tacitus records that, when the king's wife Boudica objected, the Romans flogged her, raped her two daughters, and enslaved their nobles and kinsmen. Boudica then led a failed revolt against Roman rule.
Two hundred ill-equipped men were sent to defend the provincial capital and Roman colony at Camulodunum, probably from the garrison at Londinium. The Iceni and their allies overwhelmed them and razed the city. The 9th Legion under Q. Petillius Cerialis, coming south from the Fosse Way, was ambushed and annihilated. The procurator, meanwhile, escaped with his treasure to Gaul, probably via Londinium. G. Suetonius Paulinus had been leading the 14th and 20th Legions in the invasion of Anglesey now known as the Menai massacre; hearing of the rising, he immediately returned along Watling Street with the legions' cavalry. An early historical record of London appears in Tacitus's account of his actions upon arriving and finding the state of the 9th Legion:
At first, [Paulinus] hesitated as to whether to stand and fight there. Eventually, his numerical inferiority—and the price only too clearly paid by the divisional commander's rashness—decided him to sacrifice the single city of Londinium to save the province as a whole. Unmoved by lamentations and appeals, Suetonius gave the signal for departure. The inhabitants were allowed to accompany him. But those who stayed because they were women, or old, or attached to the place, were slaughtered by the enemy.
Excavation has revealed extensive evidence of destruction by fire in the form of a layer of red ash beneath the city at this date. Suetonius then returned to the legions' slower infantry, who met and defeated the British army, slaughtering as many as 70,000 men and camp followers. There is a long-standing folklore belief that this battle took place at King’s Cross, simply because as a mediaeval village it was known as Battle Bridge. Suetonius's flight back to his men, the razing of Verulamium (St Albans), and the battle shortly thereafter at "a place with narrow jaws, backed by a forest", speaks against the tradition and no supporting archaeological evidence has been yet discovered.
After being sacked, the city was rebuilt as a planned Roman town, its streets generally adhering to a grid skewed by major roads passing from the bridgehead and by changes in alignment produced by crossings over the local streams. It recovered after about a decade. A fortified enclosure was erected at Plantation Place on Cornhill. The first forum was constructed in the 70s or 80s and has been excavated, showing it had an open courtyard with a basilica and several shops around it, altogether measuring about 100 m × 50 m (330 ft × 160 ft). The basilica would have functioned as the city's administrative heart, hearing law cases and seating the town's local senate. It formed the north side of the forum, whose south entrance was located along the north side of the intersection of the present Gracechurch, Lombard, and Fenchurch Streets. Forums elsewhere typically had a civic temple constructed within the enclosed market area; British sites usually did not, instead placing a smaller shrine for Roman services somewhere within the basilica. The first forum in Londinium seems to have had a full temple, but placed outside just west of the forum.
During the later decades of the 1st century, Londinium expanded rapidly and quickly became Roman Britain's largest city, although most of its houses continued to be made of wood. By the turn of the century, Londinium was perhaps as large as 60,000 people, and had replaced Camulodunum (Colchester) as the provincial capital. A large building discovered near Cannon Street Station has had its foundation dated to this era and is assumed to have been the governor's palace. It boasted a garden, pools, and several large halls, some of which were decorated with mosaic floors. It stood on the east bank of the now-covered Walbrook, near where it joins the Thames. London Stone may originally have been part of the palace's main entrance. Another site dating to this era is the bathhouse (thermae) at Huggin Hill, which remained in use prior to its demolition around the year 200. Brothels were legal but taxed.
The bulk of the Roman port was quickly rebuilt after Boudicca's rebellion when the waterfront was extended with gravel to permit a sturdy wharf to be built perpendicular to the shore. The port was built in four sections, starting upstream of the London Bridge and working down towards the Walbrook at the center of Londinium. Expansion of the flourishing port continued into the 3rd century. Scraps of armour, leather straps, and military stamps on building timbers suggest that the site was constructed by the city's legionaries. Major imports included fine pottery, jewelry, and wine. Only two large warehouses are known, implying that Londinium functioned as a bustling trade center rather than a supply depot and distribution center like Ostia near Rome.
Emperor Hadrian visited in 122. The impressive public buildings from around this period may have been initially constructed in preparation for his visit or during the rebuilding that followed the "Hadrianic Fire". This fire, which archaeologists have discovered destroyed much of the city, is not recorded by any surviving source and seems to have occurred in a time of relative calm in Britain; for those reasons, it is generally assumed to have been accidental.
During the early 2nd century, Londinium was at its height. London recovered from the fire and again had between 45,000 and 60,000 inhabitants around the year 140, with many more stone houses and public buildings erected. Some areas were tightly packed with townhouses (domus). The town had piped water and a "fairly-sophisticated" drainage system. The gubernatorial palace was rebuilt and an expanded forum was built around the earlier one over a period of 30 years from around 90 to 120 into an almost perfect square measuring 168 m × 167 m (551 ft × 548 ft). Its three-storey basilica was likely visible across the city and largest in the empire north of the Alps; the marketplace itself rivaled those in Rome and was the largest in the north before Augusta Treverorum (Trier, Germany) became an imperial capital. The city's temple of Jupiter was renovated, public and private bathhouses were erected, and a fort (arx) was erected around the year 120 that maintained the city garrison northwest of town. The fort was a square (with small rounded corners) measuring more than 200 m × 200 m (660 ft × 660 ft) and covering more than 12 acres (4.9 ha). Each side had a central gatehouse and stone towers were erected at the corners and at points along each wall. Londinium's amphitheatre, constructed in AD 70, is situated at Guildhall; its gladiatorial games would have been free of charge. When the ancient Romans left in the 4th century the amphitheatre lay derelict for hundreds of years. In the 11th century the area was reoccupied and by the 12th century the first Guildhall was built next to it. A large port complex on both banks near London Bridge was discovered during the 1980s. A temple complex with two Romano-British temples was excavated at Empire Square, Long Lane, Southwark in 2002/2003. A large house there may have been a guesthouse. A marble slab with a dedication to the god Mars was discovered in the temple complex. The inscription mentions the Londoners, the earliest known reference naming the people of London.
By the second half of the 2nd century, Londinium had many large, well-equipped stone buildings, some of which were richly adorned with wall paintings and floor mosaics, and had subfloor hypocausts. The Roman house at Billingsgate was built next to the waterfront and had its own bath. In addition to such structures reducing the city's building density, however, Londinium also seems to have shrunk in both size and population in the second half of the 2nd century. The cause is uncertain but plague is considered likely, as the Antonine Plague is recorded decimating other areas of Western Europe between 165 and 190. The end of imperial expansion in Britain after Hadrian's decision to build his wall may have also damaged the city's economy. Although Londinium remained important for the rest of the Roman period, no further expansion occurred. Londinium remained well populated as archaeologists have found that much of the city after this date was covered in dark earth, one that accumulated relatively undisturbed for centuries.
Some time between 190 and 225, the Romans built the London Wall, a defensive ragstone wall around the landward side of the city. Along with Hadrian's Wall and the road network, the London Wall was one of the largest construction projects carried out in Roman Britain. The wall was originally about 5 km (3 mi) long, 6 m (20 ft) high, and 2.5 m (8 ft 2 in) thick. Its dry moat (fossa) was about 2 m (6 ft 7 in) deep and 3–5 m (9.8–16.4 ft) wide. In the 19th century, Smith estimated its length from the Tower west to Ludgate at about one mile (1.6 km) and its breadth from the northern wall to the bank of the Thames at around half that.
In addition to small pedestrian postern gates like the one by Tower Hill, it had four main gates: Bishopsgate and Aldgate in the northeast at the roads to Eboracum (York) and to Camulodunum (Colchester) and Newgate and Ludgate in the west along at the road that divided for travel to Viroconium (Wroxeter) and to Calleva (Silchester) and at another road that ran along the Thames to the city's main cemetery and the old ford at Westminster. The wall partially utilized the army's existing fort, strengthening its outer wall with a second course of stone to match the rest of the course. The fort had two gates of its own—Cripplegate to the north and another to the west—but these were not along major roads. Aldersgate was eventually added, perhaps to replace the west gate of the fort. (The names of all these gates are medieval, as they continued to be occasionally refurbished and replaced until their demolition in the 17th and 18th centuries to permit widening the roads.) The wall initially left the riverbank undefended: this was corrected in the 3rd century.
Although the exact reason for the wall's construction is unknown, some historians have connected it with the Pictish invasion of the 180s. Others link it with Clodius Albinus, the British governor who attempted to usurp Septimius Severus in the 190s. The wall survived another 1,600 years and still roughly defines the City of London's perimeter.
Septimius Severus defeated Albinus in 197 and shortly afterwards divided the province of Britain into Upper and Lower halves, with the former controlled by a new governor in Eboracum (York). Despite the smaller administrative area, the economic stimulus provided by the Wall and by Septimius Severus's campaigns in Caledonia somewhat revived London's fortunes in the early 3rd century. The northwest fort was abandoned and dismantled but archaeological evidence points to renewed construction activity from this period. The London Mithraeum rediscovered in 1954 dates from around 240, when it was erected on the east bank at the head of navigation on the now-covered River Walbrook about 200 m (660 ft) from the Thames. From about 255 onwards, raiding by Saxon pirates led to the construction of a riverside wall as well. It ran roughly along the course of present-day Thames Street, which then roughly formed the shoreline. Large collapsed sections of this wall were excavated at Blackfriars and the Tower in the 1970s.
In 286, the emperor Maximian issued a death sentence against Carausius, admiral of the Roman navy's Britannic fleet (Classis Britannica), on charges of having abetted Frankish and Saxon piracy and of having embezzled recovered treasure. Carausius responded by consolidating his allies and territory and revolting. After fending off Maximian's first assault in 288, he declared a new Britannic Empire and issued coins to that effect. Constantius Chlorus's sack of his Gallic base at Gesoriacum (Boulogne), however, led his treasurer Allectus to assassinate and replace him. In 296, Chlorus mounted an invasion of Britain that prompted Allectus's Frankish mercenaries to sack Londinium. They were only stopped by the arrival of a flotilla of Roman warships on the Thames, which slaughtered the survivors. The event was commemorated by the golden "Trier Medallion", Chlorus on one side and, on the other, a woman kneeling at the city wall welcoming a mounted Roman soldier. Another memorial to the return of Londinium to Roman control was the construction of a new set of forum baths around the year 300. The structures were modest enough that they were previously identified as parts of the forum and market but are now recognized as elaborate and luxurious baths including a frigidarium with two southern pools and an eastern swimming pool.
Following the revolt, the Diocletian Reforms saw the British administration restructured. Londinium is universally supposed to have been the capital of one of them, but it remains unclear where the new provinces were, whether there were initially three or four in total, and whether Valentia represented a fifth province or a renaming of an older one. In the 12th century, Gerald of Wales listed "Londonia" as the capital of Flavia, having had Britannia Prima (Wales) and Secunda (Kent) severed from the territory of Upper Britain. Modern scholars more often list Londinium as the capital of Maxima Caesariensis on the assumption that the presence of the diocesan vicar in London would have required its provincial governor to outrank the others.
The gubernatorial palace and old large forum seem to have fallen out of use around 300, but in general the first half of the 4th century appears to have been a prosperous time for Britain, for the villa estates surrounding London appear to have flourished during this period. The London Mithraeum was rededicated, probably to Bacchus. A list of the 16 "archbishops" of London was recorded by Jocelyne of Furness in the 12th century, claiming the city's Christian community was founded in the 2nd century under the legendary King Lucius and his missionary saints Fagan, Deruvian, Elvanus, and Medwin. None of that is considered credible by modern historians but, although the surviving text is problematic, either Bishop Restitutus or Adelphius at the 314 Council of Arles seems to have come from Londinium. The location of Londinium's original cathedral is uncertain. The present structure of St Peter upon Cornhill was designed by Christopher Wren following the Great Fire in 1666 but it stands upon the highest point in the area of old Londinium and medieval legends tied it to the city's earliest Christian community. In 1995, however, a large and ornate 4th-century building on Tower Hill was discovered: built sometime between 350 and 400, it seems to have mimicked St Ambrose's cathedral in the imperial capital at Milan on a still-larger scale. It was about 100 m (330 ft) long by about 50 m (160 ft) wide. Excavations by David Sankey of MOLAS established it was constructed out of stone taken from other buildings, including a veneer of black marble. It was probably dedicated to St Paul.
From 340 onwards, northern Britain was repeatedly attacked by Picts and Gaels. In 360, a large-scale attack forced the emperor Julian the Apostate to send troops to deal with the problem. Large efforts were made to improve Londinium's defenses around the same time. At least 22 semi-circular towers were added to the city walls to provide platforms for ballistae and the present state of the river wall suggested hurried repair work around this time. In 367, the Great Conspiracy saw a coordinated invasion of Picts, Gaels, and Saxons joined with a mutiny of troops along the Wall. Count Theodosius dealt with the problem over the next few years, using Londinium—then known as "Augusta"—as his base. It may have been at this point that one of the existing provinces was renamed Valentia, although the account of Theodosius's actions describes it as a province recovered from the enemy.
In 382, Magnus Maximus organized all of the British-based troops and attempted to establish himself as emperor over the west. The event was obviously important to the Britons, as "Macsen Wledig" would remain a major figure in Welsh folklore and several medieval Welsh dynasties claimed descent from him. He was probably responsible for London's new church in the 370s or 380s. He was initially successful but was defeated by Theodosius I at the 388 Battle of the Save. A new stretch of the river wall near Tower Hill seems to have been built further from the shore at some point over the next decade.
With few troops left in Britain, many Romano-British towns—including Londinium—declined drastically over the next few decades. Many of London's public buildings had fallen into disrepair by this point, and excavations of the port show signs of rapid disuse. Between 407 and 409, large numbers of barbarians overran Gaul and Hispania, seriously weakening communication between Rome and Britain. Trade broke down. Officials went unpaid and Romano-British troops elected their own leaders. Constantine III declared himself emperor over the west and crossed the Channel, an act considered the Roman withdrawal from Britain since the emperor Honorius subsequently directed the Britons to look to their own defence rather than send another garrison force. Surviving accounts are scanty and mixed with Welsh and Saxon legends concerning Vortigern, Hengest, Horsa, and Ambrosius Aurelianus. Even archaeological evidence of Londinium during this period is minimal.
Despite remaining on the list of Roman provinces, Romano-Britain seems to have dropped their remaining loyalties to Rome. Raiding by the Irish, Picts, and Saxons continued but Gildas records a time of luxury and plenty which is sometimes attributed to reduced taxation. Archaeologists have found evidence that a small number of wealthy families continued to maintain a Roman lifestyle until the middle of the 5th century, inhabiting villas in the southeastern corner of the city and importing luxuries. Medieval accounts state that the invasions that established Anglo-Saxon England (the Adventus Saxonum) did not begin in earnest until some time in the 440s and 450s. Bede recorded that the Britons fled to Londinium in terror after their defeat at the Battle of Crecganford (probably Crayford), but nothing further is said. By the end of the 5th century, the city was largely an uninhabited ruin, its large church on Tower Hill burnt to the ground.
Over the next century, Angles, Saxons, Jutes, and Frisians arrived and established tribal areas and kingdoms. The area of the Roman city was administered as part of the Kingdom of the East Saxons – Essex, although the Saxon settlement of Lundenwic was not within the Roman walls but to the west in Aldwych. It was not until the Viking invasions of England that King Alfred the Great moved the settlement back within the safety of the Roman walls, which gave it the name Lundenburh. The foundations of the river wall, however, were undermined over time and had completely collapsed by the 11th century. Memory of the earlier settlement survived: it is generally identified as the Cair Lundem counted among the 28 cities of Britain included in the History of the Britons traditionally attributed to Nennius.
The population of Londinium is estimated to have peaked around 100 AD when it was still the capital of Britannia; at this point estimates for the population vary between about 30,000, or about 60,000 people. But there seems to have been a large decline after about 150 AD, possibly as the regional economic centres developed, and Londinium as the main port for imported goods became less significant. The Antonine Plague which swept the Empire from 165 to 180 may have had a big effect. Pottery workshops outside the city in Brockley Hill and Highgate appear to have ended production around 160, and the population may have fallen by as much as two thirds.
Londonium was an ethnically diverse city with inhabitants from across the Roman Empire, including those with backgrounds from Britannia, continental Europe, the Middle East, and Africa. Recent research of human remains in Roman cemeteries states that the "presence of people born in London with African ancestry is not an unusual or atypical result for Londinium." A 2016 study of the isotope analysis of 20 bodies from various periods suggested that at least 12 had grown up locally, with four being immigrants, and the last four unclear.
Many ruins remain buried beneath London, although understanding them can be difficult. Owing to London's own geology, which consists of a Taplow Terrace deep bed of brickearth, sand, and gravel over clay, Roman gravel roads can only be identified as such if they were repeatedly relayered or if the spans of gravel can be traced across several sites. The minimal remains from wooden structures are easy to miss and stone buildings may leave foundations, but—as with the great forum—they were often dismantled for stone during the Middle Ages and early modern period.
The first extensive archaeological review of the Roman city of London was done in the 17th century after the Great Fire of 1666. Christopher Wren's renovation of St Paul's on Ludgate Hill found no evidence supporting Camden's contention that it had been built over a Roman temple to the goddess Diana. The extensive rebuilding of London in the 19th century and following the German bombing campaign during World War II also allowed for large parts of old London to be recorded and preserved while modern updates were made. The construction of the London Coal Exchange led to the discovery of the Roman house at Billingsgate in 1848. In the 1860s, excavations by General Rivers uncovered a large number of human skulls and almost no other bones in the bed of the Walbrook. The discovery recalls a passage in Geoffrey of Monmouth's pseudohistorical History of the Kings of Britain where Asclepiodotus besieged the last remnants of the usurper Allectus's army at "Londonia". Having battered the town's walls with siegeworks constructed by allied Britons, Asclepiodotus accepted the commander's surrender only to have the Venedotians rush upon them, ritually decapitating them and throwing the heads into the river "Gallemborne". Asclepiodotus's siege was an actual event that occurred in AD 296, but further skull finds beneath the 3rd-century wall place at least some of the slaughter before its construction, leading most modern scholars to attribute them to Boudica's forces. In 1947, the city's northwest fortress of the city garrison was discovered. In 1954, excavations of what was thought to have been an early church instead revealed the London Mithraeum, which was relocated to permit building over its original site. (The building erected at the time has since been demolished, and plans to return the temple to its former location are under way.) Archaeologists began the first intensive excavation of the waterfront sites of Roman London in the 1970s. What was not found during this time has been built over making it very difficult to study or discover anything new. Another phase of archaeological work followed the deregulation of the London Stock Exchange in 1986, which led to extensive new construction in the City's financial district. From 1991, many excavations were undertaken by the Museum of London's Archaeology Service, although it was spun off into the separately-run MOLA in 2011 following legislation to address the Rose Theatre fiasco.
Major finds from Roman London, including mosaics, wall fragments, and old buildings were formerly housed in the London and Guildhall Museums. These merged after 1965 into the present Museum of London near the Barbican Centre. Museum of London Docklands, a separate branch dealing with the history of London's ports, opened on the Isle of Dogs in 2003. Other finds from Roman London continue to be held by the British Museum.
Much of the surviving wall is medieval, but Roman-era stretches are visible near Tower Hill Station, in a hotel courtyard at 8–10 Coopers Row, and in St Alphege Gardens off Wood Street. A section of the river wall is visible inside the Tower. Parts of the amphitheatre are on display at the Guildhall Art Gallery. The southwestern tower of the Roman fort northwest of town can still be seen at Noble Street. Occasionally, Roman sites are incorporated into the foundations of new buildings for future study, but these are not generally available to the public.
- Note that this image includes both the garrison fort, which was demolished in the 3rd century, and the Mithraeum, which was abandoned around the same time. The identification of the "governor's palace" remains conjectural.
- Galfredus Monumetensis [Geoffrey of Monmouth]. Historia Regnum Britanniae [History of the Kings of Britain], Vol. III, Ch. xx. c. 1136. (in Latin)
- Geoffrey of Monmouth. Translated by J.A. Giles & al. as Geoffrey of Monmouth's British History, Vol. III, Ch. XX, in Six Old English Chronicles of Which Two Are Now First Translated from the Monkish Latin Originals: Ethelwerd's Chronicle, Asser's Life of Alfred, Geoffrey of Monmouth's British History, Gildas, Nennius, and Richard of Cirencester. Henry G. Bohn (London), 1848. Hosted at Wikisource.
- Haverfield, p. 145
- This etymology was first suggested in 1899 by d'Arbois de Jubainville and is generally accepted, as by Haverfield.
- Jackson, Kenneth H. (1938). "Nennius and the 28 cities of Britain". Antiquity. 12 (45): 44–55. doi:10.1017/S0003598X00013405.
- Coates, Richard (1998). "A new explanation of the name of London". Transactions of the Philological Society. 96 (2): 203–29. doi:10.1111/1467-968X.00027.
- This is the argument made by Jackson and accepted by Coates.
- Peter Schrijver, Language Contact and the Origins of the Germanic Languages (2013), p. 57.
- Ford, David Nash. "The 28 Cities of Britain" at Britannia. 2000.
- Nennius (attrib.). Theodor Mommsen (ed.). Historia Brittonum, VI. Composed after AD 830. (in Latin) Hosted at Latin Wikisource.
- Newman, John Henry & al. Lives of the English Saints: St. German, Bishop of Auxerre, Ch. X: "Britain in 429, A. D.", p. 92. Archived 21 March 2016 at the Wayback Machine James Toovey (London), 1844.
- Bishop Ussher, cited in Newman
- Encyclopædia Britannica, 11th edition. 1911.
- White, Kevan (7 February 2016). "LONDINIVM AVGVSTA". roman-britain.co.uk. Retrieved 1 February 2018.
- Merrifield, p. 61.
- Wright, Thomas (1852). The Celt, the Roman, and the Saxon: A history of the early inhabitants of Britain, down to the conversion of the Anglo-Saxons to Christianity. London: Arthur Hall, Virtue & Co. p. 95.
- Tacitus. Ab Excessu Divi Augusti Historiarum Libri [Books of History from the Death of the Divine Augustus], Vol. XIV, Ch. XXXIII. c. AD 105. Hosted at Latin Wikisource. (in Latin)
- Latin: Londinium..., cognomento quidem coloniae non insigne, sed copia negotiatorum et commeatuum maxime celebre.
- Tacitus. Translated by Alfred John Church & William Jackson Brodribb. Annals of Tacitus, Translated into English, with Notes and Maps, Book XIV, § 33. Macmillan & Co. (London, 1876. Reprinted by Random House, 1942. Reprinted by the Perseus Project, c. 2011. Hosted at Wikisource.
- Merrifield, pp. 64–66.
- Merrifield, p. 68.
- Egbert, James. Introduction to the Study of Latin Inscriptions, p. 447. American Book Co. (Cincinnati),1896.
- Latin: P·P·BR·LON [Publicani Provinciae Britanniae Londinienses] & P·PR·LON [Publicani Provinciae Londinienses]
- Wacher, p. 85.
- Labbé, Philippe & Gabriel Cossart (eds.) Sacrosancta Concilia ad Regiam Editionem Exacta: quae Nunc Quarta Parte Prodit Actior [The Sancrosanct Councils Exacted for the Royal Edition: which the Editors Now Produce in Four Parts], Vol. I: "Ab Initiis Æræ Christianæ ad Annum CCCXXIV" ["From the Beginning of the Christian Era to the Year 324"], col. 1429. The Typographical Society for Ecclesiastical Books (Paris), 1671. (in Latin)
- Thackery, Francis. Researches into the Ecclesiastical and Political State of Ancient Britain under the Roman Emperors: with Observations upon the Principal Events and Characters Connected with the Christian Religion, during the First Five Centuries, pp. 272 ff. T. Cadell (London), 1843. (in Latin) & (in English)
- "Nomina Episcoporum, cum Clericis Suis, Quinam, et ex Quibus Provinciis, ad Arelatensem Synodum Convenerint" ["The Names of the Bishops with Their Clerics who Came Together at the Synod of Arles and from which Province They Came"] from the Consilia in Thackery
- "Living in Roman London: From Londinium to London". London: The Museum of London. Retrieved 17 February 2015.
- Hingley, Introduction
- Wallace, Leslie (2015). The Origin of Roman London. Cambridge University Press. p. 9. ISBN 978-1-107-04757-0. Retrieved 16 February 2018.
- Hingley, start of Introduction
- Merrifield, p. 40.
- It may have spanned the tidal limit of the Thames at the time, with the port in tidal waters and the bridge upstream beyond its reach. This is uncertain, however: in the Middle Ages, the Thames's tidal reach extended to Staines and today it still reaches Teddington.
- Togodumnus (2011). "Londinivm Avgvsta: Provincial Capital". Roman Britain. Archived from the original on 20 February 2015. Retrieved 16 February 2015.
- Wacher, pp. 88–90.
- Number 1 Poultry (ONE 94), Museum of London Archaeology, 2013. Archaeology Data Service, The University of York.
- Antonine Itinerary. British Routes. Routes 2, 3, & 4.
- Although three of them used the same route into town.
- "Public life: All roads lead to Londinium". Museum of London Group. Retrieved 22 February 2015.
- Margary, Ivan Donald (1967). Roman Roads in Britain (2nd ed.). London: John Baker. p. 54. ISBN 978-0-319-22942-2.
- Perring, Dominic (1991). Roman London: The Archaeology of London. Abingdon: Routledge. p. 5. ISBN 978-0-415-62010-9.
- Fearnside, William Gray; Harral, Thomas (1838). The History of London: Illustrated by Views of London and Westminster. Illustrated by John Woods. London: Orr & Co. p. 15.
- Sheppard, Francis (1998). London: A History. Oxford: Oxford University Press. pp. 12–13. ISBN 978-0-19-822922-3.
- Merrifield, Ralph (1983). London, City of the Romans. Berkeley: University of California Press. pp. 116–119. ISBN 978-0-520-04922-2.
- Merrifield, pp. 32–33.
- Margary, cited by Perring, although he notes that this remains conjectural: the known roads would not meet at the river if continued in a straight line, there is no evidence textual or archaeological at the moment for a ford at Westminster, and the Saxon ford was further upstream at Kingston. Against such doubts, Sheppard notes the known routes broadly direct towards Westminster in a way "inconceivable" if they were meant to be directed towards a ferry at Londinium and Merrifield points to routes directed towards the presumed ford from Southwark. Both include maps of the known routes around London and their proposed reconstruction of major connections now-lost.
- Rowsome, Peter (2000). Heart of the City: Roman, Medieval, and Modern London Revealed by Archaeology at 1 Poultry. Museum of London Archaeology Service. p. 18. ISBN 978-1-901992-14-4.
- As by Rowsome.
- Togodumnus (2010). "The Roman Army in Britain: Roman Military Glossary". Roman Britain Online. Archived from the original on 3 March 2015. Retrieved 22 February 2015.
- Tacitus, Annals, 12.31.
- H. H. Scullard, From the Gracchi to Nero, 1982, p. 90
- John Morris, Londinium: London in the Roman Empire, 1982, pp. 107–108
- Cassius Dio, Roman History 62.2
- Tacitus, Annals, 14.31
- Merrifield, p. 53.
- "Highbury, Upper Holloway and King's Cross", Old and New London: Volume 2 (1878:273–279). Date accessed: 26 December 2007.
- Merrifield, pp. 66–68.
- "Londinium Today: Basilica and forum". Museum of London Group. Retrieved 19 February 2015.
- Merrifield, p. 62.
- Merrifield, p. 63–64.
- Will Durant (7 June 2011). Caesar and Christ: The Story of Civilization. Simon and Schuster. pp. 468–. ISBN 978-1-4516-4760-0.
- Anne Lancashire (2002). London Civic Theatre: City Drama and Pageantry from Roman Times to 1558. Cambridge University Press. p. 19. ISBN 978-0-521-63278-2.
- Marsden, Peter (1975). "The Excavation of a Roman Palace Site in London". Transactions of the London and Middlesex Archaeological Society. 26: 1–102.
- Emerson, Giles (2003). City of Sin: London in Pursuit of Pleasure. Carlton Books. pp. 24–25. ISBN 978-1-84222-901-9.
- Hall & Merrifield.
- Fields, Nic (2011). Campaign 233: Boudicca's Rebellion AD 60–61: The Britons rise up against Rome. Illustrated by Peter Dennis. Oxford: Osprey Publishing. ISBN 978-1-84908-313-3.
- Merrifield, p. 50.
- P. Marsden (1987). The Roman Forum Site in London: Discoveries before 1985. ISBN 978-0-11-290442-7.
- Merrifield, p. 68.
- According to a recovered inscription. The location of the Temple of Jupiter has not been discovered yet.
- "Londinium Today: The fort". Museum of London Group. Retrieved 18 February 2015.
- "Londinium Today: The amphitheatre". Museum of London Group. Retrieved 21 February 2015.
- Emerson, pp. 76–77.
- Roman London Fragments, Cosmetic Cream And Bikini Bottoms
- "Londinium Today: House and baths at Billingsgate". Museum of London Group. Retrieved 20 February 2015.
- Lepage, Jean-Denis G.G. (2012). British Fortifications through the Reign of Richard III: An Illustrated History. Jefferson: McFarland & Co. p. 90. ISBN 978-0-7864-5918-6.
- "Visible Roman London: City wall and gates". Museum of London Group. Archived from the original on 19 February 2015. Retrieved 19 February 2015.
- In the 1170s, William FitzStephen mentioned seven gates in London's landward wall, but it's not clear whether this included a minor postern gate or another, now unknown, major gate. Moorgate was later counted as a seventh major gate after its enlargement in 1415, but in William's time it would have been a minor postern gate.
- "Timeline of Romans in Britain". Channel4.com. Retrieved 24 August 2012.
- "Visible Roman London: Temple of Mithras". Museum of London Group. Retrieved 19 February 2015.
- Trench, Richard; Hillman Ellis (1985). London under London: a subterranean guide. John Murray (publishers) Ltd. pp. 27–29. ISBN 978-0-7195-4080-6.
- "Londinium Today: Riverside wall". Museum of London Group. Retrieved 17 February 2015.
- The medallion is named for its mint mark from Augusta Treverorum (Trier); it was discovered in Arras, France, in the 1920s.
- Giraldus Cambriensis [Gerald of Wales]. De Inuectionibus [On Invectives], Vol. II, Ch. I, in Y Cymmrodor: The Magazine of the Honourable Society of Cymmrodorion, Vol. XXX, pp. 130–31. George Simpson & Co. (Devizes), 1920. (in Latin)
- Gerald of Wales. Translated by W.S. Davies as The Book of Invectives of Giraldus Cambrensis in Y Cymmrodor: The Magazine of the Honourable Society of Cymmrodorion, Vol. XXX, p. 16. George Simpson & Co. (Devizes), 1920.
- Denison, Simon (June 1995). "News: In Brief". British Archaeology. Council for British Archaeology. Archived from the original on 13 May 2013. Retrieved 30 March 2013.
- Keys, David (3 April 1995). "Archaeologists unearth capital's first cathedral: Giant edifice built out of secondhand masonry". The Independent. London.
- Sankey, D. (1998). "Cathedrals, granaries and urban vitality in late Roman London". In Watson, Bruce (ed.). Roman London: Recent Archaeological Work. JRA Supplementary Series. 24. Portsmouth, RI: Journal of Roman Archaeology. pp. 78–82.
- Riddell, Jim. "The status of Roman London". Archived from the original on 24 April 2008.
- "Roman London: A Brief History". Museum of London. Archived from the original on 12 September 2009.
- Giles, John Allen (ed. & trans.). "The Works of Gildas, Surnamed 'Sapiens,' or the Wise" in Six Old English Chronicles of Which Two Are Now First Translated from the Monkish Latin Originals: Ethelwerd's Chronicle, Asser's Life of Alfred, Geoffrey of Monmouth's British History, Gildas, Nennius, and Richard of Cirencester. Henry G. Bohn (London), 1848.
- Habington, Thomas (trans.). The Epistle of Gildas the most ancient British Author: who flourished in the yeere of our Lord, 546. And who by his great erudition, sanctitie, and wisdome, acquired the name of Sapiens. Faithfully translated out of the originall Latine in 8 vols. T. Cotes for William Cooke (London), 1638.
- The Ruin of Britain, Ch. 22 ff, John Allen Giles's revision of Thomas Habington's translation, hosted at Wikisource.
- Jones, Michael E.; Casey, John (1988), "The Gallic Chronicle Restored: a Chronology for the Anglo-Saxon Invasions and the End of Roman Britain", Britannia, XIX (November): 367–98, doi:10.2307/526206, JSTOR 526206, retrieved 6 January 2014[permanent dead link]
- Anderson, Alan Orr (October 1912). Watson, Mrs W.J. (ed.). "Gildas and Arthur". The Celtic Review (published 1913). VIII (May 1912 – May 1913): 149–165.
- Beda Venerabilis [The Venerable Bede]. Historia Ecclesiastica Gentis Anglorum [The Ecclesiastical History of the English People], Vol. I, Ch. XV, & Vol. V, Ch. XXIIII. 731. Hosted at Latin Wikisource. (in Latin)
- Bede. Translated by Lionel Cecil Jane as The Ecclesiastical History of the English Nation, Vol. 1, Ch. 15, & Vol. 5, Ch. 24. J.M. Dent & Co. (London), 1903. Hosted at Wikisource.
- Anonymous. Translated by James Ingram. The Saxon Chronicle, with an English Translation, and Notes, Critical and Explanatory. To Which Are Added Chronological, Topographical, and Glossarial Indices; a Short Grammar of the Anglo-Saxon Language; a New Map of England during the Heptarchy; Plates of Coins, &c., p. 15., "An. CCCCLV." Longman, Hurst, Rees, Orme, & Brown (London), 1823. (in Old English) & (in English)
- The near-contemporary 452 Gallic Chronicle recorded that "The British provinces, which to this time had suffered various defeats and misfortunes, are reduced to Saxon rule" in the year 441; Gildas described a revolt of Saxon foederati but his dating is obscure; Bede dates it to a few years after 449 and opines that invasion had been the Saxons' intention from the beginning; the Anglo-Saxon Chronicle dates the revolt to 455.
- Sheppard, 35, google books
- Sheppard, 35-36
- DNA study finds London was ethnically diverse from start, BBC, 23 November 2015
- Poinar, Hendrik N.; Eaton, Katherine; Marshall, Michael; Redfern, Rebecca C. (2017). "'Written in Bone': New Discoveries about the Lives and Burials of Four Roman Londoners". Britannia. 48: 253–277. doi:10.1017/S0068113X17000216. ISSN 0068-113X.
- Janet Montgomery, Rebecca Redfern, Rebecca Gowland, Jane Evans, Identifying migrants in Roman London using lead and strontium stable isotopes, 2016, Journal of Archaeological Science
- Grimes, Ch. I.
- Camden, William (1607), Britannia (in Latin), London: G. Bishop & J. Norton, pp. 306–7
- Clark, John (1996). "The Temple of Diana". In Bird, Joanna; et al. (eds.). Interpreting Roman London. Oxbow Monograph. 58. Oxford: Oxbow. pp. 1–9.
- Grimes, William Francis (1968). "Map of the walled city of London showing areas devastated by bombing, with sites excavated by the Excavation Council". The Excavation of Roman and Mediaeval London. Routledge. ISBN 978-1-317-60471-6.
- For a map of the locations of bombed sites in the City of London excavated by the Society of Antiquaries of London's Roman and Medieval London Excavation Council during this period, see Grimes.
- Thorpe, Lewis. The History of the Kings of Britain, p. 19. Penguin, 1966.
- Galfredus Monumetensis [Geoffrey of Monmouth]. Historia Regnum Britanniae [History of the Kings of Britain], Vol. V, Ch. iv. c. 1136. (in Latin)
- Geoffrey of Monmouth. Translated by J.A. Giles & al. as Geoffrey of Monmouth's British History, Vol. V, Ch. IV, in Six Old English Chronicles of Which Two Are Now First Translated from the Monkish Latin Originals: Ethelwerd's Chronicle, Asser's Life of Alfred, Geoffrey of Monmouth's British History, Gildas, Nennius, and Richard of Cirencester. Henry G. Bohn (London), 1848. Hosted at Wikisource.
- Merrifield, p. 57.
- Morris, John. Londinium: London in the Roman Empire, p. 111. 1982.
- Grimes, Ch. II, § 2.
- "Museum of London Act 1965". legislation.gov.uk. National Archives. Retrieved 26 February 2012.
- Billings, Malcolm (1994), London: a companion to its history and archaeology, ISBN 1-85626-153-0
- Brigham, Trevor. 1998. “The Port of Roman London.” In Roman London Recent Archeological Work, edited by B. Watson, 23–34. Michigan: Cushing–Malloy Inc. Paper read at a seminar held at The Museum of London, 16 November.
- Hall, Jenny, and Ralph Merrifield. Roman London. London: HMSO Publications, 1986.
- Haverfield, F. "Roman London." The Journal of Roman Studies 1 (1911): 141–72.
- Hingley, Richard, Londinium: A Biography: Roman London from its Origins to the Fifth Century, 2018, Bloomsbury Publishing, ISBN 1350047317, 9781350047310
- Inwood, Stephen. A History of London (1998) ISBN 0-333-67153-8
- John Wacher: The Towns of Roman Britain, London/New York 1997, p. 88–111. ISBN 0-415-17041-9
- Gordon Home: Roman London: A.D. 43–457 Illustrated with black and white plates of artefacts. diagrams and plans. Published by Eyre and Spottiswoode (London) in 1948 with no ISBN.
- Milne, Gustav. The Port of Roman London. London: B.T. Batsford, 1985.
- Sheppard, Francis, London: A History, 2000, Oxford University Press, ISBN 0192853694, 9780192853691, google books
- John Timbs (1867), "Roman London", Curiosities of London (2nd ed.), J.C. Hotten, OCLC 12878129
- Wallace, Lacey M., The Origin of Roman London, 2014, Cambridge Classical Studies, Cambridge University Press, ISBN 1107047579, 9781107047570
|Wikimedia Commons has media related to Londinium.|
- Roman London, History of World Cities
- Roman London, Encyclopædia Britannica
- A map of known and conjectural Roman roads around Londinium, from London: A History
- The eastern cemetery of Roman London: excavations 1983–90, Museum of London Archive<|endoftext|>
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# Thread: AREA under the curve
1. ## AREA under the curve
1) For $x\in[-\sqrt{2},1),$the equation $(y-1)x^2-3(y+1)x+2(y-1)=0$ defines a monotonic function $y=f(x).$Find area bounded by $y=f^{-1}(x),x=-17+12\sqrt{2},x=1$ and the x-axis.
2) Let $f(x)$ be continuous and bijective such that $f(0)=0$.If $\forall t\in R$,area bounded by $y=f(x),x=a-t,x=a$ and x-axis is equal to area bounded by $y=f(x),x=a+t,x=a$ and x-axis,then prove that
$\int_{-\lambda}^{\lambda} f^{-1}(x)=2a\lambda$
3) If $f(x)=\sin x,\forall x\in[0,\frac{\pi}{2}],$ $f(x)+f(\pi-x)=2 \forall x\in\big(\frac{\pi}{2},\pi], f(x)=f(2\pi-x),\forall x\in\big(\frac{3\pi}{2},2\pi\big]$,then find the area enclosed by $y=f(x)$ and the x-axis.
2. Originally Posted by pankaj
1) For $x\in[-\sqrt{2},1),$the equation $(y-1)x^2-3(y+1)x+2(y-1)=0$ defines a monotonic function $y=f(x).$Find area bounded by $y=f^{-1}(x),x=-17+12\sqrt{2},x=1$ and the x-axis.
Rewrite the equation in the form $y=f(x)$, sketch the curve and on that sketch shade the area definded. This will allow you to express the area as an integral of $f(x)$.
CB<|endoftext|>
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Some bats will find it harder to hunt in a warmer world, says the first study of how echolocation will be affected by climate change.
Bats home in on their prey by sending out high-pitched squeaks and listening for the echo. Rising temperatures will change how far their call can penetrate and how loudly it returns, depending on the frequency of their call. This sound attenuation means some bats will be able to “hear” further as a result; but many will be all but deafened.
Holger Goerlitz of the Max Planck Institute for Ornithology in Seewiesen, Germany, and his colleagues used the output of climate models to assess how the calls of bats will be affected as the world warms.
They found that warmer climates will be good for bats that use low call frequencies – the serotine bat (pictured), for example, which squeaks at around 28 kilohertz – because their calls will travel further through the air. Conversely, those using higher frequencies, such as the soprano pipistrelle which calls at around 56 kilohertz, will suffer.
Bats already have some adaptability because they have to cope with different temperatures during the day and between seasons. But many have only a very limited frequency range, and may find it hard to adjust their call frequency or squeak louder to compensate.
The worry, says Goerlitz, is that even small changes in their ability to catch prey may put some species at a big competitive disadvantage compared to rival hunters.
Kate Barlow of the UK’s Bat Conservation Trust says conservationists have pondered how climate change will influence bats through changing temperatures and habitat, the availability of prey, and even hibernation regimes and reproduction. Warmer, wetter winters have been good for British horseshoe bats, for instance. “But adding in this additional problem makes the issue even more complicated,” she says.
Journal reference: Journal of the Royal Society Interface, DOI: 10.1098/rsif.2013.0961
More on these topics:<|endoftext|>
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# 149 in Roman Numerals
149 in Roman Numerals is written as CXLIX. Roman Numerals are the technique of expressing numbers using the Roman alphabet. Students can download the PDF of Roman Numerals 1 to 1000 free of cost to learn the Roman Numerals for the numbers 1 to 1000 in a better way. In this article, we will discuss how to convert 149 in Roman Numerals in an efficient manner.
Number Roman Numeral 149 CXLIX
## How to Write 149 in Roman Numerals?
We can write the number 149 in Roman Numerals easily using the method given below
• First, break the number 149 into the least expandable form
• 149 = 100 + (50 – 10) + (10 – 1)
• Write the respective Roman Numerals and add/subtract them
• 149 = C + (L – X) + (X – I) = CXLIX
• Therefore, the value of 149 in Roman Numerals is CXLIX
## Frequently Asked Questions on 149 in Roman Numerals
Q1
### What is 149 in Roman Numerals?
149 in Roman Numerals is CXLIX.
Q2
### Find the value of 100 + 49 in Roman Numerals.
100 + 49 = 149. To write 149 in Roman Numerals we write the number into the simplest form i.e., 149 = 100 + (50 – 10) + (10 – 1) = C + (L – X) + (X – I) = CXLIX. Hence the value of 100 + 49 in Roman Numerals is CXLIX.
Q3
### What is the value of (50 + 50) + 149 in Roman Numerals?
(50 + 50) + 149 = 249. To denote 249 in Roman Numerals, break the number 249 into the expandable form i.e., 249 = 100 + 100 + (50 – 10) + (10 – 1) = C + C + (L – X) + (X – I) = CCXLIX. Hence, the value of (50 + 50) + 149 in Roman Numerals is CCXLIX.<|endoftext|>
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### Sample Problem
64 onions subscribe to magazines. 28 of them subscribe to Time, 41 subscribe to Reader’s Digest, and 20 subscribe to Elle. There are 10 onions who subscribe to both Time and Reader’s Digest, 12 onions who subscribe to both Time and Elle, and 12 onions who subscribe to both Reader’s Digest and Elle. How many onions subscribe to all 3 kinds of magazines?
#### Solution
Draw a Venn Diagram.
Onions that subscribe to Time but not Elle: 28 − 12 = 16.
Onions that subscribe to Reader’s Digest but not Time: 41 − 10 = 31
Onions that subscribe to Elle but not Reader’s Digest: 20 − 12 = 8
Onions that subscribe to none of the three: 16 + 31 + 8 = 55.
Onions that subscribe to all three: 64 − 55 = 9.<|endoftext|>
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Unit 4: Nationalism, Industrialism, and Imperialism
Lesson A: Birth of the Imperial World
Review and Assessment
From the 18th through the 20th centuries, Westerners practiced imperialism in many areas of the world. The original motive for this imperialism was to secure natural resources. Other motives, such as nationalism, played a role in the development of this later imperialism. The early birth of imperialism laid the foundation for over 100 years of imperialist rule and eventual international conflict over colonies.
Select the link to review the pre-assessment prior to completing the lesson assessment. (Select it a second time to hide it.)
Imperialism changed both the imperialist countries and the countries they colonized. Many reasons were used to justify imperialism. Different forms of imperialism took place, and there were different responses to the imperial system. Think about these examples as you complete the BCR below.
Brief Constructed Response - Impact of Imperialism
How did imperialism impact the world during the Age of Imperialism?
- Describe the motives and processes of imperialism.
- Analyze how this system changed both the imperialistic powers and the colonized countries.
- Include details and examples to support your answer.
Download the Student Resource: Impact of Imperialism Brief Constructed Response (BCR) (doc).
Select the link to review the Social Studies Rubric (pdf).
Submit the completed BCR to your teacher as instructed.<|endoftext|>
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Engineers are experts at understanding the mechanical advantages gained by the use of simple machines. In so many everyday applications—the design of structures, machines, products and tools—simple machines make our lives and work easier. The same physical principles and mechanical advantages of simple machines used by ancient engineers to build pyramids are exploited by today's engineers to construct modern structures such as houses, bridges and skyscrapers. Simple machines and combinations of simple machines are also important and pervasive in our modern world in the form of common devices used by everyone—wheelbarrows, bicycles, crowbars, shovels, highway ramps, jackhammers, zippers, screws, jar lids, car jack, window blind controls, rock climbing gear, gym equipment, elevators, hand truck/dolly. These complex modern devices perform much work for very little power. The student pyramid building experience parallels the modern-day engineering design and construction process, which employs the engineering design process, teamwork, creativity and problem solving.
The six simple machines are introduced in Lesson 1, examined individually in more depth in Lessons 2-5, and summarized in Lesson 6. Overview of topics by lesson: 1) overview of six types of simple machine and introduction of pyramid building scenario, starting with site selection 2) wedges, 3) wheel and axle, and lever 4) inclined plane/ramp, and screw 5) pulleys 6) use the engineering design process and knowledge of six simple machines to a design/build project.<|endoftext|>
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# PROBABILITY FOR ROLLING 2 DICE
## About "Probability for rolling 2 dice"
Probability for rolling 2 dice :
Rolling two dice always plays a key role in probability concept. Whenever we go through the stuff probability in statistics, we will definitely have examples with rolling two dice.
## Sample space when 2 dice are rolled
Look at the six faced die which is given below.
The above six faced die has the numbers 1, 2, 3, 4, 5, 6 on its faces.
When a die is rolled once, the sample space is
S = {1, 2, 3, 4, 5, 6}
So, total no. of all possible outcomes = 6
When two dice are rolled,
total no. of all possible outcomes = 6 x 6 = 36
Here, the sample space is given when two dice are rolled
Let us understand the sample space of rolling two dice.
For example,
(4, 3) stands for getting "4" on the first die and and "3" on the second die.
(1, 6) stands for getting "1" on the first die and and "6" on the second die.
## Probability for rolling 2 dice - Formula
We can use the formula from classic definition to find probability when two dice are rolled.
or
Since there are 32 outcomes in total when two dice are rolled,
we have n(S) = 36
## Probability for rolling 2 dice - Practice problems
Problem 1 :
A dice is rolled twice. What is the probability of getting a difference of 2 points?
Solution :
If an experiment results in p outcomes and if the experiment is repeated q times, then the total number of outcomes is pq.
In the present case, since a dice results in 6 outcomes and the dice is rolled twice, total no. of outcomes or elementary events is 62 or 36.
We assume that the dice is unbiased which ensures that all these 36 elementary events are equally likely.
Now a difference of 2 points in the uppermost faces of the dice thrown twice can occur in the following cases :
Thus denoting the event of getting a difference of 2 points by A, we find that the no. of outcomes favorable to A, from the above table, is 8.
By classical definition of probability, we get
P(A) = 8 / 36
P(A) = 2 / 9
Problem 2 :
Two dice are thrown simultaneously. Find the probability that the sum of points on the two dice would be 7 or more.
Solution :
If two dice are thrown then, as explained in the last problem, total no. of elementary events is 62 or 36.
Now a total of 7 or more i.e. 7 or 8 or 9 or 10 or 11 or 12 can occur only in the following combinations :
Thus the no. of favorable outcomes is 21.
Letting A stand for getting a total of 7 points or more, we have
P(A) = 21 / 36
P(A) = 7 / 12
Problem 3 :
Two dice are thrown simultaneously. Find the probability of getting a doublet.
Solution :
Let us look at the sample when two dice are rolled.
In the sample space of rolling two dice, there are six cases when in which doubles are rolled.
They are
(1, 1), (2, 2), (3, 3), (4, 4), (5, 5) and (6, 6)
Letting A stand for getting a doublet, we have
P(A) = 6 / 36
P(A) = 1 / 6
After having gone through the stuff given above, we hope that the students would have understood "Probability for rolling 2 dice".
Apart from "Probability of rolling two dice", if you need any other stuff in math, please use our google custom search here.
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On September 19, 1889, part of the rocky promontory above rue Champlain at the west end of terrasse Dufferin collapsed, crushing seven houses. The toll: 35 deaths, numerous injuries, and significant property damage. This part of the Québec promontory, better known as 'cap Diamant', was one of the most dangerous inhabited areas of the region, where rockslides claimed at least 85 victims during the nineteenth century. The area around rue du Petit-Champlain has also seen some dramatic rockslides, the most memorable of which took place in 1841 and 1889.
The promontory was formed by the interplay of tectonic uplift and erosion that created the Appalachians. As a result, the layers of sedimentary rock that make up this feature now dip parallel to the slope, and in places are even vertically inclined. As a result, gravity is able to cause some layers to slide along other layers. Rockslides often occur after heavy rains, when water infiltrates the crevices, or during periods of successive freezing and thawing.
The Montmorency River is known for more than its famous falls. It is also renowned for the spectacular ice jams that regularly form along its length. During a winter or spring thaw, the ice cover breaks up into blocks that are transported by the current and pile up where the river narrows. In this way, a temporary dam can form, causing the water to rise and upstream shoreline areas to flood. This hummocky ice can sometimes reach heights of 5 to 7 m, or twice that of a house. Occasionally, an ice dam breaks suddenly under the pressure of the water behind it, sending an outburst flood surging downstream to inundate the lower reaches.
Spring ice dams on the Montmorency river, Île Enchanteresse (Courtesy of MRNQ)
Water-front residences of Île Enchanteresse threatened by the ice flow on the Montmorency river (Courtesy of MRNQ)<|endoftext|>
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# How do you convert -3.5 to degrees?
Aug 3, 2015
use the conversion $\pi = 180$
#### Explanation:
-3.5 is a radian measure which needs to be converted to degrees.
for that we need a simple conversion factor: $\pi = 180$
Now we are going to setup an equation with the conversion factor. A nice way to do this is to take advantage of the proportionality of the problem. So we set the equations up like this:
$- 3.5 = x$ in the same proportion as $\pi = 180$
$- \frac{3.5}{\pi} = \frac{x}{180}$
to solve we simply undo the division on $x$ with a multiply
$180 \cdot - \frac{3.5}{\pi} = x$
$- \frac{630}{\pi} = x$
now it is a matter of how many decimals. Let us say that we are using the standard 3.14 approximation and need two decimal places.
$- \frac{630}{3.14} = x$
$x = - 200.64$
(we changed the direction at the end to follow convention)
this should make sense because 3.5 is a little bigger than 3.14 and 200 is a little bigger than 180. And don't forget the sign!
NOTE: always put the same units in the same fraction
$\frac{r a d}{r a d} = \frac{\mathrm{de} g r e e s}{\mathrm{de} g r e e s}$
and that makes us happy. :)<|endoftext|>
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# Samacheer Kalvi 9th Maths Solutions Chapter 2 Real Numbers Ex 2.2
## Tamilnadu Samacheer Kalvi 9th Maths Solutions Chapter 2 Real Numbers Ex 2.2
9th Maths Exercise 2.2 Samacheer Kalvi Question 1.
Express the following rational numbers into decimal and state the kind of decimal expansion.
(i) $$\frac { 2 }{ 7 }$$
(ii) $$-5 \frac{3}{11}$$
(iii) $$\frac { 22 }{ 3 }$$
(iv) $$\frac { 327 }{ 200 }$$
Solution:
(i) $$\frac { 2 }{ 7 }$$
$$\frac{2}{7}=0 . \overline{285714}$$
Nen-terminating and recurring
(ii) $$-5 \frac{3}{11}$$
$$-5 \frac{3}{11}=-5 . \overline{27}$$
Nen-terminating and recurring
(iii) $$\frac { 22 }{ 3 }$$
$$\frac{22}{3}=7 . \overline{3}$$
Nen-terminating and recurring
(iv) $$\frac { 327 }{ 200 }$$
$$\frac { 327 }{ 200 }$$ = 1.635, Terminating.
9th Maths Exercise 2.2 In Tamil Question 2.
Express $$\frac { 1 }{ 13 }$$ in decimal form. Find the length of the period of decimals.
Solution:
$$\frac{1}{13}=0 . \overline{076923}$$ has the length of the period of decimals = 6.
9th Standard Maths Exercise 2.2 Question 3.
Express the rational number $$\frac { 1 }{ 13 }$$ in recurring decimal form by using the recurring decimal expansion of $$\frac { 1 }{ 11 }$$ . Hence write $$\frac { 71 }{ 33 }$$ in recurring decimal form.
Solution:
The recurring decimal expansion of $$\frac { 1 }{ 11 }$$ = 0.09090909…. = $$0.\overline { 09 }$$
Maths 9th Class Chapter 2 Real Numbers Question 4.
Express the following decimal expression into rational numbers.
(i) $$0.\overline { 24 }$$
(ii) $$2.\overline { 327 }$$
(iii) -5.132
(iv) $$3.1\overline { 7 }$$
(v) $$17.\overline { 215 }$$
(vi) $$-21.213\overline { 7 }$$
Solution:
(i) $$0.\overline { 24 }$$
Let x = $$0.\overline { 24 }$$ = 0.24242424……… ….(1)
(Here period of decimal is 2, multiply equation (1) by 100)
100x = 24.242424 ………. ….(2)
(2) – (1)
100x – x = 24.242424…. – 0.242424….
99x = 24
x = $$\frac { 24 }{ 99 }$$
(ii) $$2.\overline { 327 }$$
Let x = 2.327327327…… …………. (1)
(Here period of decimal is 3, multiply equation (1) by 1000)
1000x = 2327.327… ……………. (2)
(2) – (1)
1000x – x = 2327.327327… – 2.327327….
999x = 2325
x = $$\frac { 2325 }{ 999 }$$
(iii) -5.132
$$x=-5.132=\frac{-5132}{1000}=\frac{-1283}{250}$$
(iv) $$3.1\overline { 7 }$$
Let x = 3.1777 ……. ………… (1)
(Here the repeating decimal digit is 7, which is the second digit after the decimal point, multiply equation (1) by 10)
10x = 31.7777 …….. …………. (2)
(Now period of decimal is 1, multiply equation (2) by 10)
100x = 317.7777…….. …………….. (3)
(3) – (2)
100x – 10x = 317.777…. – 31.777….
90x = 286
$$x=\frac{286}{90}=\frac{143}{45}$$
(v) $$17.\overline { 215 }$$
Let x = 17.215215 ……. ………. (1)
1000x = 17215.215215…… …………. (2)
(2) – (1)
1000x – x = 17215.215215… – 17.215…
999x = 17198
x = $$\frac { 17198 }{ 999 }$$
(vi) $$-21.213\overline { 7 }$$
Let x = -21.2137777… ……….. (1)
10x = -212.137777…… ……….. (2)
100x = -2121.37777…… ………… (3)
1000x = -21213.77777…. ……….. (4)
10000x = 212137.77777….. ………… (5)
(Now period of decimal is 1, multiply equation (4) it by 10)
(5) – (4)
10000x – 1000x = (-212137.7777…) – (-21213.7777…)
9000x = -190924
x = –$$\frac { 190924 }{ 9000 }$$
Class 9 Maths Chapter 2 Real Numbers Question 5.
Without actual division, find which of the following rational numbers have terminating decimal expansion.
(i) $$\frac { 7 }{ 128 }$$
(ii) $$\frac { 21 }{ 15 }$$
(iii) 4$$\frac { 9 }{ 35 }$$
(iv) $$\frac { 219 }{ 2200 }$$
Solution:
(i) $$\frac { 7 }{ 128 }$$
So $$\frac{7}{128}=\frac{7}{2^{7} 5^{0}}$$
This of the form 4m, n ∈ W
So $$\frac { 7 }{ 128 }$$ has a terminating decimal expansion.
(ii) $$\frac { 21 }{ 15 }$$
So $$\frac { 21 }{ 15 }$$ has a terminating decimal expansion.
(iii) 4$$\frac { 9 }{ 35 }$$ = $$\frac { 149 }{ 35 }$$
$$\frac{49}{35}=\frac{149}{5^{1} 7^{1}}$$
∴ This is not of the form $$\frac{p}{5^{1} 7^{1}}$$
So 4$$\frac { 9 }{ 35 }$$ has a non-terminating recurring decimal expansion.
(iv) $$\frac { 219 }{ 2200 }$$
$$\frac{219}{2200}=\frac{219}{2^{3} 5^{2} 11^{1}}$$
∴ This is not of the form $$\frac{p}{2^{m} 5^{n}}$$
So $$\frac { 219 }{ 2200 }$$ has a non-terminating recurring decimal expansion.<|endoftext|>
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# 2004 AMC 12A Problems/Problem 21
## Problem
If $\sum_{n = 0}^{\infty}{\cos^{2n}}\theta = 5$, what is the value of $\cos{2\theta}$?
$\text {(A)} \frac15 \qquad \text {(B)} \frac25 \qquad \text {(C)} \frac {\sqrt5}{5}\qquad \text {(D)} \frac35 \qquad \text {(E)}\frac45$
## Solutions
### Solution 1
This is an infinite geometric series, which sums to $\frac{\cos^0 \theta}{1 - \cos^2 \theta} = 5 \Longrightarrow 1 = 5 - 5\cos^2 \theta \Longrightarrow \cos^2 \theta = \frac{4}{5}$. Using the formula $\cos 2\theta = 2\cos^2 \theta - 1 = 2\left(\frac 45\right) - 1 = \frac 35 \Rightarrow \mathrm{(D)}$.
### Solution 2
$$\sum_{n = 0}^{\infty}{\cos^{2n}}\theta = \cos^{0}\theta + \cos^{2}\theta + \cos^{4}\theta + ... = 5$$
Multiply both sides by $\cos^{2}\theta$ to get:
$$\cos^{2}\theta + \cos^{4}\theta + \cos^{6}\theta + ... = 5*\cos^{2}\theta$$
Subtracting the two equations, we get:
$$\cos^{0}\theta=5-5*\cos^{2}\theta$$
After simplification, we get $cos^{2}\theta=\frac{4}{5}$. Using the formula $\cos 2\theta = 2\cos^2 \theta - 1 = 2\left(\frac 45\right) - 1 = \frac 35 \Rightarrow \mathrm{(D)}$.<|endoftext|>
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Name: ___________________Date:___________________
Email us to get an instant 20% discount on highly effective K-12 Math & English kwizNET Programs!
### Grade 3 - Mathematics3.7 Counting by 1000s
Method: Read the numbers given. Notice the pattern in the sequence of numbers. Guess the number that should come next. Example 5000, 6000, 7000, 8000,___ Look at the numbers in the above sequence. Here the difference between one number to the next is 1000, therefore, it is skip counting by 1000. That is, to each number we have to add 1000 to get the next number. In the above example, 6000 - 5000 = 1000 7000 - 6000 = 1000 8000 - 7000 = 1000 Hence, to each number we have to add 1000 to get the next number. Answer: 9000 Example 2345, 3345, 4345, 5345,___ Here: 3345 - 2345 = 1000 4345 - 3345 = 1000 5345 - 4345 = 1000 Hence, to each number we have to add 1000 to get the next number. 5345 + 1000 = 6345 Answer: 6345 Directions: Find the missing number. Also write at least ten examples of your own.
Name: ___________________Date:___________________
### Grade 3 - Mathematics3.7 Counting by 1000s
Q 1: What is the missing number?92266, 93266, 94266, 95266, ___97266942669526696266 Q 2: What is the missing number?30722, 31722, 32722, 33722, ___34722357223372232722 Q 3: What is the missing number?95829, 96829, 97829, 98829, ___998291008299882997829 Q 4: What is the missing number?65169, 66169, 67169, 68169, ___70169691696816967169 Q 5: What is the missing number?5090, 6090, 7090, 8090, ___10090709080909090 Question 6: This question is available to subscribers only! Question 7: This question is available to subscribers only! Question 8: This question is available to subscribers only! Question 9: This question is available to subscribers only! Question 10: This question is available to subscribers only!<|endoftext|>
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Dahlia is trying to decide which bank she should use for a loan she wants to take out. in either case, the principal of the loan will be \$19,450, and dahlia will make monthly payments. bank p offers a nine-year loan with an interest rate of 5.8%, compounded monthly, and assesses a service charge of \$925.00. bank q offers a ten-year loan with an interest rate of 5.5%, compounded monthly, and assesses a service charge of \$690.85. which loan will have the greater total finance charge, and how much greater will it be? round all dollar values to the nearest cent.
## THIS IS THE BEST ANSWER 👇
To solve this, we are going to use the loan payment formula:
place
the payment
the current debt
whether the interest rate is in decimal form
is the number of payments per year
the time for years
We know from our problem that the principal of the loan, therefore, is \$ 19,450. We also know that Bank P lends nine years with an interest rate of 5.8%, multiplied monthly, so and so. Because Dahlia makes monthly payments, and its 12 months a year,. Let’s replace the values ​​in our formula:
Now we know that Dahlia’s monthly payment is \$ 231.59. Knowing that she is going to make 12 monthly payments for 9 years, we can calculate the value of the future loan by multiplying the amount of monthly payments (\$ 231.59) by the number of payments monthly (12) by the number of years (9):
Now we know that she is about to pay \$ 25,011.72 for her loan. Finally, to calculate the total finance charge, we subtract the original loan (\$ 19,450) from the value of the future loan (\$ 25,011.72), and then we add the service charge (\$ 925.00):
The total financial charge of P bank is \$ 6,486.72.
We are going to repeat the same procedure as before.
,,, and. Let’s replace the values ​​in our formula:
Now that we have our monthly payment, we can calculate the value of the future loan by multiplying the amount of monthly payments (\$ 211.08) by the number of monthly payments (12) by the number of years (10):
Just as before, to calculate the total finance charge, we are going to subtract the original loan (\$ 19,450) from the value of the future loan (\$ 25,329.6), and then we are going to add the service charge (\$ 690.85) with:
The total financial charge of bank Q is \$ 6570.45.
Note that the financial charge on prohibition Q is greater than the financial charge of bank P, so we are going to subtract the financial charge of bank Q from the bank charge of bank P:
We can conclude that the financial charge of Loan Q will be \$ 83.73 more than Loan P. Therefore, the correct answer is to
Step by step explanation:<|endoftext|>
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- If you exercise on a hot day, you are likely to lose a lot of water in sweat. Then, for the next several hours, you may notice that you do not pass urine as often as normal and that your urine is darker than usual.
- Do you know why this happens? Your body is low on water and trying to reduce the amount of water lost in urine. The amount of water lost in urine is controlled by the kidneys, the main organs of the excretory system.
- Excretion is the process of removing wastes and excess water from the body. It is one of the major ways the body maintains homeostasis.
- Although the kidneys are the main organs of excretion, several other organs also excrete wastes.
- They include the large intestine, liver, skin, and lungs. All of these organs of excretion, along with the kidneys, make up the excretory system.
- The roles of the other excretory organs are summarized below:
- The large intestine eliminates solid wastes that remain after the digestion of food.
- The liver breaks down excess amino acids and toxins in the blood.
- The skin eliminates excess water and salts in sweat.
- The lungs exhale water vapor and carbon dioxide.
- The kidneys are part of the urinary system, which is shown in Figure below.
- The main function of the urinary system is to filter waste products and excess water from the blood and excrete them from the body.
Kidneys and Nephrons
- The kidneys are a pair of bean-shaped organs just above the waist. A cross-section of a kidney is shown in Figure below.
- The function of the kidney is to filter blood and form urine.
- Urine is the liquid waste product of the body that is excreted by the urinary system. Nephrons are the structural and functional units of the kidneys.
- A single kidney may have more than a million nephrons!
Filtering Blood and Forming Urine
- As shown in Figure below, each nephron is like a tiny filtering plant. It filters blood and forms urine in the following steps:
- Blood enters the kidney through the renal artery, which branches into capillaries. When blood passes through capillaries of the glomerulus of a nephron, blood pressure forces some of the water and dissolved substances in the blood to cross the capillary walls into Bowman’s capsule.
- The filtered substances pass to the renal tubule of the nephron. In the renal tubule, some of the filtered substances are reabsorbed and returned to the bloodstream. Other substances are secreted into the fluid.
- The fluid passes to a collecting duct, which reabsorbs some of the water and returns it to the bloodstream. The fluid that remains in the collecting duct is urine.
Excretion of Urine
- From the collecting ducts of the kidneys, urine enters the ureters, two muscular tubes that move the urine by peristalsis to the bladder (see Figure above). The bladder is a hollow, sac-like organ that stores urine.
- When the bladder is about half full, it sends a nerve impulse to a sphincter to relax and let urine flow out of the bladder and into the urethra.
- The urethra is a muscular tube that carries urine out of the body. Urine leaves the body through another sphincter in the process of urination.
- This sphincter and the process of urination are normally under conscious control.
Kidneys and Homeostasis
- The kidneys play many vital roles in homeostasis. They filter all the blood in the body many times each day and produce a total of about 1.5 liters of urine.
- The kidneys control the amount of water, ions, and other substances in the blood by excreting more or less of them in urine.
- The kidneys also secrete hormones that help maintain homeostasis. Erythropoietin, for example, is a kidney hormone that stimulates bone marrow to produce red blood cells when more are needed.
- The kidneys themselves are also regulated by hormones. For example, antidiuretic hormone from the hypothalamus stimulates the kidneys to produce more concentrated urine when the body is low on water.
Kidney Disease and Dialysis
- A person can live a normal, healthy life with just one kidney. However, at least one kidney must function properly to maintain life.
- Diseases that threaten the health and functioning of the kidneys include kidney stones, infections, and diabetes.
- Kidney stones are mineral crystals that form in urine inside the kidney. They may be extremely painful. If they block a ureter, they must be removed so urine can leave the kidney and be excreted.
- Bacterial infections of the urinary tract, especially the bladder, are very common. Bladder infections can be treated with antibiotics prescribed by a doctor. If untreated, they may lead to kidney damage.
- Uncontrolled diabetes may damage capillaries of nephrons. As a result, the kidneys lose much of their ability to filter blood. This is called kidney failure.
- The only cure for kidney failure is a kidney transplant, but it can be treated with dialysis. Dialysis is a medical procedure in which blood is filtered through a machine.
- JPSC Mains Tests and Notes Program
- JPSC Prelims Exam 2018- Test Series and Notes Program
- JPSC Prelims and Mains Tests Series and Notes Program
- JPSC Detailed Complete Prelims Notes<|endoftext|>
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# Finish Line & Beyond
## • Basic Terms and Definitions
• Intersecting Lines and Non-intersecting Lines
• Pairs of Angles
• Parallel Lines And A Transversal
• Lines Parallel To The Same Line
• Angle Sum Property of A Triangle
(a) Segment: - A part of line with two end points is called a line-segment.
## A ray is denoted by AB.
We can denote a line-segment AB, a ray AB and length AB and line AB by the
same symbol AB.
(c) Collinear points: - If three or more points lie on the same line, then they
are called
collinear points, otherwise they are called non-collinear points.
(c) Angle: - An angle is formed by two rays originating from the same end point.
The rays making an angle are called the arms of the angle and the end-points are
called the vertex of the angle.
(d)Types of Angles:-
(i) Acute angle: - An angle whose measure lies between 0° and 90°, is
called an acute angle.
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(ii) Right angle: - An angle, whose measure is equal to 90°, is called a right
angle.
(iii) Obtuse angle: - An angle, whose measure lies between 90° and 180°, is
called an obtuse angle.
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## (iv) Straight angle: - The measure of a straight angle is 180°.
(v) Reflex angle: - An angle which is greater than 180° and less than 360°,
is called the reflex angle.
(vi) Complimentary angle: - Two angles, whose sum is 90°, are called
complimentary angle.
(vii) Supplementary angle: - Two angles whose sum is 180º, are called
supplementary angle.
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(viii) Adjacent angle: - Two angles are adjacent, if they have a common
vertex, a
common vertex, common arm and their non-common arms are on
different sides of the common arm.
In the above figure ∠ABD and ∠DBC are adjacent angle. Ray BD is their
common arm and point B is their common vertex. Ray BA and ray BC are non-
common arms.
When the two angles are adjacent, then their sum is always equal to the
angle formed by the two non-common arms.
## Thus, ∠ABC = ∠ABD + ∠DBC .
Here we can observe that ∠ABC and ∠DBC are not adjacent angles, because
their non-common arms BD and AB lie on the same side of the common arm BC.
(ix) Linear pair of angles: - If the sum of two adjacent angles is 180º, then
their non-common lines are in the same straight line and two adjacent
angles form a linear pair of angles.
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In the fig. ∠ABD and ∠CBD form a linear pair of angles because
## (x) Vertically opposite angles: - When two lines AB and CD intersect at a
point O, the vertically opposite angles are formed.
Here are two pairs of vertically opposite angles. One pair is ∠AOD and ∠BOC
and the second pair is ∠AOC and ∠BOD
The vertically opposite angles are always equal.
## So, ∠AOD = ∠BOC and ∠AOC = ∠BOD
(e) Intersecting lines and non-intersecting lines: - Two lines are intersecting
if they have one point in common. We have observed in the above figure that
lines AB and CD are intersecting lines, intersecting at O, their point of
intersection.
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## Parallel lines: - If two lines do not meet at a point if extended to both
directions, such lines are called parallel lines.
P Q
R S
Lines PQ and RS are parallel lines.
## The length of the common perpendiculars at different points on these parallel
lines is same. This equal length is called the distance between two parallel
lines.
Axiom 1. If a ray stands on a line, then the sum of two adjacent angles so formed is
180º.
Conversely if the sum of two adjacent angles is 180º, then a ray stands
on a line
(i.e., the non-common arms form a line).
Axiom 2. If the sum of two adjacent angles is 180º, then the non-common arms of
the angles form a line. It is called Linear Pair Axiom.
(f) Theorem 1. If two lines intersect each other, then the vertically
opposite angles are equal.
Sol. Given: Two lines AB and CD intersect each other at O.
To Prove: -
∠AOC = ∠BOD
and ∠AOD = ∠BOC
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Proof: -
⇒ ∠AOC = ∠BOD
Now, Again
## ⇒ ∠BOC + ∠BOD = ∠AOD + ∠BOD
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⇒ ∠BOC = ∠AOD
Hence Proved.
## Parallel Lines And A Transversal
In the above figure m and n are two parallel lines and l is the transversal, which
intersect the parallel line m and n at points P and Q respectively.
## Here Exterior Angles are: - ∠1 , ∠ 2 , ∠ 7 and ∠8
Interior Angles are: - ∠3 , ∠ 4 , ∠5 and ∠ 6
## (i) ∠1 and ∠5 (ii) ∠ 2 and ∠ 6 (iii) ∠ 4 and ∠8 (iv) ∠3 and ∠ 7
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## If a transversal intersects two parallel lines, then each pair of corresponding
angle is equal. (Corresponding Angles Axiom) – Axiom 3.
## Axiom 4. If a transversal intersects two lines such that a pair of
corresponding angles is equal, then the two lines are parallel to each
other.
## Alternate Exterior Angles: - (i) ∠1 and ∠ 7 (ii) ∠ 2 and ∠8
If a transversal intersects two parallel lines, then each pair of alternate interior
and exterior angles are equal.
## Alternate Exterior Angles: - (i) ∠1 = ∠7 (ii) ∠ 2 = ∠8
Interior angles on the same side of the transversal line are called the consecutive
interior angles or allied angles or co-interior angles.
## Theorem 2. If a transversal intersects two parallel lines, then each pair
of alternate interior angles is equal.
Sol. Given: Let PQ and RS are two parallel lines and AB be the transversal which
intersects them on L and M respectively.
## To Prove: - ∠PLM = ∠SML
And ∠LMR = ∠MLQ
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## Proof: - ∠PLM = ∠RMB ……….equation (i) {corresponding angle}
And ∠RMB = ∠SML ……….equation (ii) {vertically opposite angle are equal}
∠PLM = ∠SML
Similarly,
## ∠LMR = ∠ALP ……….equation (iii) {corresponding angle}
And ∠ALP = ∠MLQ ……….equation (iv) {vertically opposite angle are equal}
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∠LMR = ∠MLQ
Hence Proved.
## Theorem 3. If a transversal intersects two lines such that a pair of alternate
interior angles is equal, then the two lines are parallel.
## Sol. Given: - A transversal AB intersects two lines PQ and RS such that
∠PLM = ∠SML .
To Prove: - PQ║RS
## From equations (i) & (ii),
∠PLM = ∠RMB
But these are corresponding angles.
We know that if a transversal intersects two lines such that a pair of corresponding
angles is equal, then the two lines ate parallel to each other.
## Hence, PQ║RS Proved.
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## Theorem 4. If a transversal intersects two parallel lines, then each pair of
interior angles on the same side of the transversal is supplementary.
Solution:
respectively.
## From equations (i) & (ii),
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∠ 2 + ∠3 = 180º
∠1 + ∠ 4 = 180º
## Theorem 5. If a transversal intersects two lines such that a pair of interior
angles on the same side of the transversal is supplementary, then the two
lines are parallel.
Sol.
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∠1 + ∠ 2 = 180º
To Prove: AB║CD
## From equations (i) & (ii),
∠1 + ∠ 2 = ∠1 + ∠3
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⇒ ∠1 + ∠ 2 - ∠1 = ∠3
⇒ ∠ 2 = ∠3
But these are alternate interior angles. We know that if a transversal intersects two
lines such that the pair of alternate interior angles are equal, then the lines are
parallel.
## Hence, AB║CD Proved.
Theorem 6. Lines which are parallel to the same line are parallel to each
other.
Sol.
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Given: - Three lines AB, CD and EF are such that AB║CD, CD║EF.
To Prove: - AB║EF.
Construction: - Let us draw a transversal GH which intersects the lines AB, CD and
EF at P, Q and R respectively.
∠1 = ∠3
## But these are corresponding angles.
We know that if a transversal intersects two lines such that a pair of corresponding
angles is equal, then the two lines ate parallel to each other.
## Angle Sum Property of Triangle: -
Theorem 7. The sum of the angles of a triangle is 180º.
Sol.
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Given: - A ∆ ABC.
To Prove: - ∠1 + ∠ 2 + ∠3 = 180º.
## By adding equation (i) & (ii),
∠1 + ∠ 2 = ∠ 4 + ∠5 ………..equation (iii)
## Now by adding ∠3 to both sides of equation (iii), we get
∠1 + ∠ 2 + ∠3 = ∠ 4 + ∠5 + ∠3
## ∠ 4 + ∠5 + ∠3 = 180º {Linear Pair}
∴ ∠1 + ∠ 2 + ∠3 = 180º
Hence Proved.
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## Theorem 8. If a side of a triangle is produced, then the exterior angle so
formed is equal to the sum of the two interior opposite angles.
Sol.
## Given: - A ∆ ABC in which side BC is produced to D forming exterior angle ∠ACD of
∆ ABC.
To Prove: - ∠ 4 = ∠1 + ∠ 2 .
## From equations (i) & (ii),
∠1 + ∠ 2 + ∠3 = ∠3 + ∠ 4
⇒ ∠1 + ∠ 2 + ∠3 - ∠3 = ∠ 4
⇒ ∠1 + ∠ 2 = ∠ 4
Hence, ∠ 4 = ∠1 + ∠ 2 Proved.
Finish Line & Beyond
EXERCISE 1
Q1. In the Fig. lines AB and CD intersect at O. If ∠AOC + ∠BOE = 70º AND
∠BOD = 40º, find ∠BOE and reflex ∠COE .
## ⇒ ∠BOE = 70º - 40º
⇒ ∠BOE = 30º
∴ ∠BOE = 30º
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⇒ ∠COE = 110º
## And Reflex ∠COE = 250º
Q2. In the following figure, lines XY and MN intersect at O. If ∠POY = 90º and a : b
=
2 : 3, find c.
## Sol. Given: - ∠POY = 90º
And a : b = 2 : 3.
a 2
∴ =
b 3
2
⇒ a= b ………….equation (i)
3
## Now, ∠POX + ∠POY = 180º
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## ⇒ ∠POX = 180º - 90º
⇒ ∠POX = 90º
⇒ a + b = 90º { ∠POX = a + b}
2
⇒ b + b = 90º
3
2b + 3b
⇒ = 90º
3
⇒ 2b + 3b = 90º ×3
⇒ 5b = 270º
270°
⇒ b=
5
⇒ b = 54º
2
a= b
3
2
⇒ a= x 54º
3
⇒ a = 2 x 18º
⇒ a = 36º
## Now, b + c = 180º {Angles at a common point on a line}
⇒ 54º + c = 180º
⇒ c = 180º- 54º
⇒ c = 126º
So, c = 126º.
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Q3. In the figure, ∠PQR = ∠PRQ , then prove that ∠PQS = ∠PRT .
## ⇒ ∠PQR + ∠PQS - ∠PQR = ∠PRT
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⇒ ∠PQS = ∠PRT
Given: - x+y=w+z
## But, x + y + w + z = 360º {Angles around a point}
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⇒ (x + y) + (w + z) = 360º
## ⇒ (x + y) + (x + y) = 360º {from equation (i)}
⇒ x + y + x + y = 360º
⇒ 2x + 2y = 360º
⇒ 2(x + y) = 360º
360°
⇒ x+y=
2
⇒ x + y = 180º
## Hence, AOB is a straight line. Proved.
Q5. In the given figure, POQ is a line. Ray OR is perpendicular to line PQ. OS is
another ray lying between rays OP and OR.
1
Prove that ∠ROS = ( ∠QOS - ∠POS )
2
P O Q
OR ⊥ PQ
## Ray OS meets line PQ at O.
1
To Prove: - ∠ROS = ( ∠QOS - ∠POS )
2
## Proof: - ∠QOS = ∠ROS + ∠ROQ …………….equation (i)
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## ⇒ ∠QOS - ∠POS = 2 ∠ROS
1
⇒ ( ∠QOS - ∠POS ) = ∠ROS
2
1
⇒ ∠ROS = ( ∠QOS - ∠POS )
2
1
Hence, ∠ROS = ( ∠QOS - ∠POS ) Proved.
2
Q6. It is given that ∠XYZ = 64º and XY is produced to a point P. Draw a figure from
the given information. If ray YQ bisects ∠ZYP , find ∠XYQ and reflex ∠QYP .
Sol.
## ⇒ 64º + ∠ZYP = 180º
Finish Line & Beyond
⇒ ∠ZYP = 116º
## Since, YQ bisects ∠ZYP .
1 116°
∴ ∠ZYQ = ∠PYQ = ∠ZYP = = 58°
2 2
## Now reflex ∠QYP = 360º - 58° = 302º.
EXERCISE 2
1. In the following figure find the value of x and y, the show that AB║CD.
## ∠APM + ∠APN = 180° ( Angles on the same side of a line)
⇒ ∠APN = 180° − 50° = 130° = x ----------------------(1)
Finish Line & Beyond
## ∠CQN = ∠DQM (Opposite Angles)
Now,
⇒ ∠DQM = 130° = y ------------------------------------(2)
## As we have seen ∠APN = ∠CQN
So by the theorem of corresponding angles on one side of the transversal it is clear
that
AB║CD
2. In the following figure if AB║CD and CD║EF and y:z=3:7, find the value of x
## Now ∠DPO + ∠CPO = 180°
⇒ 3 x + 7 x = 180°
⇒ 10x = 180°
⇒ x = 18°
Putting the value of x in the given ratio we get following values:
Finish Line & Beyond
∠DPO = 126°
∠CPO = 54°
## Now it is given that AB║CD,
So, ∠DPO = ∠AOP = 126° = x
## 3. In the following figure if AB║CD , EF ⊥ CD and ∠GED = 126° , find
∠AGE , ∠GEF , and∠FGE
## ∠GEF = ∠GED − ∠FED
⇒ ∠GEF = 126° − 90° = 36°
As AB║CD,
So, ∠EFG =∠FED = 90°
⇒ ∠FGE = 180° − (90° + 36°) = 54°
## Now,∠AGE + ∠FGE = 180° (Angles on the same side of a line)
⇒ ∠AGE = 180° − 54° = 126°
4. In the following figure PQ║ST, values of ∠PQR = 110° and ∠RST = 130° , find the
value of ∠QRS .
Finish Line & Beyond
## Now, ∠RST + ∠BRS = 180°
And, ∠PQR + ∠ARQ = 180°
Because, internal angles on the one side of the transversal are complementary
angles.
Hence, ∠BRS = 180° − 130° = 50°
∠ARQ = 180° − 110° = 70°
Now, it is clear that ∠ARQ + ∠QRS + ∠BRS = 180°
⇒ 70° + ∠QRS + 50° = 180°
⇒ ∠QRS = 60°
5. In the following figure AB║CD, ∠APQ = 50° and ∠PRD = 127° , find values of x
and y.
## Answer: ∠BPR + ∠PRD = 180° (Internal Angles on one side of transversal)
⇒ ∠BPR = 180° − 127° = 53°
Finish Line & Beyond
## ∠PRD + PRQ = 180°
On the line CD
⇒ ∠PRQ = 180° − 127° = 53°
## ∠APQ + ∠QPR + ∠BPR = 180°
On the line AB,
⇒ ∠QPR = 180° − (50° + 53°) = 77°
## In ΔPQR, ∠PQR + ∠QPR + ∠PRQ = 180° (Sum of angles of a Triangle)
⇒ ∠PQR = 180° − (77° + 53°) = 50°
⇒ x = 50° and y = 77°
6. In the given figure PQ and RS are two mirrors placed parallel to each other. An
incident ray AB strikes the mirror PQ, the reflected ray moves along the path BC and
strikes the mirror RS. The second mirror reflects the ray along CD. Prove that
AB║CD.
Answer: From the theory of reflection in Physics we know that angle of incidence is
equal to angle of reflection.
Here, In the case of mirror PQ, Angle of incidence i = ∠ABP
And angle of reflection r = ∠QBC
## In the case of mirror RS, Angle of incidence i = ∠BCR
And angle of reflection r = ∠SCD
## Required evidence to prove AB║CD
We need to check if ∠ABC = ∠BCD (Alternate angles)
## On line PQ,∠ABP + ∠ABC + ∠QBC = 180°
⇒ i + ∠ABC + r = 180°
⇒ i + i + ∠ABC = 180° ………………………..(1)
Similarly, on line RS it can be observed that
i + i + ∠BCD + 180° ……………………………(2)
## From the question it is given that PQ║RS
Hence, ∠QBC = ∠BCR (Alternate Angles)
Finish Line & Beyond
## Hence, values of angles of incidence for both mirrors are same.
Correlating this finding with equations (1) and (2) it is clear that
∠ABC = ∠BCD
So, AB║CD Proved.
## Theorem: The sum of the angles of a triangle is 180º.
Construction: Let us draw a triangle PQR and draw a line XY║QR so that it touches
the vertex P of the triangle. For convenience let us name angles as 1, 2, 3, 4 and 5.
## Proof:∠1 = ∠ 4 (Alternate angles)
∠3 = ∠5 (Alternate angles)
∠1 + ∠ 2 + ∠3 = 180°
## Substituting the values of ∠1 and ∠ 2 we get
∠ 4 + ∠ 2 + ∠5 = 180°
## Theorem: If a side of a triangle is produced, then the exterior angle so
formed is equal to the sum of the two interior opposite angles.
Construction: Let us construct a triangle PQR and extend it base to S. Let us name
angles as 1, 2, 3 and 4 for convenience.
Finish Line & Beyond
Required Proof: ∠ 4 = ∠1 + ∠ 2
## Evidence: From earlier theorem we know that,
∠1 + ∠ 2 + ∠3 = 180° (Sum of angles of a triangle) ………………..(1)
On line QS,
∠3 + ∠ 4 = 180° (Angles on the same side of a line) …………………..(2)
## From equation (1) and (2) it is clear,
∠ 4 = ∠1 + ∠ 2
Exercise 3:
1. In the following figure, sides QP and RQ of ΔPQR are produced to points S and T
respectively. If ∠SPR = 135° and ∠PQT = 110° , find the value of ∠PRQ
## Answer: On line QS, ∠QPR + ∠SPR = 180°
⇒ ∠QPR = 180° − 135° = 45°
Similarly on line TR, ∠TQP + PQR = 180°
Finish Line & Beyond
## ⇒ ∠PQR = 180° − 110° = 70°
Now we have values of two angles of the given triangle so value of the third angle
can be calculated as follows:
∠PRQ = 180° − (70° + 45°) = 65°
2. In the following figure ∠X = 62° and ∠XYZ = 54° . If YO and ZO are the bisectors
of ∠XYZ and ∠XZY respectively of ΔXYZ, find values of ∠OZY and ∠YOZ
∠XYZ + ∠YXZ + ∠XZY = 180°
⇒ ∠XZY = 180° − (62° + 54°) = 64°
As per the question YO and ZO are bisectors of ∠XYZ and ∠XZY respectively
1
Hence, ∠OYZ = ∠XYZ = 54 ÷ 2 = 27°
2
1
And, ∠OZY = ∠XZY = 64 ÷ 2 = 32°
2
## Now, for ΔOYZ,
∠YOZ = 180° − (∠OYZ + ∠OZY )
=180°-(27°+32°)=121°
Requires answers are 32° and 121°
3. In the following figure AB║DE, ∠BAC = 35° and ∠CDE = 53° , find the value of
∠DCE
Finish Line & Beyond
## In ΔDCE, ∠DCE + ∠CDE + ∠CED = 180° (Sum of angles of a triangle)
Hence, ∠DCE = 180° − (53° + 35° ) = 92°
4. In the following figure lines PQ and RS intersect at point T, such that ∠PRT = 40°
, ∠RPT = 95° and ∠TSQ = 75° . Find the value of ∠SQT .
∠PRT + ∠RPT + ∠PTR = 180°
⇒ ∠PTR = 180° − (95° + 40°) = 45°
## As we know opposite angles are equal so ∠PTR = ∠STQ = 45°
Now, in ΔQST,
Finish Line & Beyond
## ∠QST + ∠STQ + ∠SQT = 180°
⇒ ∠SQT = 180° − (75° + 45°) = 60°
5. In the following figure, PQ ⊥ PS, PQ║SR, ∠SQR = 28° and ∠QRT = 65° . Find the
values of x and y.
## Answer: On the line ST,
∠QRS + ∠QRT = 180°
⇒ ∠QRS = 180° − 65° = 115°
In ΔQRS,
∠QSR + ∠SQR + ∠QRS = 180°
∠QSR = 180° − (28° + 115°) = 27°
## Now,∠SPQ = ∠PSR (Complementary Angles on the inner side of transversal)
∴ ∠PSQ = 90° − 27° = 63°
In ΔSPQ,
∠PQS = 180° − (∠SPQ + ∠PSQ)
=180°-(90°+63°) = 27°
So, x=27°
Y=63°
## 6. In the following figure, the side QR of ΔPQR is produced to a point S. If the
1
bisectors of ∠PQR and ∠PRS meet at point T, then prove that ∠QTR = ∠QPR
2
Finish Line & Beyond
## ∠SRP = ∠QPR + ∠PQR
1 1 1
∴ ∠SRP = ∠QPR + ∠PQR
2 2 2
1 1 1
or, ∠QPR = ∠SRP − ∠PQR …………………………..(1)
2 2 2
In Δ TQR
∠SRT = ∠QTR + ∠TQR
1 1
Or, ∠SRP = ∠QTR + ∠PQR
2 2
1 1
Or, ∠QTR = ∠SRP − ∠PQR ……………………………..(2)
2 2
As RHS of both equations are same So, following can be written:
1
∠QTR = ∠QPR
2<|endoftext|>
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Rules, Recognize, Report, Respond, Refuse, Replace. What do these 6 R’s have in common? Bullying prevention, of course!
The Essential 6 R’s of Bullying Prevention
[PDF] is a free resource and great guide for creating a safe learning environment. Author Dr. Michele Borba, Ed.D., stresses that bullying prevention is not a quick fix, but an ongoing process. Her solutions are unique in that they help several people within the bullying cycle: The bully, the bullied, and the bystanders.
Here are a few highlights from Dr. Borba’s guide to help you get started:
- Know where you are now and set goals.
Evaluate yourself, your staff and coworkers, and your school’s climate. Reflect on what bullying prevention looks like now and set a goal of what it should look like. This can help you develop next steps. (See page 7 of the guide for more details.)
- Conduct an informal survey.
This will help you get an idea of your students’ opinions about bullying at your school. Your survey should include all students in order to reach the bullies, the bullied, and the bystanders. (See page 13 for an example survey.)
- Identify patterns and hot spots.
From your survey, you may identify specific places or times that bullying takes place. This can help your school create a bullying hot-spots map. Understanding where and when bullying takes place can help teachers, administrators, and other supervisors be extra vigilant about high-frequency areas. (See page 15 for an example image of bullying hot spots.)
These are just a few ways to get started with the 6 R’s. It’s important to keep in mind that bullying prevention is a continuous effort and often involves a culture change within your school’s environment. Take a look at the complete guide to learn more, and be sure to head on over to 31 for 31 and check out our School Bullying Prevention Difference Makers<|endoftext|>
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There's enough partially molten rock teeming beneath Yellowstone's surface to fill the Grand Canyon eleven times -- with some left over.
University of Utah seismologists say they've produced a full rendering of the famed Yellowstone supervolano's underground plumbing for the first time, and have revealed, in the process, that its deep magma reservoir is nearly four and half times larger than the smaller, shallower chamber we already knew about.
Scientists have suspected for awhile now that a vast, deep chamber, but this is the first time anyone's been able to prove it. The researchers used a technique called seismic tomography -- sort of like a CT scan that substitutes earthquake waves for x-rays -- to measure rock density and map out the reservoir. Their study, published in the journal Science, provides a comprehensive understanding of how the whole system works:
The green lines represent the boundary of Yellowstone National Park. The study focused on Earth's crust, shows the previously known magma chamber (orange) about 3 to 9 miles beneath the surface, and reveals a previously unknown magma reservoir (red) at a depth of 12 to 28 miles. Beneath that is the Yellowstone hotspot plume (yellow) which brings hot rock up from deep within Earth's mantle (Hsin-Hua Huang, University of Utah Department of Geology and Geophysics).
The researchers say this new information explains, at long last, why Yellowstone is such a major emitter of carbon dioxide. It can also, they say, help broaden our understanding of the potential risks of earthquakes and, in the very, very worst case scenario, eruptions. It's happened three times before, the last time 640,000 years ago, but the researchers say the risk of it happening again in our lifetimes remains extremely low -- scientists with the U.S. Geological Survey say there's a 99.9 percent chance that the volcano will remain dormant throughout the 21st century.<|endoftext|>
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# 1.3 Algebraic Expressions
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1 1.3 Algebraic Expressions A polynomial is an expression of the form: a n x n + a n 1 x n a 2 x 2 + a 1 x + a 0 The numbers a 1, a 2,..., a n are called coefficients. Each of the separate parts, such as a 3 x 3 or a 1 x, is called a term of the polynomial. If there is only one term, it is called a monomial. For two terms, it is called a binomial. For three terms, a trinomial. The degree of the polynomial is n: the highest power of x. To add (or subtract) polynomials, we combine like terms (those that have the same variables raised to the same powers). Example: Find the difference (x 3 4x 2 + 6x + 12) (2x 3 + x 2 4x 6) To multiply polynomials, we use the distributive laws. In particular, to multiply two binomials, we use the FOIL method. Example: Multiply (x 7)(2x + 3)(x 1) Special Product and Factoring Formulas 1. Difference of Squares (A + B)(A B) = A 2 B 2 2. Perfect Squares (a) (A + B) 2 = A 2 + 2AB + B 2 (b) (A B) 2 = A 2 2AB + B 2 3. Cubing a Sum or Difference (a) (A + B) 3 = A 3 + 3A 2 B + 3AB 2 + B 3 (b) (A B) 3 = A 3 3A 2 B + 3AB 2 B 3 4. Sum or Difference of Cubes (a) A 3 + B 3 = (A + B)(A 2 AB + B 2 ) (b) A 3 B 3 = (A B)(A 2 + AB + B 2 ) 1
2 Examples: Expand the following. (2 x + 3)(2 x 3) (x 3 4) 2 (x 2 2) 3 Factoring Steps 1. Factor out all common factors. 2. See if you can use a special factoring formula. 3. See if you can factor by grouping. 4. Use trial and error. Examples: Factor the following. 8x 2 24x x 2 25 x
3 2x x 21 x 3 + 3x 2 x 3 (a 2 + 2a) 2 2(a 2 + 2a) 3 To factor expressions with rational exponents, first factor out, if possible, the smallest power of x. Factor x 3/2 + 2x 1/2 + x 1/2 Its easy to check if you factored correctly. Just multiply back out to double check. 3
4 1.4 Rational Expressions A rational expression is a fractional expression where both the numerator and denominator are polynomials. The domain of any algebraic expression is the set of values that the variable can be. So far, we have two things to look for to determine the domain: The denominator can t be zero. If a value of x makes the denominator zero, we must exclude it from the domain. For even powered roots (square roots, fourth roots, etc), whatever is under the radical must be 0. Examples: Find the domains of the following rational expressions. 1. x 2x 3 2. x x x 2 + 3x x + 3 Working with rational expressions is just like working with fractions. To multiply rational expressions, factor the numerator and denominator, multiply, and then simplify by cancelling common factors in the numerator and denominator. To divide rational expressions, multiply by the reciprocal. To add or subtract rational expressions, you MUST have a COMMON DENOMINATOR (just like with fractions). Examples x 2 1 x 2 7x + 12 x2 x 6 x + 1 4
5 4y 2 9 2y 2 + 9y 18 2y2 + y 3 y 2 + 5y 6 x x 2 + x 2 2 x 2 5x + 4 Compound fractions are fractional expressions where the numerator and/or denominator are themselves fractional expressions. Examples: x x+y x y + y x 3(1 + x)1/3 x(1 + x) 2/3 (1 + x) 2/3 5
6 Rationalizing the Denominator or Numerator As before, rationalizing means to get rid of any radicals. We do this by multiplying by the conjugate. The conjugate is usually found by just changing the sign of the second term. For example, the conjugate of A + B C is A B C. What s the point? When you multiply these two expressions together you get an expression that has NO radicals: (A + B C)(A B C) = A 2 B 2 C Examples What are the conjugates of: x 3 9 x + y Rationalize the denominator of x 4 x. Rationalize the numerator of 3 + x Equations Linear equations can always be written in the form ax + b = 0. To solve linear equations, use basic algebra, but remember what you do to one side, you MUST do to the other side. Example: Solve the following linear equation for the variable x. ax + b cx + d = 2 6
7 Quadratic equations can be written in the form ax 2 + bx + c = 0 with a 0. To solve a quadratic equation by factoring, we use the Zero-Product Property, which tells us that if AB = 0, then A = 0 or B = 0. Note: This does NOT work if the right hand side is not 0. So, if AB = 2, it does NOT mean that A = 2 or B = 2. Example: Solve this equation by factoring: 2x 2 x = 3 To solve quadratics that are written as perfect squares, just take square roots of both sides. Example: Solve (x 5) 2 = 17 If an equation does not easily factor, we can use the method of completing the square. The idea is to get the equation in the form of the example above, where the equation has a perfect square in it. To make x 2 + bx a perfect square, add ( b 2 )2. Example: Solve this equation by completing the square: 3x 2 6x 1 = 0 What happens if a quadratic is not factorable or you can t remember how to factor it? Use THE QUADRATIC FORMULA. It ALWAYS works. The roots (or solutions) of the quadratic equation ax 2 + bx + c = 0 where a 0, are: x = b ± b 2 4ac 2a Example: Solve the equation 3x 2 6x 1 = 0 by using the Quadratic Formula. 7
8 The discriminant of a quadratic equation is D = b 2 4ac. It is the part of the quadratic formula that is underneath the square root. The discriminant tells us how many real solutions the equation has. There are 3 cases. 1. D > 0 2. D = 0 3. D < 0 Example: How many real solutions does 4x 2 5x + 2 = 0 have? An object thrown up at an initial speed of v 0 ft/s will reach a height of h feet after t seconds based on the formula h = 16t 2 + v 0 t Suppose I throw a ball up with a speed of 48 ft/s. a) When will the ball fall back down to me? b) When will it reach a height of 32 ft.? c) Will it reach a height of 64 ft.? 8
9 Other types of equations Equations with Fractionial Expressions. Multiply both sides by the LCD and then solve. x + 5 x 2 = 5 x x 2 4 However, we must then check for extraneous solutions, which are potential solutions that do not actually satisfy the original equation. Whenever we multiply an equation by something with the variable, or whenever we square both sides of an equation, we must check for extraneous solutions. In general, it is a good idea to check your answers. Equations with Radicals: Isolate the square root on one side of the equation and then square both sides. Remember to check for extraneous solutions. 5 x + 1 = x 2 9
10 Equations of Quadratic Type: The idea is to get the equation in the form of a quadratic, which we know how to solve. After solving, again check for extraneous solutions. x 4 5x = 0 x 4/3 5x 2/3 + 6 = 0 Absolute Value Equations: Isolate the absolute value and separate into cases. 4x + 7 = 2 10
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### Students Currently in Algebra 2 Maine East Math Placement Exam Review Problems
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### Algebra I. In this technological age, mathematics is more important than ever. When students
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In all of world history, there are only a handful of examples of a “David” taking on a “Goliath” and effectively changing the course of history. Many of these events have become part of the collective public memory through means other than our history books. Film has enabled us to remember the 300 Spartans at the Battle of Thermopylae, and the musical “Les Miserables” has helped us remember the spirit at the barricades of the French Revolution. Remembrance of such courageous acts is important, as it encourages their repetition.
Forgotten among these rare acts of eminent heroism that changed the course of history are the unsurpassed acts of bravery of the Greek people against Hitler’s seemingly unstoppable Axis Forces.
The world cowered in fear as the greatest military force the world had known crushed the world’s previously greatest military powers and country after country. In this foreboding environment, the people of little Greece refused to surrender and bravely stepped forward to fight. In doing so, Greece inflicted a fatal wound on Axis forces at a crucial moment in World War II, forcing Hitler to change his timeline, delaying the attack on Russia where the Axis Forces met defeat. Winston Churchill said, “If there had not been the virtue and courage of the Greeks, we do not know which the outcome of World War II would have been.”
It was one of the most consequential “David vs. Goliath” victories for freedom and democracy in the modern world and, at the time, an act that inspired and gave hope to the free world. Yet over time, this story of Greece’s victory has become forgotten.
The Washington Oxi Day Foundation’s events and ongoing activities will resurrect the incredible Oxi Day story and assure it becomes part of the collective public memory. Through participation in our ceremonies and their nomination of prospective winners of various Oxi Day awards, hundreds of national policymakers and opinion leaders with a national audience will learn of modern Greece’s heroic feat.<|endoftext|>
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Key Difference – Endosome vs Lysosome
The key difference between the Endosome and the Lysosome is based upon its formation and its function in the cell. Endosome is formed by endocytosis, whereas the lysosome is a membrane bound vesicle containing degrading hydrolytic enzymes.
The endosomal and the lysosomal systems are important in cellular degradation. When a molecule is captured by endocytosis, they form the endosome. Endosome is a membrane bound compartment in eukaryotic cells. The endosome then fuses with the lysosome to degrade the molecule by lysosomal hydrolytic enzymes.
What is an Endosome?
Endosomes are membrane bound compartments derived from plasma membrane due to the process of endocytosis. Endocytosis is the process by which fluid matter, solutes, different macromolecules, plasma membrane components and various other particles are internalized. The plasma membrane forms invaginations, and they form vesicles through membrane fission. These vesicles are called Endosomes. Endosomes are primarily involved in regulating the trafficking of proteins and lipids in the cell.
Endosomes can be categorized as early endosomes, late endosomes, and recycling endosomes. Early endosomes are the first to be formed. Upon maturation by the release of different substances such as acids, they convert into late endosomes. Late endosomes then fuse with lysosomes to form endolysosomes. This fusion will then result in the degradation of the molecule.
Recycling endosomes contain a fine tubular network and are involved in re-shuttling the molecules back to the plasma membrane. This is vital in protein recycling.
What is Lysosome?
Lysosomes are membrane bound organelles present in eukaryotic cells. Lysosomes contain acid hydrolases that have the ability to degrade biomolecules. These enzymes function only at the acidic pH.
When molecules are captured via endocytosis, they form endosomes. Thus the endosomes then fuse with the lysosomes to initiate degradation. Endolysosomes are formed as a result of this fusion. Precisely, the late endosomes that have an acidic pH fuse with the lysosomes. Thus, the lowered acidic pH will, in turn, activate the hydrolases that would degrade the molecules.
In addition to endocytosis, phagocytosis and autophagy can also activate lysosomal systems. Phagocytic cells can fuse with lysosomes forming Phagolysosomes which then undergoes degradation. During autophagy, the intracellular components are compartmentalized into autophagosomes. These autophagosomes fuse with lysosomes to undergo degradation of the compounds resulting in gradual cell death.
What are the Similarities Between Endosome and Lysosome?
- Both endosomes and lysosomes are present in eukaryotic cells.
- Both are membrane bound structures and found in the cell cytoplasm.
- Both participate in the degradation of compounds.
What is the Difference Between Endosome and Lysosome?
Endosome vs Lysosome
|Endosomes are plasma membrane based invaginations formed by the process of endocytosis.||Lysosomes are membrane bound organelles that contain hydrolytic enzymes.|
|Endosomes are formed as a result of endocytosis, where the plasma membrane formed invaginations by capturing a molecule. The plasma membrane fission results in endosomes.||Lysosomes are naturally present as membrane bound organelles in the cell cytoplasm.|
|Early endosome, late endosome, recycling endosomes are the three types of endosomes.||Endolysosome, Phagolysosome, Autophagolysosome are the three types of lysosomes.|
|Capture of biomolecules, fluids, and solutes and direct them for degradation, protein recycling is the functions of endosomes.||Degradation of molecules taken up by endosomes and phagocytes, degradation or intracellular matter taken up by autophagy are the functions of lysosomes.|
Summary – Endosome vs Lysosome
Endosomes and Lysosomes are found in eukaryotes. Endosomes are formed as a result of endocytosis which engulfs components such as proteins and lipids to form plasma membrane based vesicles known as endosomes. Lysosomes, in contrast, are organelles containing acid hydrolases and participate in the degradation of biomolecules when fused with endosomes, phagosomes or autophagosomes. This is the difference between endosomes and lysosomes.
1.Marisa Otegui, and Francisca C. Reyes. “Endosomes in Plants.” Nature News, Nature Publishing Group. Available here
2.Cooper, Geoffrey M. “Lysosomes.” The Cell: A Molecular Approach. 2nd Edition., U.S. National Library of Medicine, 1 Jan. 1970. Available here
1.’Endocytic pathway of animal cells showing EGF receptors, transferrin receptors and mannose-6-phosphate receptors’By Matthew R G Russell – Own work, (CC BY-SA 3.0) via Commons Wikimedia
2.’Lysosome’By lumoreno – Own work, (CC BY-SA 3.0) via Commons Wikimedia<|endoftext|>
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# Show that the product of the areas of the floor
Question:
Show that the product of the areas of the floor and two adjacent walls of a cuboid is the square of its volume.
Solution:
Suppose that the length, breadth and height of the cuboidal floor are $l \mathrm{~cm}, b \mathrm{~cm}$ and $h \mathrm{~cm}$, respectively.
Then, area of the floor $=l \times b \mathrm{~cm}^{2}$
Area of the wall $=b \times h \mathrm{~cm}^{2}$
Area of its adjacent wall $=l \times h \mathrm{~cm}^{2}$
Now, product of the areas of the floor and the two adjacent walls $=(l \times b) \times(b \times h) \times(l \times h)=l^{2} \times b^{2} \times h^{2}=(l \times b \times h)^{2}$
Also, volume of the cuboid $=l \times b \times h \mathrm{~cm}^{2}$
$\therefore$ Product of the areas of the floor and the two adjacent walls $=(l \times b \times h)^{2}=(\text { volume })^{2}$<|endoftext|>
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How to find equivalent fractions for 3/9. How would you find three fractions equivalent to 3/9? Equivalent fractions of 3/9? How do you know if two fractions are equivalent?
### The first 10 equivalent fractions to 3/9 are:
1/3, 2/6, 3/9, 4/12, 5/15, 6/18, 7/21, 8/24, 9/27, 10/30, and so on ...
### How do you do equivalent fractions? What is an equivalent fraction?
Two frations are equivalent when they are both equal when written in lowest terms. The fraction 3/9 is equal to 1/3 when reduced to lowest terms (equivalent fractions definition). To find equivalent fractions, you just need to multiply the numerator and denominator of that reduced fraction (1/3) by the same interger number, ie, multiply by 2, 3, 4, ...
• 2/6 is equivalent to 3/9 once 1 x 2 = 2 and 3 x 2 = 6 and so on ...
At a glance, equivalent fractions look different, but if you reduce then to the lowest terms you will get the same value showing that they are equivalent. If a given fraction is not reduced to lowest terms, you can find other equivalent fractions by dividing both numerator and denominator by the same number.
### How do you know if two fractions are equivalent?
If you want to check if two fractions are equivalent, you can use this Rule:
Two fractions (a/b and c/d) are equivalent only if the product (multiplication) of the numerator (a) of the first fraction and the denominator (d) of the other fraction is equal to the product of the denominator (b) of the first fraction and the numerator (c) of the other fraction.
In other words, if you cross-multiply (a/b = c/d) the equality will remain, i.e, a.d = b.c.
Here are some examples:
- 1/3 is equivalent to 3/9 once 1 x 9 = 3 x 3 = 9
- 2/6 is equivalent to 3/9 once 2 x 9 = 6 x 3 = 18
- 4/12 is equivalent to 3/9 once 4 x 9 = 12 x 3 = 36 and so on ...
### Other ways people ask this question:
• What fraction is equivalent to 3/9?
• What fractions are equivalent to 3/9?
• What is the equivalent fraction of 3/9?
• 3/9 is equal to what fraction?
• Equivalent fractions of 3/9.
• What fraction is equal to 3/9?
Reference:
Equivalent fractions calculator at: Equivalent Fractions Calculator<|endoftext|>
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# Math
posted by on .
For problems 1 and 2, determine how many solutions there are for each triangle. You do not have to solve the triangle.
1. A = 29°, a = 13, c = 27
2. A = 100.1°, a = 20, b = 11
For problems 3-6, solve each triangle using the Law of Sines. If there is no solution, write “no solution.” Round each answer to the nearest tenth.
3. A = 41°, B =61°, c = 19
4. A = 125.4°, a = 33, b = 41
5. A = 76°, a = 15, b = 5
6. A = 97.5°, a = 13, b = 9
• Math - ,
Your exercise probably deals with the
"ambiguous case", that is, the information usually consists of two sides and a non-contained angle.
make sketches marking the given information, even though in reality the triangle may not be possible.
e.g. #1
by the sine law:
Sin C/27 = sin 29°/13
sinC = 1.0069
this is not possible since the sine of any angle must be between -1 and +1.
So, no solution is possible
#2,
sinB/11 = sin100.1/20
sinB = .54147
angle B = 32.784
but the sine is + in quadrants I or II
so angle B could also have been 180-32.784 = 147.726°.
However, a triangle cannot have two obtuse angles, so the second case is not possible.
Find the third side and angle in the usual way.
#5
sinB/5 = sin76/15
sinB = .3234
angle B = 18.9° or 161.1°
In this case the sum of 161.1 + 76 is already > 180, so we have to go with the solution of
angle B = 18.9°.
The rest of the question is routine.<|endoftext|>
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Issue: March 20, 2004
Last fall I cleaned my garden and made a compost pile of the dead plants and leaves from the trees in the yard. When I went to collect compost from the compost pile, there were still a lot of old leaves and plant stems but not much compost. Why didnāt the compost pile make compost?Answer:
To form compost quickly, several things are needed. These are a proper balance of carbon and nitrogen in the material being composted, oxygen, and water. Even when things are not optimal, composting occurs slowly.
In New Mexico, water is often a limiting factor. Rain is often not sufficient to keep the composting material sufficiently moist to allow rapid composting. Even in the winter, water must often be added. The proper moisture level is that which keeps the material in the compost pile feeling like a wet sponge after the water has been squeezed out (damp but not wet).
Oxygen is needed because the fungi and bacteria that convert dead plant material into useful compost need the oxygen to respire as they "eat" the garden wastes. In respiration, some of the materials in the compost are converted into sugar, then into carbon dioxide and water by the plant. This process requires oxygen. Energy is also produced in this process, resulting in growth and multiplication of the bacteria and fungi. Some of this energy is the heat that allows composting to continue through the winter and kills weed seeds and disease organisms. Frequent turning (mixing) of the composting material helps mix air into the compost and speeds composting. If you don't turn the compost pile, it will slowly make compost. A proper mix of coarse and fine particles also helps with aeration. If the composting materials are too fine, the pile becomes anaerobic and foul smells develop. The composting process may slow under these conditions. If the particles are too coarse, then there is too much air exchange, and drying and loss of heat slow the composting process.
Finally, a proper balance of carbon containing materials and nitrogen containing materials is necessary. Dead, dry (often brown) plant material is usually high in carbon. The carbon is necessary, but it must be present in the proper ratio with the nitrogen (green plant material, manure, and nitrogen fertilizer). This ratio should be about 20 to 30 parts carbon for 1 part nitrogen. If the balance is too high in nitrogen, ammonia smells develop and useful nitrogen is lost into the atmosphere. If carbon is too high, composting proceeds at a slow rate.
For more information you can find publications on composting at your local Cooperative Extension Service office or online at http://www.cahe.nmsu.edu/pubs/_h. Dr. George Dickerson, NMSU Extension Horticulture Specialist has written the following publications that will be helpful: H-110: Backyard Composting and H-164: Vermicomposting (composting with worms).back to top
Please join us on Southwest Yard & Garden, a weekly program made for gardeners in the Southwest. It airs on KRWG in Las Cruces Saturdays at 4:30 p.m., on KENW in Portales on Saturdays at 10 a.m., and on KNME in Albuquerque on Saturdays at 9:30 a.m.
Send your gardening questions to
Yard and Garden, ATTN: Dr. Curtis Smith
NMSU Cooperative Extension Service
9301 Indian School Road, NE, Suite 112
Albuquerque, NM 87112
Curtis W. Smith, Ph.D., is an Extension Horticulture Specialist with New Mexico State University's Cooperative Extension Service. New Mexico State University is an equal opportunity/affirmative action employer and educator.<|endoftext|>
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Jackie Gammaro
Centers of Triangles
We have already studied the centroid and how it is a point of balance for a triangle. We then took a detailed look at the location of orthocenters.
Lets look again at these centers and also the Circumcenter , the circumcircle, the incenter, the incircle, and finally the
I – Centroid
Vocab: Median – the median of a triangle is the segment from a vertex to the midpoint of the opposite side
How to Construct a Centroid:
1. Construct triangle ABC
2. Construct the midpoint of each side of the triangle. Let the midpoint of AB is D, the midpoint of BC is E, and the midpoint of AC is F.
3. Connect midpoint D with vertex C, to create median DC and likewise, create medians EA and FB.
4. The centroid is the point of concurrency for the three medians, point G.
II) Orthocenter
III) Circumcenter and Circumcircle
Vocab: circumcenterthe point in the plane equidistant from the three vertices of the triangle.
The circumcenter sits on the three perpendicular bisectors of a triangle.
The circumcenter is also the center of the circumcircle – the circumscribed circle of the triangle.
Constructing a Circumcenter
1. Construct triangle ABC.
2. Construct perpendicular bisectors of each side of the triangle, by constructing the midpoint of each side, then constructing perpendicular lines to each side through each midpoint.
3. The point of concurrency for the three perpendicular bisectors is the circumcenter.
4. To construct a circumcircle, you know the vertices of the triangle lie on the circumference of the circumcircle.- construct a circle that has center at the circumcenter and a point on the circle pass through one of the triangles vertices.
Creating an Incenter:
The Incenter is the point of concurrency for the three angle bisectors of each angle of the triangle.
To create the incenter, you have to create the three angle bisectors for each triangle. If I recall back to high school geometry, I remember how to create an angle bisector, by creating an arc that passes through both side of the angle.
Then create another arc, along one side of the angle, create an equidistant arc on the other side of the angle. Where those two arcs intersect is the a point that lies on the angle bisector of the given angle.
Shown here is triangle ABC, with incenter, I. The circle incscribed within the triangle is called the incircle. Recall, that the incenter is the point on the interior of the triangle that is equidistant from the three sides. That being said, then the incircle's radius has to be perpendicular to each side at the circle's and triangle's point of tangency. The dashed lines are perpendicular lines created, the points of intersection with the perpendcular lines and the sides of the triangle are points on the incircle's circumference. The orange circle is the incircle.
Euler Line
Here is shown the centers of triangle DEF.
H - orthocenter
I - Incenter
G - Centroid
C - Circumcenter
Notice H, G and C are collinear. The line they sit along is called the Euler Line.
RETURN<|endoftext|>
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Question
Updated on:30/05/2023
# A bag contains 5 red balls and some blue balls. If the probability of drawing a blue ball is double that of a red ball, determine the number of blue balls in the bag.
लिखित उत्तर
A
10
B
5
C
15
D
20
Solution
Let number of blue balls is x.
Number of red balls =5
So, total number of balls =5+x
We are given,
Probability of drawing a blue ball is double that of a red ball.
So,x5+x=2(55+x)
x=10
So, number of blue balls are 10.
Transcript
hi students are question is that a bag contains 55 red balls and some Blue Balls if the probability of drawing a blue ball is double that of a red ball determine the number of blue balls in the bag rights of firstly let number of Blue Balls so let let number of Blue Balls BX right b x now we have given that red balls are red balls are five so number of red balls are 52 total number of balls total number of balls will be total number of balls will be equal to X + 5 right now probability of happen of acquiring an event is number of favourable outcomes number of favourable outcomes divided by total outcomes write favourable
out divided by total outcomes for number of favourable now probability that the ball drawn is a blue ball so it is equal to favourable outcomes means number of Blue Balls that is X and total number of balls are X + 5 so this is the probability that the blue ball is drawn know the probability that the ball drawn is a red ball so it will be equal to number of favourable outcomes for number of red balls are 5 and total number of balls are X + 5 now adding to question it is given that the probability of drawing a blue ball is doubled then that of A Red Ball So according to question we can say twice of the probability of red ball will be equal to the probability of blue balls X + X + 5 right now will cross multiply and solve the question so this is equal to
this is equal to write first in Multiplex multiply 5 into 10 into 10 divided by X + 5 is equal to X Into X + now in this step vehicross multiplying so 10 into X + 5 will be equal to X Into X + 5 right X Into X + 5 if you can if you divide X + 5 you transfer the right hand digits to the left hand side so or to the left hand side to the right inside so it will be equal to 10 into X + 5 / X Into X + 5 which is equal to now one so this 5 x + 5 will be cancelled so x is equal to 10 write the value of x will be equal to 10 number of Blue Balls number of blue balls are 10 and this is the required answer for the given question<|endoftext|>
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General solution of partial differential equation$$y^2 \frac{\delta z}{\delta x} - xy\frac{\delta z}{\delta y} = x (z - 2y)$$ will be _______, where ϕ is an arbitrary function.
This question was previously asked in
Junior Executive (ATC) Official Paper 3: Held on Nov 2018 - Shift 3
View all JE ATC Papers >
1. ϕ (x3 - x2y, x + y + z) = 0
2. ϕ (x2 + y2, y2 - yz) = 0
3. ϕ (x2 + xy, y + z) = 0
4. ϕ (x2 - y3, y - z) = 0
Option 2 : ϕ (x2 + y2, y2 - yz) = 0
Free
Junior Executive (ATC) Official Paper 1: Held on Nov 2018 - Shift 1
20549
120 Questions 120 Marks 120 Mins
Detailed Solution
Given partial differential equation as:
$${y^2}\frac{{\partial z}}{{\partial x}} - xy\frac{{\partial z}}{{\partial y}} = x\left( {z - 2y} \right)$$
The auxiliary simultaneous equation is:
$$\frac{{dx}}{{{y^2}}} = - \frac{{dy}}{{xy}} = \frac{{dz}}{{x\left( {z - 2y} \right)}}$$
From $$\frac{{dx}}{{{y^2}}} = - \frac{{dy}}{xy}$$
⇒ $$\frac{{dx}}{y} = - \frac{{dy}}{x}$$
Integrating both sides, we get
$$\smallint xdx = - \smallint ydy$$
⇒ $$\frac{{{x^2}}}{2} = - \frac{{{y^2}}}{2} + c$$
⇒ x2 + y2 = constant
Again, from
$$- \frac{{dy}}{{xy}} = \frac{{dz}}{{x\left( {z - 2y} \right)}}$$
$$- \frac{{dy}}{y} = \frac{{dz}}{{z - 2y}}$$
⇒ z dy - 2y dy + y dz = 0
⇒ z dy + y dz - 2y dy = 0
⇒ y2 – yz = constant
hence, general solution is
ϕ (x2 + y2, y2 - yz) = 0<|endoftext|>
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The Warrior Brain
On the 11th hour of the 11th day of the 11th month in 1918, an armistice was declared between the Allied nations and Germany during the First World War. Later, in 1938, we declared November 11th a national holiday for veterans.
Acts of bravery are not all glory, they come with sacrifice. Witnessing and/or experiencing a traumatic life event can have lasting effects; flashbacks, distressing dreams, and involuntary memory relapses are just some of the symptoms of Post-traumatic Stress Disorder (PTSD, 2013), a disorder that effects more individuals than we come to realize. This disorder is characterized by a heightened sensitivity to potential threats, including those related and/or not related to the event. The nervous system of individuals who have been affected by trauma responds differently than those who have not. Responses such as immobilization, physical and/or verbal aggression, reckless or self-destructive behavior are commonly seen. PTSD victims may also have issues with sleep, trouble falling asleep or staying asleep, due to their dream patterns.
PTSD is not a rarity. In fact, 6 out of every 10 people go through a least one traumatic event throughout their lifetime (PTSD, 2013). These events may include but are not limited to sexual assault, child abuse, physical assault, and exposure to war as a combatant or a civilian. For military personal, being a perpetrator, witnessing atrocities, or killing the enemy may be some of the environmental peritraumatic factors that can effect the severity of the PTSD they experience (DSM-V, 2014).
The challenge with this severe stress disorder is to teach the brain to turn off mental triggers (i.e. visual reminders, places, sounds, people, past flashbacks). In a recent neurofeedback study involving veterans with PTSD, many veterans showed an elimination of one or more of their symptoms, abnormal behaviors, and/or had a significant decrease in their blood stress hormone levels following treatment (Veterans Day, 2009). Another study (PTSD, 2013) showed an 86 percent improvement after only 10 neurofeedback sessions due to the brain being trained to produce healthier patterns, thus significantly reducing symptoms without the use of medication.
As we just celebrated Veteran’s Day, it is time for us all to be appreciative of the men and women who put their lives on the line, and to be aware of those whom suffer from the effects of this disorder. By raising awareness and educating others about PTSD, we can serve those who are in need with different methods of healing. It is never too late to correct and reteach the brain.
American Psychiatric Association. (2013). Diagnostic and Statistical Manual of Mental Disorders: DSM-5 (5th ed.)
PTSD | Treatment for Post Traumatic Stress Disorder. (2013). Retrieved November 10, 2015, from http://ptsd-treatment.info
Veterans Day. (2009). Retrieved November 10, 2015, from http://www.history.com/topics/holidays/history-of-veterans-day<|endoftext|>
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MIT scientists have now designed a system that could store renewable energy such as the wind and solar power, deliver it back into an electric grid on demand. The system which is described in the Environmental Science and Journal of Energy may be designed to power a small city not just when the sun is up, or the wind is high but around the entire clock.
The new design stores heat which is generated by excess electricity from wind or solar power in large tanks of white-hot molten silicon, and then converts the light from the glowing metal black into electricity when it is much more needed.
The researchers from the MIT in the US estimates that such a system that would be vastly more affordable than lithium-ion batteries which have been proposed as a viable, though expensive, a method to store renewable energy. They also estimate that the system would cost about half as much as pumped hydroelectric storage, which is the cheapest form of grid-scale energy storage to date.
“Even if we wanted to run the grid on renewables right now we couldn’t because you’d need fossil-fuelled turbines to make up for the fact that the renewable supply cannot be dispatched on demand,” said Asegun Henry, Associate Professor at MIT. “We are developing a new technology that, if successful, would solve this most important and critical problem in energy and climate change, namely, the storage problem,” Henry said. The new storage system stems from a project in which the researchers looked for ways to increase the efficiency of a form of renewable energy known as concentrated solar power.
“The reason that technology is interesting is, once you do this process of focusing the light to get heat, you can store heat much more cheaply than you can store electricity,” Henry said.<|endoftext|>
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# The sum of two natural numbers equals 120, in which the multiplication of the square of one of them by the other number is to be as maximum as possible, how do you find the two numbers?
Dec 30, 2016
a = 80, b=40
#### Explanation:
let say the two numbers are a and b.
$a + b = 120$
$b = 120 - a$
let say that a is a number to be squared.
$y = {a}^{2} \cdot b$
$y = {a}^{2} \cdot \left(120 - a\right)$
$y = 120 {a}^{2} - {a}^{3}$
$\frac{\mathrm{dy}}{\mathrm{dx}} = 240 a - 3 {a}^{2}$
max or min when $\frac{\mathrm{dy}}{\mathrm{dx}} = 0$
$240 a - 3 {a}^{2} = 0$
$a \left(240 - 3 a\right) = 0$
$a = 0 \mathmr{and} 80$
$b = 120 \mathmr{and} 40$
$\frac{{d}^{2} y}{{\mathrm{dx}}^{2}} = 240 - 6 a$
when a =0,
$\frac{{d}^{2} y}{{\mathrm{dx}}^{2}} = 240$. minimum
when a =80,
$\frac{{d}^{2} y}{{\mathrm{dx}}^{2}} = - 240$. maximum.
the answer is a = 80 and b =40.<|endoftext|>
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A new study from the Australian National University examines defects in rocks found below the earth’s surface in an attempt to better understand how seismic waves travel through the mantle and to gather information that will help geologists better interpret seismological models of the Earth’s internal structure.
Defects found in rocks below the Earth’s surface have a major impact on the transmission of seismic waves, such as those caused by earthquakes, researchers at The Australian National University have discovered.
Professor Ian Jackson, from the Research School of Earth Sciences, part of the ANU College of Physical and Mathematical Sciences, said the team’s research allows us to better understand the way seismic waves travel through the mantle deep below the Earth’s surface.
“We found that defects, known as ‘dislocations’, in the structures of mantle rocks slow down the passage of seismic waves through the mantle. This new information will help us better interpret seismological models of the Earth’s internal structure,” Professor Jackson said.
“These defects have long been considered responsible for the motions of the Earth’s mantle, which have facilitated the movement of tectonic plates over millions of years. This is the first systematic study of their influence over the much shorter timescales of seismic waves.
“The rocks of the Earth’s mantle behave differently at different time scales. At periods of microseconds to nanoseconds, typical of seismic waves, they are quite rigid. However, over periods of millions of years they lose their rigidity entirely, behaving instead like fluid. Previous research has shown that dislocations contribute to this interesting change in behavior.”
To investigate the impacts of these defects on the passage of seismic waves, the team made synthetic materials in the laboratory to represent the mantle rocks below the surface. They then deformed the synthetic rocks to introduce dislocations into the materials, and tested them with novel techniques at 1-1000 second seismic timescales.
Co-author Dr John Fitzgerald, also from the Research School of Earth Sciences, said that these findings allow for a better understanding of how the materials below the Earth’s surface transmit seismic waves, such as those associated with earthquakes.
“It tells us that the same defects that allow the long-term movement of the tectonic plates also have an important influence on the way seismic waves travel through the Earth’s mantle. Such insights from the laboratory will help ‘calibrate’ the seismological probe of the Earth’s internal structure – yielding tighter constraints on its thermal regime and long-term evolution,” Dr Fitzgerald said.
The paper, Dislocation Damping and Anisotropic Seismic Wave Attenuation in the Earth’s Upper Mantle, was published today in Science. A copy is available from the ANU media office. The work was completed by ANU scientists Robert Farla (now at Yale University), Professor Ian Jackson and Dr John Fitzgerald, in collaboration with researchers in Boston and Minnesota.
Image: Norris Geyser Basin from Shutterstock<|endoftext|>
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# combinations, how many ways are there?
how many ways are there to put 36 non-distinguishable balls in 15 distinguishable buckets? This is what I thought: suppose the balls are distinguishable. every time you want to put a ball in a bucket, you have 15 possibilities. so if you have to do this 36 times, you have 15^36 ways to do this. Suppose then that the balls are non-distinguishable, then you have 15^36/15!.
-
$15^{36}/15!$ does not look like integer ... – Santosh Linkha Jan 9 '13 at 19:29
@ThomasAndrews sorry I edited my comment, could you add it as an answer? – Santosh Linkha Jan 9 '13 at 19:36
This is a version of the stars-and-bars problem in combinatorics.
For any pair of natural numbers $n$ and $k$, the number of distinct $n$-tuples ($15$ buckets for balls) of non-negative integers whose sum is $k$ ($36$ total balls) is given by the binomial coefficient:
$$\binom{n + k - 1}{k} = \binom{n+k-1}{n-1}.$$
So the number of ways to put $36$ balls in $15$ distinct buckets is given by:
$$\displaystyle \binom{15+36 - 1}{36} =\binom{15 + 36 - 1}{15-1} = \frac{50\,!}{14\,!\;36\,!}$$
-
As commenter ExperimentX notes above, your number is not an integer. You can see this because $15^{36}$ is odd while $15!$ is even.
This is a stars-and-bars problem.
Let $x_i$ be the number of balls that end up in bucket $i$, $i=1,...,15$. Then you have $0\leq x_i$ for all $i$ and $x_1+x_2+...x_{15}=36$.
Using the formula from the Wikipedia article, we get:
$$\binom{36+15-1}{15-1}$$
Note that the process does not make all of these results equally likely, if you were counting probabilities, but this count gives you the number of distinct results.
-<|endoftext|>
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# Continuity
## Understanding Continuity in Functions
In the realm of mathematics, continuity in functions is a fundamental concept to comprehend, particularly when studying calculus. It refers to the smoothness and unbrokenness of a function at a specific point. In this article, we will dive into the intuitive understanding of continuity and its formal definition, as well as how to check for continuity at a given point.
### Intuitive Understanding of Continuity
To better grasp the concept of continuity, it may be helpful to think of it as "drawing a function without lifting your pencil." In other words, there should be no gaps or breaks in the function's graph. Let's explore this further by examining a single point, p.
Firstly, we must establish that the function is indeed defined at p. If it is not, we cannot draw it without interrupting the line, hence the phrase "assume exists." Additionally, the limits from the left and right of p must be equal for the function to be continuous at p. If they are unequal, the function is not continuous at that point.
Therefore, it is not enough for a function to have a value at p and equal limits from the left and right. It must also have the same limit as its function value. Let's now formalize this understanding into a definition.
### Defining Continuity
A function is continuous at p if and only if the following three conditions are met: 1. exists, 2. both limits from the left and right are equal, and 3. the function value is equal to the limit. If any of these conditions are not fulfilled, the function is considered discontinuous at p.
The phrase "if and only if" represents a logical statement, indicating that both sides must be true for the statement to hold.
### Steps for Checking Continuity
To determine if a function is continuous at a particular point, we can follow these simple steps:
• Step 1: Verify that the function is defined at the given point. If not, the function is not continuous at that point.
• Step 2: Examine if exists. If not, the function is not continuous at that point.
• Step 3: Compare the limit and function value at the given point. If they are not equal, the function is not continuous at that point.
Note that some may use the term "discontinuity" to describe a function that does not exhibit continuity, but they convey the same meaning.
### Examples of Continuous Functions
Let's explore some examples to solidify our understanding of continuity.
Example 1: Is the function continuous at ?
If we try to evaluate the function at x=2, we encounter division by zero, indicating that the function is undefined at this point. Therefore, the function is not continuous at x=2. This can also be observed in the graph, where a vertical asymptote at x=2 signifies a discontinuity.
Example 2: Is the function continuous at ?
In this case, the function is defined at x=2, but the limit from the left and right either do not exist or are not equal. Hence, the function is not continuous at x=2, despite having a defined value at that point.
Example 3: Is the function continuous at ?
If we slightly alter the function, we can observe its effect on continuity. In this case, the limit as x approaches 2 is not infinity, rendering the function not continuous at x=2.
Example 4: Is the function continuous at ?
The key here is to pay attention to the question's context. Although the function's definition changes at x=2, the question pertains to continuity at x=3. In this scenario, the function is discontinuous at x=3, as the two definitions do not seamlessly merge at that point.
## The Concept of Continuity in Calculus
In calculus, the concept of continuity refers to the smoothness and connectedness of a function. Let's take a look at an example to understand continuity at a point.
Consider the function f(x) = 2x + 3 and let's determine if it is continuous at the point x = 2.
• Step 1: The function is defined at x = 2, as we can substitute 2 for x and get an output of 7.
• Step 2: When we approach x = 2 from the left, the limit is 2, and from the right, it is also 2. Therefore, the limits from both sides are equal.
• Step 3: The function value at x = 2 is also 7, hence the function value and the limit value align.
As all three conditions are met, we can conclude that the function f(x) = 2x + 3 is continuous at x = 2.
## Another Illustration of Continuity
Let's consider another example with a slightly different function: f(x) = 5x + 2 and point x = 3.
• Step 1: The function is defined at x = 3, as we can substitute 3 for x and get an output of 17.
• Step 2: When we approach x = 3 from the left, the limit is 13, while from the right, it is 17. Since these two values are not equal, the limit at x = 3 does not exist.
As the condition in step 2 is not met, we do not need to proceed to step 3. Therefore, we can conclude that the function f(x) = 5x + 2 is not continuous at x = 3.
## The Significance of Continuity
You may question why it is essential to determine if a function is continuous or not. Continuity can provide us with significant information about the behavior of a function. For instance, if a function is discontinuous at a specific point, it indicates that something significant occurred at that point.
For example, let's consider a population model with time (x) measured in years and the function given by f(x) = 3000/x. If we conclude that this function is not continuous at x = 0, it implies that a noteworthy event took place in the population being studied. In this scenario, it could be a sudden decline that requires further investigation.
## Types of Continuity
So far, we have only discussed continuity at a single point, but what about continuity over an interval or the entire real line? To learn more about these types of continuity, take a look at "Continuity over an Interval" and "Theorems of Continuity."
## Key Takeaways
• A function is continuous if the limit of the function at a given point is the same as the function value at that point.
• To determine continuity at a point, we need to check that the function is defined at the point, the limit from both sides is equal, and the function value aligns with the limit value.
• A function that is discontinuous at a point is considered to be discontinuous.
• Continuity is crucial in comprehending the behavior of a function and can provide insight into significant events.
## In Conclusion
To summarize, continuity is a fundamental concept in calculus that can offer valuable information about a function's behavior. By following a simple three-step process, we can determine if a function is continuous at a given point. It is essential to have a strong grasp on continuity before moving on to more complex mathematical concepts.<|endoftext|>
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