question: How are the infinite digits following a decimal point of a number such as Pi determined? Like how do we know that the 1000th number of Pi is a 9?
comment: Because someone took the time to perform such a detailed calculation. Today’s computers can obviously perform such calculations much faster, but because it’s infinite we will never know every digit. It would be like trying to count to the highest number, it would be impossible to finish because there’s literally no end. I’m not sure how many digits of PI have been calculated but I’m sure google knows. We don’t know what the numbers after that calculation are. Edit: here’s an old article from 2010 saying that it had been calculated to the two-quadrillionth digit: URL_0
passages: Until the 20th century, the number of digits of pi which mathematicians had the stamina to calculate by hand remained in the hundreds, so that memorization of "all" known digits at the time was possible. In 1949 a computer was used to calculate π to 2000 places, presenting one of the earliest opportunities for a more difficult challenge.
   By the 5th century CE, was known to about seven digits in Chinese mathematics, and to about five in Indian mathematics. Further progress was not made for nearly a millennium, until the 14th century, when Indian mathematician and astronomer Madhava of Sangamagrama, founder of the Kerala school of astronomy and mathematics, discovered the infinite series for , now known as the Madhava–Leibniz series, and gave two methods for computing the value of . One of these methods is to obtain a rapidly converging series by transforming the original infinite series of . By doing so, he obtained the infinite series
   Ancient civilizations required fairly accurate computed values to approximate for practical reasons, including the Egyptians and Babylonians. Around 250 BC the Greek mathematician Archimedes created an algorithm for calculating it. In the 5th century AD Chinese mathematics approximated to seven digits, while Indian mathematics made a five-digit approximation, both using geometrical techniques. The historically first exact formula for , based on infinite series, was not available until a millennium later, when in the 14th century the Madhava–Leibniz series was discovered in Indian mathematics. In the 20th and 21st centuries, mathematicians and computer scientists discovered new approaches that, when combined with increasing computational power, extended the decimal representation of to many trillions of digits after the decimal point. Practically all scientific applications require no more than a few hundred digits of , and many substantially fewer, so the primary motivation for these computations is the quest to find more efficient algorithms for calculating lengthy numeric series, as well as the desire to break records. The extensive calculations involved have also been used to test supercomputers and high-precision multiplication algorithms.
   where is  (Gelfond's constant), is an odd number, and are certain rational numbers that Plouffe computed.
  
  Section::::Modern quest for more digits.:Monte Carlo methods.
  
  Monte Carlo methods, which evaluate the results of multiple random trials, can be used to create approximations of . Buffon's needle is one such technique: If a needle of length is dropped times on a surface on which parallel lines are drawn units apart, and if of those times it comes to rest crossing a line (  0), then one may approximate based on the counts:
   As individual terms of this infinite series are added to the sum, the total gradually gets closer to , and – with a sufficient number of terms – can get as close to as desired. It converges quite slowly, though – after 500,000 terms, it produces only five correct decimal digits of .
  
  An infinite series for (published by Nilakantha in the 15th century) that converges more rapidly than the Gregory–Leibniz series is:
  
  The following table compares the convergence rates of these two series:
   The calculation of was revolutionized by the development of infinite series techniques in the 16th and 17th centuries. An infinite series is the sum of the terms of an infinite sequence. Infinite series allowed mathematicians to compute with much greater precision than Archimedes and others who used geometrical techniques. Although infinite series were exploited for most notably by European mathematicians such as James Gregory and Gottfried Wilhelm Leibniz, the approach was first discovered in India sometime between 1400 and 1500 AD. The first written description of an infinite series that could be used to compute was laid out in Sanskrit verse by Indian astronomer Nilakantha Somayaji in his "Tantrasamgraha", around 1500 AD. The series are presented without proof, but proofs are presented in a later Indian work, "Yuktibhāṣā", from around 1530 AD. Nilakantha attributes the series to an earlier Indian mathematician, Madhava of Sangamagrama, who lived c. 1350 – c. 1425. Several infinite series are described, including series for sine, tangent, and cosine, which are now referred to as the Madhava series or Gregory–Leibniz series. Madhava used infinite series to estimate to 11 digits around 1400, but that value was improved on around 1430 by the Persian mathematician Jamshīd al-Kāshī, using a polygonal algorithm.
   In 1989, the Chudnovsky brothers computed to over 1 billion decimal places on the supercomputer IBM 3090 using the following variation of Ramanujan's infinite series of :
  
  In 1999, Yasumasa Kanada and his team at the University of Tokyo computed to over 200 billion decimal places on the supercomputer HITACHI SR8000/MPP (128 nodes) using another variation of Ramanujan's infinite series of . In October 2005, they claimed to have calculated it to 1.24 trillion places.
   This formula, the Gregory–Leibniz series, equals when evaluated with  = 1. In 1699, English mathematician Abraham Sharp used the Gregory–Leibniz series for formula_13 to compute to 71 digits, breaking the previous record of 39 digits, which was set with a polygonal algorithm. The Gregory–Leibniz for formula_14 series is simple, but converges very slowly (that is, approaches the answer gradually), so it is not used in modern calculations.
  
  In 1706 John Machin used the Gregory–Leibniz series to produce an algorithm that converged much faster:
   which computes a further eight decimal places of with each term in the series. His series are now the basis for the fastest algorithms currently used to calculate . See also Ramanujan–Sato series.
  
  From the mid-20th century onwards, all calculations of have been done with the help of calculators or computers.
  
  In 1944, D. F. Ferguson, with the aid of a mechanical desk calculator, found that William Shanks had made a mistake in the 528th decimal place, and that all succeeding digits were incorrect.
   Pi is a fundamental constant of nature. Archimedes discovered that the area of the circle equals the square of its radius times pi. Archimedes was the first to accurately calculate pi by using polygons with 96 sides both inside and outside a circle then measuring the line segments and finding that pi was between formula_5 and formula_6. A Chinese calculation used polygons with 3,000 sides and calculated pi accurately to five decimal places. The Chinese also found that formula_7 was an accurate estimate of pi to within 6 decimal places and was the most accurate estimate for 1,000 years until arabic numerals were used for arithmetic. 
   There are many ways to memorize , including the use of "piems" (a portmanteau, formed by combining pi and poem), which are poems that represent in a way such that the length of each word (in letters) represents a digit. Here is an example of a piem: "Now I need a drink, alcoholic of course, after the heavy lectures involving quantum mechanics." Notice how the first word has three letters, the second word has one, the third has four, the fourth has one, the fifth has five, and so on. In longer examples, 10-letter words are used to represent the digit zero, and this rule is extended to handle repeated digits in so-called Pilish writing. The short story "Cadaeic Cadenza" records the first 3834 digits of in this manner, and a 10,000-word novel, "Not A Wake", has been written accordingly.
   The English amateur mathematician William Shanks, a man of independent means, spent over 20 years calculating to 707 decimal places. This was accomplished in 1873, with the first 527 places correct. He would calculate new digits all morning and would then spend all afternoon checking his morning's work. This was the longest expansion of until the advent of the electronic digital computer three-quarters of a century later.
  
  Section::::20th and 21st centuries.
  
  In 1910, the Indian mathematician Srinivasa Ramanujan found several rapidly converging infinite series of , including
   In the early years of the computer, an expansion of to decimal places was computed by Maryland mathematician Daniel Shanks (no relation to the above-mentioned William Shanks) and his team at the United States Naval Research Laboratory in Washington, D.C. In 1961, Shanks and his team used two different power series for calculating the digits of . For one, it was known that any error would produce a value slightly high, and for the other, it was known that any error would produce a value slightly low. And hence, as long as the two series produced the same digits, there was a very high confidence that they were correct. The first 100,265 digits of were published in 1962. The authors outlined what would be needed to calculate to 1 million decimal places and concluded that the task was beyond that day's technology, but would be possible in five to seven years.
   In this example, the spelling of Archimedes is normalised to nine. (Although 'Archimedes' is, today, a more correct spelling of the ancient Greek mathematician's name in English, Archimede is also often seen when this mnemonic is given, since Archimède is the more correct spelling in some languages, such as French.)
  
  Longer mnemonics employ the same concept. This example created by Peter M. Brigham incorporates twenty decimal digits:
  
  Section::::Examples in English.:Poems.
   In 1996, Simon Plouffe derived an algorithm to extract the th decimal digit of (using base10 math to extract a base10 digit), and which can do so with an improved speed of time. The algorithm requires virtually no memory for the storage of an array or matrix so the one-millionth digit of can be computed using a pocket calculator. However, it would be quite tedious and impractical to do so.
  
  The calculation speed of Plouffe's formula was improved to by Fabrice Bellard, who derived an alternative formula (albeit only in base2 math) for computing .
  
  Section::::Efficient methods.
   The early string of six 9's is also the first occurrence of four and five consecutive identical digits. The next appearance of four consecutive identical digits is of the digit 7 at position 1,589.
  
  The next sequence of six consecutive identical digits is again composed of 9's, starting at position 193,034. The next distinct sequence of six consecutive identical digits starts with the digit 8 at position 222,299, while strings of nine 9's next occur at position 590,331,982 and 640,787,382.
   In August 2009, a Japanese supercomputer called the T2K Open Supercomputer more than doubled the previous record by calculating to roughly 2.6 trillion digits in approximately 73 hours and 36 minutes.
  
  In December 2009, Fabrice Bellard used a home computer to compute 2.7 trillion decimal digits of . Calculations were performed in base 2 (binary), then the result was converted to base 10 (decimal). The calculation, conversion, and verification steps took a total of 131 days.
   Since the advent of computers, a large number of digits of have been available on which to perform statistical analysis. Yasumasa Kanada has performed detailed statistical analyses on the decimal digits of and found them consistent with normality; for example, the frequencies of the ten digits 0 to 9 were subjected to statistical significance tests, and no evidence of a pattern was found. Any random sequence of digits contains arbitrarily long subsequences that appear non-random, by the infinite monkey theorem. Thus, because the sequence of 's digits passes statistical tests for randomness, it contains some sequences of digits that may appear non-random, such as a sequence of six consecutive 9s that begins at the 762nd decimal place of the decimal representation of . This is also called the "Feynman point" in mathematical folklore, after Richard Feynman, although no connection to Feynman is known.
   The iterative algorithms were widely used after 1980 because they are faster than infinite series algorithms: whereas infinite series typically increase the number of correct digits additively in successive terms, iterative algorithms generally "multiply" the number of correct digits at each step. For example, the Brent-Salamin algorithm doubles the number of digits in each iteration. In 1984, brothers John and Peter Borwein produced an iterative algorithm that quadruples the number of digits in each step; and in 1987, one that increases the number of digits five times in each step. Iterative methods were used by Japanese mathematician Yasumasa Kanada to set several records for computing between 1995 and 2002. This rapid convergence comes at a price: the iterative algorithms require significantly more memory than infinite series.
   Section::::Modern quest for more digits.:Rapidly convergent series.
  
  Modern calculators do not use iterative algorithms exclusively. New infinite series were discovered in the 1980s and 1990s that are as fast as iterative algorithms, yet are simpler and less memory intensive. The fast iterative algorithms were anticipated in 1914, when the Indian mathematician Srinivasa Ramanujan published dozens of innovative new formulae for , remarkable for their elegance, mathematical depth, and rapid convergence. One of his formulae, based on modular equations, is
   After five terms, the sum of the Gregory–Leibniz series is within 0.2 of the correct value of , whereas the sum of Nilakantha's series is within 0.002 of the correct value of . Nilakantha's series converges faster and is more useful for computing digits of . Series that converge even faster include Machin's series and Chudnovsky's series, the latter producing 14 correct decimal digits per term.
  
  Section::::History.:Irrationality and transcendence.
   Section::::Outside mathematics.:Memorizing digits.
  
  Piphilology is the practice of memorizing large numbers of digits of , and world-records are kept by the "Guinness World Records". The record for memorizing digits of , certified by Guinness World Records, is 70,000 digits, recited in India by Rajveer Meena in 9 hours and 27 minutes on 21 March 2015. In 2006, Akira Haraguchi, a retired Japanese engineer, claimed to have recited 100,000 decimal places, but the claim was not verified by Guinness World Records.
   It has long been speculated how Madhava might have found these correction terms. They are the first three convergents of a finite continued fraction which, when combined with the original Madhava's series evaluated to "n" terms, yields about 3"n"/2 correct digits:
  
  The absolute value of the correction term in next higher order is 
  
  He also gave a more rapidly converging series by transforming the original infinite series of π, obtaining the infinite series
  
  By using the first 21 terms to compute an approximation of π, he obtains a value correct to 11 decimal places (3.14159265359).
  
  The value of
   These approximations have so many digits that they are no longer of any practical use, except for testing new supercomputers. Properties like the potential normality of  will always depend on the infinite string of digits on the end, not on any finite computation.
  
  Section::::Development of efficient formulae.:Miscellaneous approximations.
  
  Historically, base 60 was used for calculations. In this base, can be approximated to eight (decimal) significant figures with the number 3:8:29:44, which is
  
  In addition, the following expressions can be used to estimate :
  
  BULLET::::- accurate to three digits:
  
  BULLET::::- accurate to three digits:
  
  BULLET::::- accurate to four digits:
   The correct digits are underlined in the above example. In particular, is correct to 12 decimal places. We see that the number of correct digits after the decimal point increases from 2 (for ) to 5 and 10, illustrating the quadratic convergence.
  
  Section::::Pseudocode.
  
  The following is an example of an implementation of the Newton's Method for finding a root of a function codice_1 which has derivative codice_2.
  
  The initial guess will be and the function will be so that .
  
raw_presuppositions: People know the infinite digits following a decimal point of a number.
raw_corrections: People will never know all of the infinite digits following a decimal point of a number.