math_formula_dataset.jsonl

{"shape": "Square", "formula": "A = l²", "variables": [{"symbol": "l", "description": "length of side"}]}
{"shape": "Rectangle", "formula": "A = w × h", "variables": [{"symbol": "w", "description": "width"}, {"symbol": "h", "description": "height"}]}
{"shape": "Triangle", "formula": "A = (b × h)/2", "variables": [{"symbol": "b", "description": "base"}, {"symbol": "h", "description": "height"}]}
{"shape": "Rhombus", "formula": "A = (D × d)/2", "variables": [{"symbol": "D", "description": "large diagonal"}, {"symbol": "d", "description": "small diagonal"}]}
{"shape": "Trapezoid", "formula": "A = ((B + b)/2) × h", "variables": [{"symbol": "B", "description": "large side"}, {"symbol": "b", "description": "small side"}, {"symbol": "h", "description": "height"}]}
{"shape": "Regular polygon", "formula": "A = (P/2) × a", "variables": [{"symbol": "P", "description": "perimeter"}, {"symbol": "a", "description": "apothem"}]}
{"shape": "Circle", "formulas": [{"name": "Area", "formula": "A = πr²"}, {"name": "Perimeter", "formula": "P = 2πr"}], "variables": [{"symbol": "r", "description": "radius"}]}
{"shape": "Cone (lateral surface)", "formula": "A = πr × s", "variables": [{"symbol": "r", "description": "radius"}, {"symbol": "s", "description": "slant height"}]}
{"shape": "Sphere (surface area)", "formula": "A = 4πr²", "variables": [{"symbol": "r", "description": "radius"}]}
{"category":"Volumes","topic":"Cube","formula":"V = s³","variables":[{"symbol":"s","description":"side"}]}
{"category":"Volumes","topic":"Parallelepiped","formula":"V = l × w × h","variables":[{"symbol":"l","description":"length"},{"symbol":"w","description":"width"},{"symbol":"h","description":"height"}]}
{"category":"Volumes","topic":"Regular prism","formula":"V = b × h","variables":[{"symbol":"b","description":"base"},{"symbol":"h","description":"height"}]}
{"category":"Volumes","topic":"Cylinder","formula":"V = πr² × h","variables":[{"symbol":"r","description":"radius"},{"symbol":"h","description":"height"}]}
{"category":"Volumes","topic":"Cone (or pyramid)","formula":"V = ⅓b × h","variables":[{"symbol":"b","description":"base"},{"symbol":"h","description":"height"}]}
{"category":"Volumes","topic":"Sphere","formula":"V = ⁴⁄₃πr³","variables":[{"symbol":"r","description":"radius"}]}
{"category":"Functions","topic":"Directly Proportional","formulas":[{"expression":"y = kx"},{"expression":"k = y/x"}]}
{"category":"Functions","topic":"Inversely Proportional","formulas":[{"expression":"y = k/x"},{"expression":"k = yx"}]}
{"category":"Functions","topic":"Quadratic formula","formula":"x = [-b ± √(b²-4ac)]/2a"}
{"category":"Functions","topic":"Concavity","cases":[{"condition":"a > 0","result":"Concave up"},{"condition":"a < 0","result":"Concave down"}]}
{"category":"Functions","topic":"Discriminant","formula":"Δ = b² - 4ac"}
{"category":"Functions","topic":"Vertex of parabola","formulas":[{"standard":"V(-b/2a, -Δ/4a)"},{"general":"V(h,k)"}]}
{"category":"Functions","topic":"Zero-product property","rule":"A×B=0 ⇔ A=0 ∨ B=0","example":"(x+2)(x-1)=0 ⇔ x=-2 ∨ x=1"}
{"category":"Functions","topic":"Difference of squares","identity":"(a-b)(a+b)=a²-b²","example":"(x-2)(x+2)=x²-4"}
{"category":"Functions","topic":"Perfect square trinomial","identity":"(a+b)²=a²+2ab+b²","example":"(2x+3)²=4x²+12x+9"}
{"category":"Functions","topic":"Binomial theorem","formula":"(x+y)ⁿ = Σ[nCk·xⁿ⁻ᵏ·yᵏ]"}
{"category":"Exponents","rule":"Product (same base)","formula":"aᵐ × aⁿ = aᵐ⁺ⁿ","example":"3⁵ × 3² = 3⁷"}
{"category":"Exponents","rule":"Product (same exponent)","formula":"aᵐ × bᵐ = (a×b)ᵐ","example":"3⁵ × 2⁵ = 6⁵"}
{"category":"Exponents","rule":"Quotient (same base)","formula":"aᵐ ÷ aⁿ = aᵐ⁻ⁿ","example":"3⁷ ÷ 3² = 3⁵"}
{"category":"Exponents","rule":"Quotient (same exponent)","formula":"aᵐ ÷ bᵐ = (a÷b)ᵐ","examples":["6⁵ ÷ 2⁵ = 3⁵","5³ ÷ 2³ = (5/2)³"]}
{"category":"Exponents","rule":"Power of power","formula":"(aᵐ)ᵖ = aᵐˣᵖ","example":"(5²)³ = 5⁶"}
{"category":"Exponents","rule":"Zero exponent","formula":"a⁰ = 1","example":"8⁰ = 1"}
{"category":"Exponents","rule":"Negative exponents","formula":"a⁻ⁿ = (1/a)ⁿ","examples":["3⁻² = (⅓)²","(⅔)⁻⁴ = (³⁄₂)⁴"]}
{"category":"Exponents","rule":"Fractional exponents","formula":"aᵖ/ᵃ = ᵃ√(aᵖ)","example":"2⁴⁄₃ = ³√(2⁴)"}
{"category":"Radicals","operation":"Multiplication","rule":"ⁿ√x × ⁿ√y = ⁿ√(x×y)","example":"³√2 × ³√5 = ³√10"}
{"category":"Radicals","operation":"Division","rule":"ⁿ√x ÷ ⁿ√y = ⁿ√(x/y)","example":"⁴√8 ÷ ⁴√3 = ⁴√(8/3)"}
{"category":"Radicals","operation":"Addition","rule":"aⁿ√x + bⁿ√x = (a+b)ⁿ√x","example":"4³√5 - 2³√5 = 2³√5"}
{"category":"Radicals","operation":"Exponents","rule":"(ⁿ√x)ᵖ = ⁿ√xᵖ","example":"(³√2)³ = ³√8"}
{"category":"Radicals","operation":"Nested Radicals","rule":"ⁿ√(ᵐ√x) = ᵐⁿ√x","example":"³√5 = ³√5 (identity)"}
{"category":"Radicals","operation":"Exponentiation","rules":[{"form":"Radical to Exponent","rule":"ⁿ√aᵐ = aᵐ⁄ₙ","example":"³√4⁵ = 4⁵⁄₃"},{"form":"Radical Power","rule":"(ⁿ√a)ⁿ = a","example":"(³√3)² = 3"}]}
{"category":"Radicals","operation":"Simplification","rule":"(ⁿ√a)ᵐ = ⁿ√aᵐ","example":"(³√4)⁵ = ³√4⁵"}
{"category":"Geometry","theorem":"Euler's Polyhedral Formula","formula":"F + V = E + 2","variables":[{"symbol":"F","description":"Faces"},{"symbol":"V","description":"Vertices"},{"symbol":"E","description":"Edges"}]}
{"category":"Geometry","theorem":"Interior Angles","formula":"Sᵢ = (n-2)×180°","variables":[{"symbol":"n","description":"Number of sides"}]}
{"category":"Geometry","theorem":"Pythagorean","formula":"H² = C₁² + C₂²","variables":[{"symbol":"H","description":"Hypotenuse"},{"symbol":"C₁,C₂","description":"Legs"}],"note":"Right triangles only"}
{"category":"Geometry","operation":"Distance","formula":"AB = √[(x₁-x₂)² + (y₁-y₂)²]","example":{"points":"A(8,2), B(4,-1)","solution":"√[(8-4)² + (2+1)²] = 5"}}
{"category":"Geometry","operation":"Midpoint","formula":"M( (x₁+x₂)/2, (y₁+y₂)/2 )","example":{"points":"A(2,6), B(4,-2)","solution":"M(3,2)"}}
{"category":"Geometry","form":"Slope-Intercept","equation":"y = mx + b","variables":[{"symbol":"m","description":"Slope"},{"symbol":"b","description":"Y-intercept"}]}
{"category":"Geometry","form":"Vector Equation","equation":"(x,y,z) = (x₀,y₀,z₀) + k(u₁,u₂,u₃)","components":["Point: (x₀,y₀,z₀)","Direction vector: u(u₁,u₂,u₃)"]}
{"category":"Geometry","form":"Cartesian Line","equation":"(x-x₀)/u₁ = (y-y₀)/u₂ = (z-z₀)/u₃"}
{"category":"Geometry","form":"Parametric Line","equations":["x = x₀ + ku₁","y = y₀ + ku₂","z = z₀ + ku₃"]}
{"category":"Geometry","form":"Plane Equation","standard":"n₁(x-x₀) + n₂(y-y₀) + n₃(z-z₀) = 0","scalar":"n₁x + n₂y + n₃z + d = 0"}
{"category":"Geometry","shape":"Circle","equation":"(x-x₀)² + (y-y₀)² = r²","parameters":["Center: (x₀,y₀)","Radius: r"]}
{"category":"Geometry","shape":"Sphere","equation":"(x-x₀)² + (y-y₀)² + (z-z₀)² = r²"}
{"category":"Geometry","shape":"Ellipse","equation":"[(x-h)/a]² + [(y-k)/b]² = 1","parameters":["Center: (h,k)","Semi-axes: a, b"]}
{"category":"Logic","operator":"Conjunction","symbol":"∧","truth_table":[[1,1,1],[1,0,0],[0,1,0],[0,0,0]]}
{"category":"Logic","operator":"Disjunction","symbol":"∨","truth_table":[[1,1,1],[1,0,1],[0,1,1],[0,0,0]]}
{"category":"Logic","operator":"Implication","symbol":"⇒","truth_table":[[1,1,1],[1,0,0],[0,1,1],[0,0,1]]}
{"category":"Logic","law":"Non-Contradiction","statement":"p ∧ ¬p ⇔ F"}
{"category":"Logic","law":"Excluded Middle","statement":"p ∨ ¬p ⇔ T"}
{"category":"Logic","law":"Double Negation","rule":"¬(¬p) ⇔ p"}
{"category":"Logic","property":"Commutativity","rules":["p∧q ⇔ q∧p","p∨q ⇔ q∨p"]}
{"category":"Logic","property":"Associativity","rules":["(p∧q)∧r ⇔ p∧(q∧r)","(p∨q)∨r ⇔ p∨(q∨r)"]}
{"category":"Logic","property":"Identity","rules":["p∧T ⇔ p","p∨F ⇔ p","p∧F ⇔ F","p∨T ⇔ T"]}
{"category":"Logic","property":"Idempotence","rules":["p∧p ⇔ p","p∨p ⇔ p"]}
{"category":"Logic","property":"Distributivity","rules":["p∧(q∨r) ⇔ (p∧q)∨(p∧r)","p∨(q∧r) ⇔ (p∨q)∧(p∨r)"]}
{"category":"Logic","rule":"Implication","equivalences":["p⇒q ⇔ ¬p∨q","¬(p⇒q) ⇔ p∧¬q"]}
{"category":"Logic","rule":"Contrapositive","statement":"(p⇒q) ⇔ (¬q⇒¬p)"}
{"category":"Logic","rule":"De Morgan's","laws":["¬(p∧q) ⇔ ¬p∨¬q","¬(p∨q) ⇔ ¬p∧¬q"]}
{"category":"Logic","quantifier":"Universal","negation":"¬(∀x P(x)) ⇔ ∃x ¬P(x)"}
{"category":"Logic","quantifier":"Existential","negation":"¬(∃x P(x)) ⇔ ∀x ¬P(x)"}
{"category":"Vectors","concept":"Notation","definition":"AB→ = B - A = (b₁-a₁, b₂-a₂)","example":{"points":"A(3,2), B(4,5)","calculation":"(4-3,5-2) = (1,3)"}}
{"category":"Vectors","concept":"Magnitude","formula":"‖u→‖ = √(u₁² + u₂²)","example":{"vector":"u→(3,2)","calculation":"√(3²+2²) = √13"}}
{"category":"Vectors","concept":"Magnitude Square","identity":"(u→)² = ‖u→‖²","example":{"vector":"u→(4,3)","magnitude":5,"calculation":"5² = 25"}}
{"category":"Vectors","operation":"Point-Vector Addition","formula":"A + u→ = (a₁+u₁, a₂+u₂)","example":{"point":"A(4,5)","vector":"u→(3,2)","result":"(7,7)"}}
{"category":"Vectors","operation":"Vector Addition","formula":"u→ + v→ = (u₁+v₁, u₂+v₂)","example":{"vectors":["u→(6,3)","v→(2,1)"],"result":"(8,4)"}}
{"category":"Vectors","operation":"Scalar Multiplication","formula":"k×u→ = (k×u₁, k×u₂)","example":{"scalar":2,"vector":"u→(3,4)","result":"(6,8)"}}
{"category":"Vectors","operation":"Dot Product","formulas":["u→·v→ = u₁v₁ + u₂v₂","‖u→‖×‖v→‖×cos(θ)"],"example":{"vectors":["u→(2,1)","v→(0,3)"],"calculation":"2×0 + 1×3 = 3"}}
{"category":"Vectors","concept":"Angle Between Vectors","formula":"cosα = |u→·v→|/(‖u→‖×‖v→‖)","note":"Extends to 3D with third coordinate"}
{"category":"Statistics","rule":"Summation","formulas":["∑ᵢ₌ₚⁿλ = (n-p+1)λ","∑λxᵢ = λ∑xᵢ","∑(xᵢ+yᵢ) = ∑xᵢ + ∑yᵢ","∑ᵢ₌₁ⁿxᵢ = ∑ᵢ₌₁ᵖxᵢ + ∑ᵢ₌ₚ₊₁ⁿxᵢ"]}
{"category":"Statistics","concept":"Data Description","terms":["Sample: x = (x₁,x₂,...,xₙ)","Size: N","Abs. Freq: nᵢ","Rel. Freq: fᵢ = nᵢ/N","Cum. Freq: Nᵢ, Fᵢ"]}
{"category":"Statistics","calculation":"Mean","formulas":["Ungrouped: x̄ = (∑xᵢ)/N","Weighted: x̄ = (∑nᵢxᵢ)/N","Grouped: x̄ = ∑fᵢxᵢ"]}
{"category":"Statistics","calculation":"Median","rules":["Odd N: Me = xₖ where k=(N+1)/2","Even N: Me = (xₖ+xₖ₊₁)/2 where k=N/2"]}
{"category":"Statistics","property":"Sum of Deviations","theorem":"∑(xᵢ - x̄) = 0"}
{"category":"Statistics","calculation":"Sum of Squares","formulas":["SSₓ = ∑(xᵢ - x̄)²","Ungrouped: SSₓ = ∑xᵢ² - kx̄²","Grouped: SSₓ = ∑(xᵢ - x̄)²nᵢ"]}
{"category":"Statistics","calculation":"Variance","formula":"S² = SSₓ/(N-1)"}
{"category":"Statistics","calculation":"Standard Deviation","formula":"Sₓ = √(SSₓ/(N-1))"}
{"category":"Sequences","type":"Arithmetic","properties":["Common difference: r = uₙ₊₁ - uₙ","nth term: uₙ = u₁ + (n-1)r","Monotonicity: Increasing if r>0, Decreasing if r<0","Sum: Sₙ = (u₁+uₙ)/2 × n"]}
{"category":"Sequences","type":"Geometric","properties":["Common ratio: r = uₙ₊₁/uₙ","nth term: uₙ = u₁×rⁿ⁻¹","Monotonicity: Increasing if u₁>0∧r>1, Decreasing if u₁<0∧r>1, Non-monotonic if r<0","Sum: Sₙ = u₁×(1-rⁿ)/(1-r)"]}
{"category":"Sequences","application":"Simple Interest","formula":"FV = P×(1 + r×t)","variables":[{"symbol":"FV","description":"Future Value"},{"symbol":"P","description":"Principal"},{"symbol":"r","description":"Interest rate"},{"symbol":"t","description":"Time"}]}
{"category":"Sequences","application":"Compound Interest","formula":"FV = P×(1 + r)ᵗ"}
{"category":"Calculus","concept":"Average Rate of Change","formula":"SSL = [f(b)-f(a)]/(b-a)","description":"Slope of secant line between points [a,b]"}
{"category":"Calculus","concept":"Instantaneous Rate","formulas":["f'(x₀) = lim_(x→x₀)[f(x)-f(x₀)]/(x-x₀)","f'(x₀) = lim_(h→0)[f(x₀+h)-f(x₀)]/h"]}
{"category":"Calculus","rule":"Constant","formula":"a' = 0","example":"4' = 0"}
{"category":"Calculus","rule":"Constant Multiple","formula":"(mx)' = m","example":"(3x)' = 3"}
{"category":"Calculus","rule":"Power Rule","formula":"(uⁿ)' = n×uⁿ⁻¹×u'","example":"((6x)⁵)' = 5(6x)⁴×6"}
{"category":"Calculus","rule":"Root","formula":"(ⁿ√u)' = u'/(n×ⁿ√uⁿ⁻¹)","example":"(√(2x))' = 1/√(2x)"}
{"category":"Calculus","rule":"Exponential","formulas":["(aᵘ)' = u'×aᵘ×ln(a)","(eᵘ)' = u'×eᵘ"],"examples":["(7ⁿ²)' = 3×7ⁿ²×ln7","(e²ˣ)' = 2e²ˣ"]}
{"category":"Calculus","rule":"Sum/Difference","formula":"(u±v)' = u'±v'","example":"(2x+5)' = 2"}
{"category":"Calculus","rule":"Product","formula":"(uv)' = u'v + uv'","example":"(x²eˣ)' = 2xeˣ + x²eˣ"}
{"category":"Calculus","rule":"Quotient","formula":"(u/v)' = (u'v-uv')/v²","example":"((x+1)/2x)' = [(1)(2x)-(x+1)(2)]/(2x)²"}
{"category":"Calculus","rule":"Chain","formula":"(g∘f)' = g'(f)×f'","example":"g(x)=2x², f(x)=2x ⇒ (g∘f)'=4(2x)×2"}
{"category":"Calculus","rule":"Trigonometric","formulas":["(sin u)' = u'cos u","(cos u)' = -u'sin u","(tan u)' = u'/cos² u"],"examples":["(sin6x)'=6cos6x","(cos3x)'=-3sin3x"]}
{"category":"Calculus","rule":"Logarithmic","formulas":["(logₐu)' = u'/(u ln a)","(ln u)' = u'/u"],"examples":["(log₄6x)'=1/(x ln4)","(ln5x)'=1/x"]}
{"category":"Probability","law":"Commutative","rules":["A∪B = B∪A","A∩B = B∩A"]}
{"category":"Probability","law":"Associative","rules":["A∪(B∪C) = (A∪B)∪C","A∩(B∩C) = (A∩B)∩C"]}
{"category":"Probability","property":"Identity","rules":["A∪∅ = A","A∩E = A","A∪E = E","A∩∅ = ∅"]}
{"category":"Probability","law":"Distributive","rules":["A∪(B∩C)=(A∪B)∩(A∪C)","A∩(B∪C)=(A∩B)∪(A∩C)"]}
{"category":"Probability","law":"De Morgan's","rules":["A̅∩B̅ = A̅∪B̅","A̅∪B̅ = A̅∩B̅"]}
{"category":"Probability","rule":"Laplace","formula":"P(A) = (# favorable outcomes)/(# total outcomes)"}
{"category":"Probability","rule":"Complement","formula":"P(A̅) = 1 - P(A)"}
{"category":"Probability","rule":"Union","formula":"P(A∪B) = P(A)+P(B)-P(A∩B)"}
{"category":"Probability","rule":"Conditional","formula":"P(A|B) = P(A∩B)/P(B)"}
{"category":"Probability","rule":"Independence","conditions":["P(A|B)=P(A)","P(A∩B)=P(A)×P(B)"]}
{"category":"Probability","counting":"Permutations","types":[{"name":"Simple","formula":"Pₙ=n!","example":"4! = 24"},{"name":"Without Repetition","formula":"ⁿAₚ=n!/(n-p)!","example":"⁶A₂=30"},{"name":"With Repetition","formula":"ⁿA'ₚ=nᵖ","example":"⁵A'₃=125"}]}
{"category":"Probability","counting":"Combination","formula":"ⁿCₚ = n!/((n-p)!×p!)","example":"⁵C₄ = 5"}
{"category":"Probability","distribution":"Properties", "metrics":[{"name":"Mean","formula":"μ = Σxᵢpᵢ"},{"name":"Standard Deviation","formula":"σ = √[Σpᵢ(xᵢ-μ)²]"}]}
{"category":"Probability","distribution":"Binomial","formula":"P(X=k)=ⁿCₖ×pᵏ×(1-p)ⁿ⁻ᵏ","example":"B(10,0.6) P(X=3)=¹⁰C₃×0.6³×0.4⁷"}
{"category":"Logarithms","definition":"logₐb=x ⇔ b=aˣ","example":"3ˣ=15 ⇔ x=log₃15"}
{"category":"Logarithms","identity":"Special Cases","rules":[{"case":"logₐ1=0","example":"log₃1=0"},{"case":"logₐa=1","example":"log₁₀10=1"},{"case":"logₐaᵇ=b","example":"lne²=2"}]}
{"category":"Logarithms","rule":"Product","formula":"logₐ(u×v)=logₐu + logₐv","example":"log₆10 + log₆2 = log₆20"}
{"category":"Logarithms","rule":"Quotient","formula":"logₐ(u/v)=logₐu - logₐv","example":"log₄9 - log₄3 = log₄3"}
{"category":"Logarithms","rule":"Power","formula":"logₐuᵛ = v×logₐu","example":"log₄36 = 2×log₄6"}
{"category":"Logarithms","rule":"Change of Base","formula":"logₐu = logᵦu/logᵦa","application":"log₄5 × log₅6 = log₄6"}
{"category":"Calculus","type":"Limit","expression":"lim(x→∞) aˣ/xᵖ = +∞","conditions":"a,p ∈ ℝ"}
{"category":"Calculus","type":"Limit","expression":"lim(x→∞) logₐx/x = 0","conditions":"a>1, a∈ℝ"}
{"category":"Calculus","type":"Limit","expression":"lim(x→0) (eˣ-1)/x = 1"}
{"category":"Calculus","type":"Limit","expression":"lim(x→0) ln(x+1)/x = 1"}
{"category":"Calculus","type":"Limit","expression":"lim(x→0) sinx/x = 1"}
{"category":"Calculus","type":"Limit","expression":"lim(x→∞) sinx/x = 0"}
{"category":"Calculus","type":"Limit","expression":"lim(uₙ→∞) (1 + k/uₙ)ᵘⁿ = eᵏ"}
{"category":"Calculus","type":"Limit","expression":"lim(n→∞) (1 + 1/n)ⁿ = e","conditions":"n∈ℕ"}
{"category":"Calculus","type":"Primitive","formula":"∫1 dx = x + c","constraint":"c∈ℝ"}
{"category":"Calculus","type":"Power Rule","formula":"∫u(x)ᵃ·u'(x) dx = u(x)ᵃ⁺¹/(a+1) + c","constraints":["a≠-1","c∈ℝ"]}
{"category":"Calculus","type":"Logarithmic","formula":"∫u'(x)/u(x) dx = ln|u(x)| + c","constraint":"c∈ℝ"}
{"category":"Calculus","type":"Exponential","formula":"∫eᵘ⁽ˣ⁾·u'(x) dx = eᵘ⁽ˣ⁾ + c","constraint":"c∈ℝ"}
{"category":"Calculus","type":"Trigonometric","formulas":["∫sin(u(x))·u'(x) dx = -cos(u(x)) + c","∫cos(u(x))·u'(x) dx = sin(u(x)) + c"],"constraint":"c∈ℝ"}
{"category":"Calculus","type":"Linearity","rules":["∫(f+g) dx = ∫f dx + ∫g dx","∫k·f(x) dx = k∫f(x) dx"]}
{"category":"Calculus","type":"Definite Integral","properties":["∫ₐᵇ f(x) dx = -∫ₐᵇ f(x) dx","∫ₐᵃ f(x) dx = 0","∫ₐᵇ f(x) dx = ∫ₐᶜ f(x) dx + ∫ᶜᵇ f(x) dx"]}
{"category":"Calculus","type":"Barrow's Rule","formula":"∫ₐᵇ f(x) dx = F(b)-F(a)","description":"F is primitive of f on [a,b]"}
{"category":"Complex Numbers","form":"Algebraic","definition":"z = a + bi","operations":["Conjugate: z̄ = a - bi","Negation: -z = -a - bi","Equality: z₁=z₂ ⇔ a=c ∧ b=d"]}
{"category":"Complex Numbers","operation":"Addition","formula":"(a+bi)+(c+di) = (a+c)+(b+d)i"}
{"category":"Complex Numbers","operation":"Subtraction","formula":"(a+bi)-(c+di) = (a-c)+(b-d)i"}
{"category":"Complex Numbers","operation":"Multiplication","formula":"(a+bi)×(c+di) = (ac-bd)+(ad+bc)i"}
{"category":"Complex Numbers","operation":"Division","formula":"(a+bi)/(c+di) = (ac+bd)/(c²+d²) + (bc-ad)i/(c²+d²)"}
{"category":"Complex Numbers","operation":"Inverse","formulas":["z⁻¹ = 1/z","z⁻¹ = z̄/|z|²"]}
{"category":"Complex Numbers","property":"Magnitude","formulas":["|z| = √(a²+b²)","|z|² = z·z̄"]}
{"category":"Complex Numbers","property":"Components","formulas":["Re(z) = (z+z̄)/2","Im(z) = (z-z̄)/2i"]}
{"category":"Complex Numbers","form":"Polar","conversion":["|z| = √(a²+b²)","θ = arctan(b/a)","z = |z|(cosθ + isinθ)"]}
{"category":"Complex Numbers","form":"Exponential","definition":"z = |z|e^(iθ)","operations":["Conjugate: z̄ = |z|e^(-iθ)","Negation: -z = |z|e^(i(θ+π))"]}
{"category":"Complex Numbers","operation":"Polar Multiplication","formula":"z₁×z₂ = |z₁||z₂|e^(i(θ₁+θ₂))"}
{"category":"Complex Numbers","operation":"Polar Division","formula":"z₁/z₂ = (|z₁|/|z₂|)e^(i(θ₁-θ₂))"}
{"category":"Complex Numbers","operation":"Exponentiation","formula":"zⁿ = |z|ⁿe^(inθ)"}
{"category":"Complex Numbers","operation":"Roots","formula":"ⁿ√z = ⁿ√|z|e^(i(θ+2πk)/n","conditions":"k∈{0,...,n-1}, n∈ℕ"}
{"category":"Algebra","expression_type":"Monomial","definition":"Single term expression","examples":["2x","6y"]}
{"category":"Algebra","expression_type":"Binomial","definition":"Two-term expression","examples":["6xy+5","xy+y²"]}
{"category":"Algebra","expression_type":"Polynomial","definition":"Multi-term expression with non-negative exponents","examples":["6x²+4x+7","3y³+5y+15"]}
{"category":"Algebra","formula_type":"Basic Identities","formulas":["(a+b)² = a² + 2ab + b²","(a-b)² = a² - 2ab + b²","a² - b² = (a-b)(a+b)","a² + b² = (a-b)² + 2ab"]}
{"category":"Algebra","formula_type":"Trinomial Expansion","formulas":["(a+b+c)² = a²+b²+c²+2ab+2ac+2bc","(a-b-c)² = a²+b²+c²-2ab-2ac+2bc"]}
{"category":"Algebra","formula_type":"Cubic Identities","formulas":["(a+b)³ = a³ + 3a²b + 3ab² + b³","(a-b)³ = a³ - b³ + 3ab² - 3a²b","a³-b³ = (a²+ab+b²)(a-b)","a³+b³ = (a²-ab+b²)(a+b)"]}
{"category":"Algebra","formula_type":"Quadratic Products","formulas":["(x+a)(x+b) = x² + (a+b)x + ab","(x+a)(x-b) = x² + (a-b)x - ab","(x-a)(x+b) = x² + (b-a)x - ab","(x-a)(x-b) = x² - (a+b)x + ab"]}
{"category":"Algebra","formula_type":"Higher Degree","formulas":["a⁴ - b⁴ = (a²+b²)(a²-b²)","a⁵ - b⁵ = (a-b)(a⁴+a³b+a²b²+ab³+b⁴)","a⁵ + b⁵ = (a+b)(a⁴-a³b+a²b²-ab³+b⁴)"]}
{"category":"Algebra","formula_type":"Quadratic Formula","formula":"x = [-b ± √(b²-4ac)]/2a","application":"Solves ax²+bx+c=0","note":"Discriminant Δ = b²-4ac determines nature of roots"}
{"category":"Algebra","formula_type":"Distance Formula","formula":"d = √[(x₂-x₁)² + (y₂-y₁)²]","application":"Distance between points (x₁,y₁) and (x₂,y₂)"}
{"category":"Algebra","formula_type":"Slope Formula","formula":"m = (y₂-y₁)/(x₂-x₁)","application":"Slope of line through (x₁,y₁) and (x₂,y₂)"}
{"category":"Algebra","formula_type":"Exponential Growth/Decay","formulas":["Growth: A = A₀e^(rt)","Decay: A = A₀e^(-rt)"],"variables":["A₀=initial amount","r=rate","t=time"]}
{"category":"Algebra","formula_type":"Logarithmic","formula":"log_b(y) = x ⇔ y = b^x","common_bases":["Base 10","Base e (natural log)"]}
{"category":"Algebra","formula_type":"Pythagorean Theorem","formula":"c² = a² + b²","application":"Right triangle hypotenuse calculation"}
{"category":"Algebra","formula_type":"Factorial","formula":"n! = 1×2×3×...×n","examples":["5! = 120","0! = 1"]}
{"category":"Algebra","formula_type":"Laws of Exponents","rules":["aᵐ×aⁿ = aᵐ⁺ⁿ","(ab)ᵐ = aᵐbᵐ","(aᵐ)ⁿ = aᵐⁿ"]}
{"category":"Algebra","formula_type":"Fractional Exponents","rules":["a⁰ = 1","aᵐ/aⁿ = aᵐ⁻ⁿ","a⁻ᵐ = 1/aᵐ","aᵐ = 1/a⁻ᵐ"]}
{"category":"Algebra","formula_type":"Special Factorizations","formulas":["x³+y³+z³-3xyz = (x+y+z)(x²+y²+z²-xy-yz-zx)","x²+y²+z²-xy-yz-zx = ½[(x-y)²+(y-z)²+(z-x)²]"]}
{"category":"Trigonometry","type":"Function Relationships","formulas":["sinθ = 1/cscθ","cosθ = 1/secθ","tanθ = 1/cotθ","cscθ = 1/sinθ","secθ = 1/cosθ","cotθ = 1/tanθ","tanθ = sinθ/cosθ","cotθ = cosθ/sinθ"]}
{"category":"Trigonometry","type":"Pythagorean Identities","formulas":["sin²θ + cos²θ = 1","tan²θ + 1 = sec²θ","cot²θ + 1 = csc²θ"]}
{"category":"Trigonometry","type":"Double Angle","formulas":["sin2θ = 2sinθcosθ","cos2θ = cos²θ - sin²θ = 1-2sin²θ = 2cos²θ-1","tan2θ = 2tanθ/(1-tan²θ)"]}
{"category":"Trigonometry","type":"Triple Angle","formulas":["sin3θ = 3sinθ - 4sin³θ","cos3θ = 4cos³θ - 3cosθ","tan3θ = (3tanθ - tan³θ)/(1-3tan²θ)"]}
{"category":"Trigonometry","type":"Opposite Angle","formulas":["sin(-θ) = -sinθ","cos(-θ) = cosθ","tan(-θ) = -tanθ","cot(-θ) = -cotθ","sec(-θ) = secθ","csc(-θ) = -cscθ"]}
{"category":"Trigonometry","type":"Half Angle","formulas":["sin(θ/2) = ±√[(1-cosθ)/2]","cos(θ/2) = ±√[(1+cosθ)/2]","tan(θ/2) = ±√[(1-cosθ)/(1+cosθ)] = (1-cosθ)/sinθ = sinθ/(1+cosθ)"]}
{"category":"Trigonometry","type":"Power Reducing","formulas":["sin²θ = (1-cos2θ)/2","cos²θ = (1+cos2θ)/2","tan²θ = (1-cos2θ)/(1+cos2θ)"]}
{"category":"Trigonometry","type":"Law of Cosines","formulas":["a² = b² + c² - 2bc·cosA","b² = a² + c² - 2ac·cosB","c² = a² + b² - 2ab·cosC"]}
{"category":"Trigonometry","type":"Law of Tangents","formula":"(a-b)/(a+b) = tan[(A-B)/2]/tan[(A+B)/2]"}
{"category":"Trigonometry","type":"Cofunction","formulas":["sinθ = cos(π/2-θ)","cosθ = sin(π/2-θ)","tanθ = cot(π/2-θ)","cotθ = tan(π/2-θ)","secθ = csc(π/2-θ)","cscθ = sec(π/2-θ)"],"note":"Valid in Quadrant I"}
{"category":"Trigonometry","type":"Angle Addition","formulas":["sin(A±B) = sinAcosB ± cosAsinB","cos(A±B) = cosAcosB ∓ sinAsinB","tan(A+B) = (tanA+tanB)/(1-tanAtanB)","tan(A-B) = (tanA-tanB)/(1+tanAtanB)"]}
{"category":"Trigonometry","type":"Product-to-Sum","formulas":["sinA·sinB = ½[cos(A-B)-cos(A+B)]","cosA·cosB = ½[cos(A-B)+cos(A+B)]","sinA·cosB = ½[sin(A+B)+sin(A-B)]","cosA·sinB = ½[sin(A+B)-sin(A-B)]"]}
{"category":"Trigonometry","type":"Sum-to-Product","formulas":["sinA±sinB = 2sin[(A±B)/2]cos[(A∓B)/2]","cosA+cosB = 2cos[(A+B)/2]cos[(A-B)/2]","cosA-cosB = -2sin[(A+B)/2]sin[(A-B)/2]"]}