Math Formulas

Euler's Identity and Formula

Reference for Euler's identity e^(i*pi)+1=0 and Euler's formula e^(ix)=cos(x)+i*sin(x). Explains the link between complex exponentials and trigonometry.

Law of Sines Formula

The Law of Sines relates triangle sides to their opposite angles. Learn the formula, when to use it, ambiguous cases, and worked examples.

Logarithm Change of Base Formula

Learn the change of base formula for logarithms. Convert any logarithm to base 10 or natural log for easy calculation, with worked examples.

Divergence Theorem (Gauss's Theorem)

The divergence theorem converts a surface flux integral into a volume integral of the divergence. Used in electromagnetism, fluid mechanics, and heat transfer.

Eigenvalue and Eigenvector Formula

How to find eigenvalues and eigenvectors of a matrix. Fundamental to linear algebra with applications in PCA, quantum mechanics, and vibration analysis.

Euler's Totient Function

Euler's totient function phi(n) counts the integers from 1 to n that are coprime to n. Essential for RSA encryption and number theory.

Fermat's Little Theorem

Fermat's little theorem: if p is prime and gcd(a,p)=1, then a^(p-1) is congruent to 1 mod p. The foundation of RSA encryption and primality testing.

Fourier Transform Formula

The Fourier transform converts a time-domain signal into its frequency components. Learn the formula, inverse transform, and real-world applications.

Gamma Function

The Gamma function extends the factorial to real and complex numbers. Learn the formula, key properties, and worked examples.

Green's Theorem

Green's theorem relates a line integral around a closed curve to a double integral over the enclosed region. A fundamental result in vector calculus.

Jacobian Matrix Formula

The Jacobian matrix contains all first-order partial derivatives of a vector-valued function. Used in coordinate transformations, optimization, and robotics.

Newton's Method (Newton-Raphson) Formula

Newton's Method is an iterative algorithm for finding the roots of a function. It converges rapidly and is used throughout numerical mathematics and computing.

Skewness and Kurtosis Formulas

Skewness measures the asymmetry of a distribution; kurtosis measures the heaviness of the tails. Both are essential for understanding non-normal data.

Stirling's Approximation

Stirling's approximation estimates the factorial of large numbers using logarithms, making calculations in statistics and physics tractable.

Stokes' Theorem

Stokes' theorem relates the line integral of a vector field around a closed curve to the surface integral of its curl over the enclosed surface.

Vieta's Formulas

Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots. Includes examples for quadratic and cubic equations.

Cauchy-Schwarz Inequality

Reference for Cauchy-Schwarz inequality |u·v| ≤ ||u|| ||v|| with proofs for vectors, finite sums, and integrals. Covers linear algebra and real analysis.

Chain Rule Formula

Apply the chain rule for derivatives: d/dx[f(g(x))] = f'(g(x)) times g'(x). Essential calculus technique for composite functions.

Fourier Series

Reference for the Fourier series formula decomposing periodic functions into sine and cosine components. Includes coefficient formulas and worked examples.

Golden Ratio

Reference for the golden ratio phi = (1 + sqrt5)/2. Explore its presence in art, architecture, nature, and its direct connection to the Fibonacci sequence.

L'Hopital's Rule

Reference for L'Hopital's Rule for evaluating 0/0 and infinity/infinity indeterminate limits. Covers conditions, repeated application, and worked examples.

Law of Cosines

Reference for the Law of Cosines: c squared equals a squared plus b squared minus 2ab times cos(C). Solve SSS and SAS triangles with worked examples.

Vector Cross Product Formula

Calculate the cross product of two 3D vectors. Find the perpendicular vector, its magnitude, and direction using the right-hand rule.

Absolute Value Properties

Reference for absolute value properties and rules. Covers the triangle inequality, product rule, quotient rule, and solving absolute value equations.

Arithmetic Mean Formula

Reference for the arithmetic mean formula (sum / count). Compares arithmetic, geometric, and harmonic means with examples for statistics and data analysis.

Completing the Square

Reference for completing the square: converting ax² + bx + c to a(x + h)² + k form to solve quadratics and find vertex form. Includes worked examples.

Complex Number Operations

Reference for complex number operations: addition, multiplication, division, polar form, modulus, argument, and conjugate rules with fully worked examples.

Distance, Rate, and Time Formula

Reference for d = r x t, the distance-rate-time formula. Solves for distance, speed, or time with driving, running, and unit conversion examples.

Geometric Mean Formula

Geometric mean formula GM = (x1 × x2 × ... × xn)^(1/n) for central tendency of growth rates, investment returns, and multiplicative data sets.

Infinite Series Formulas

Learn key infinite series formulas including geometric, harmonic, and p-series. Understand convergence tests with worked examples.

Integration by Parts

Integration by parts formula: the integral of u dv equals uv minus the integral of v du. Essential for integrating products of functions.

Lagrange Interpolation Formula

Lagrange interpolation finds a polynomial passing through given data points. Learn the formula with step-by-step worked examples.

Laplace Transform

Common Laplace transforms table and the definition L{f(t)} = integral of e^(-st)f(t)dt. Used to solve differential equations.

Laws of Exponents

Complete reference for exponent laws: product, quotient, power of a power, zero rule, and negative exponents. With worked examples and common mistakes.

Maclaurin Series

A Maclaurin series expands a function as an infinite polynomial centered at zero. Learn the formula with examples for common functions.

Matrix Operations

Reference for matrix addition, subtraction, multiplication, determinant, inverse, and transpose. Includes 2x2 and 3x3 worked examples for linear algebra.

Partial Fraction Decomposition

Partial fraction decomposition breaks a complex rational expression into simpler fractions, essential for integration and Laplace transforms.

Pascal's Triangle

Pascal's Triangle reveals binomial coefficients and combinatorial patterns. Learn the construction rules and formulas with examples.

Polar Coordinates Formulas

Convert between polar and cartesian coordinates using r, theta, x, and y. Learn polar coordinate formulas with worked examples.

Quadratic Discriminant

The discriminant b²−4ac determines the number and type of roots in a quadratic. Returns two real, one real, or complex conjugate roots with worked examples.

Series Convergence Tests

Key tests for determining if an infinite series converges or diverges. Ratio test, comparison test, integral test with worked examples.

Taylor Series Formula

The Taylor series expands a function as an infinite polynomial around any point. Learn the general formula with practical examples.

Vector Operations Formulas

Essential vector formulas for dot product, cross product, and magnitude. Learn vector math with clear explanations and worked examples.

Absolute Value

Reference for absolute value |x|. Covers equations |ax+b|=c, inequalities, distance on the number line, complex modulus, and properties with examples.

Binomial Theorem

Reference for the binomial theorem (a+b)^n expansion using binomial coefficients C(n,k). Covers Pascal's triangle, algebra, probability, and combinatorics.

Euler's Formula

The beautiful equation linking exponentials, trigonometry, and complex numbers. Often called the most elegant formula in math.

Fibonacci Sequence

Generate the Fibonacci sequence where each number is the sum of the two before it. Found throughout nature and mathematics.

Summation Formulas

Reference for arithmetic series Sn = n(a₁+aₙ)/2 and geometric Sn = a₁(1-rⁿ)/(1-r). Covers sum of integers n(n+1)/2, squares, cubes, and infinite series.

Arithmetic Sequence Formulas

Find the nth term and sum of an arithmetic sequence. Formulas: aₙ = a₁ + (n-1)d and S = n/2 × (a₁ + aₙ) with worked examples.

Combinations Formula

Calculate the number of ways to choose items with C(n,r) = n! / (r!(n-r)!). Learn when order does not matter in counting problems.

Distance Formula

Calculate the distance between two points in a coordinate plane using the distance formula d = √((x₂-x₁)² + (y₂-y₁)²). Includes step-by-step examples.

Exponent Rules

Complete guide to exponent rules: product, quotient, power, zero, and negative exponents. Simplify expressions with these essential rules.

Factorial Formula

Calculate n factorial (n!) — the product of all positive integers from 1 to n. Used in permutations, combinations, and probability.

Geometric Sequence Formulas

Find the nth term and sum of a geometric sequence. Formulas: aₙ = a₁ × r^(n-1) and S = a₁(1-rⁿ)/(1-r) with worked examples.

Logarithm Properties

Essential logarithm rules including product, quotient, power, and change of base. Learn how to simplify and solve log expressions.

Midpoint Formula

Reference for midpoint formula M = ((x₁+x₂)/2, (y₁+y₂)/2). Covers 2D and 3D midpoints, bisecting line segments, and coordinate geometry proofs.

Percentage Formula

Calculate percentages using Percentage = (Part / Whole) × 100. Includes formulas for percentage increase, decrease, and finding the whole.

Permutations Formula

Permutations formula P(n,r) = n! / (n−r)! counts ordered arrangements of r items from n. Learn when order matters versus combinations with worked examples.

Quadratic Formula

The quadratic formula solves any equation of the form ax² + bx + c = 0. Find both roots instantly with this essential algebra formula.

Slope Formula

Calculate the slope of a line between two points using m = (y2-y1)/(x2-x1). Covers positive, negative, zero, and undefined slopes with real-world examples.