Verified formula reference library

Common Math Formulas

Browse 31 reviewed formulas with explicit numeric inputs, step-by-step calculations, verified outputs, assumptions, and copyable LaTeX.

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Algebra

Algebra formulas

Each listed formula includes explicit input, output, worked steps, verification, assumptions, and non-applicability boundaries.

Browse 6 reviewed references

Algebra

Arithmetic Sequence Formula

a_n=a_1+(n-1)d

Finds the nth term of a sequence with a constant difference.

Algebra

Arithmetic Series Sum

S_n=\frac n2(a_1+a_n)

Computes the sum of the first n terms of an arithmetic sequence.

Algebra

Distance Formula

d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}

Calculates the straight-line distance between two points in a coordinate plane.

Algebra

Geometric Sequence Formula

a_n=a_1r^{n-1}

Finds the nth term of a sequence with a constant ratio.

Algebra

Quadratic Formula

x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}

Finds the roots of a quadratic equation ax² + bx + c = 0 when a is nonzero.

Algebra

Slope-Intercept Form

Expresses a straight line using its slope and vertical intercept.

Applied Math

Applied Math formulas

Each listed formula includes explicit input, output, worked steps, verification, assumptions, and non-applicability boundaries.

Browse 2 reviewed references

Applied Math

Compound Interest Formula

A=P\left(1+\frac{r}{n}\right)^{nt}

Calculates a balance with interest compounded a fixed number of times per year.

Applied Math

Percentage Change Formula

\%\text{ change}=\frac{\text{new}-\text{old}}{|\text{old}|}\times100\%

Measures relative increase or decrease from an original nonzero value.

Calculus

Calculus formulas

Each listed formula includes explicit input, output, worked steps, verification, assumptions, and non-applicability boundaries.

Browse 5 reviewed references

Calculus

Chain Rule

\frac{d}{dx}f(g(x))=f'(g(x))g'(x)

Differentiates a composite function by combining outer and inner derivatives.

Calculus

Derivative Definition

f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}

Defines the instantaneous rate of change of a function as a limit.

Calculus

Fundamental Theorem of Calculus

\int_a^b f(x)\,dx=F(b)-F(a),\quad F'=f

Connects definite integration with antiderivatives.

Calculus

Integration by Parts

\int u\,dv=uv-\int v\,du

Transforms an integral of a product using the product rule in reverse.

Calculus

Product Rule

(fg)'=f'g+fg'

Differentiates the product of two differentiable functions.

Differential Equations

Differential Equations formulas

Each listed formula includes explicit input, output, worked steps, verification, assumptions, and non-applicability boundaries.

Browse 1 reviewed references

Differential Equations

Laplace Transform

F(s)=\int_0^{\infty}f(t)e^{-st}\,dt

Maps a time-domain function to a complex-frequency function useful for initial-value problems.

Geometry

Geometry formulas

Each listed formula includes explicit input, output, worked steps, verification, assumptions, and non-applicability boundaries.

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Geometry

Area of a Circle

Calculates the area enclosed by a circle of radius r.

Geometry

Area of a Triangle

A=\frac{1}{2}bh

Calculates triangle area from a base and its perpendicular height.

Geometry

Circumference of a Circle

C=2\pi r=\pi d

Calculates the distance around a circle from its radius or diameter.

Geometry

Haversine Distance

d=2R\arcsin\sqrt{\sin^2\frac{\Delta\varphi}{2}+\cos\varphi_1\cos\varphi_2\sin^2\frac{\Delta\lambda}{2}}

Approximates great-circle distance between two latitude–longitude points on a sphere.

Geometry

Law of Sines

\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}

Relates each side of a triangle to the sine of its opposite angle.

Geometry

Pythagorean Theorem

Relates the side lengths of a right triangle.

Geometry

Volume of a Sphere

V=\frac{4}{3}\pi r^3

Calculates the volume enclosed by a sphere.

Linear Algebra

Linear Algebra formulas

Each listed formula includes explicit input, output, worked steps, verification, assumptions, and non-applicability boundaries.

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Linear Algebra

2×2 Matrix Determinant

\det\begin{pmatrix}a&b\\c&d\end{pmatrix}=ad-bc

Calculates the determinant of a two-by-two matrix.

Linear Algebra

Cross Product Components

\mathbf a\times\mathbf b=\begin{bmatrix}a_2b_3-a_3b_2\\a_3b_1-a_1b_3\\a_1b_2-a_2b_1\end{bmatrix}

Calculates the three-dimensional vector cross product.

Linear Algebra

Vector Dot Product

\mathbf{a}\cdot\mathbf{b}=\sum_{i=1}^{n}a_ib_i=\|\mathbf{a}\|\|\mathbf{b}\|\cos\theta

Combines two vectors into a scalar and relates their components to the angle between them.

Probability

Probability formulas

Each listed formula includes explicit input, output, worked steps, verification, assumptions, and non-applicability boundaries.

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Probability

Bayes’ Theorem

P(A\mid B)=\frac{P(B\mid A)P(A)}{P(B)}

Updates the probability of an event after observing evidence.

Probability

Binomial Probability Formula

P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}

Finds the probability of exactly k successes in n independent Bernoulli trials.

Probability

Conditional Probability

P(A\mid B)=\frac{P(A\cap B)}{P(B)}

Calculates the probability of A when B is known to have occurred.

Statistics

Statistics formulas

Each listed formula includes explicit input, output, worked steps, verification, assumptions, and non-applicability boundaries.

Browse 4 reviewed references

Statistics

Sample Mean

\bar{x}=\frac{1}{n}\sum_{i=1}^{n}x_i

Calculates the arithmetic average of n observed values.

Statistics

Sample Variance

s^2=\frac{1}{n-1}\sum_{i=1}^{n}(x_i-\bar{x})^2

Measures sample spread using squared deviations and Bessel’s correction.

Statistics

Standard Error of the Mean

SE_{\bar{x}}=\frac{s}{\sqrt{n}}

Estimates the standard deviation of the sample mean’s sampling distribution.

Statistics

Z-Score Formula

z=\frac{x-\mu}{\sigma}

Expresses how many population standard deviations a value lies from the mean.

Browse 219 supporting formula references

These pages remain useful as a catalog, but they are excluded from the launch index until they contain formula-specific worked inputs, outputs, checks, source evidence, and boundaries.

Geometry

Arc Length from a Radian Angle

Calculates the length of a circular arc from radius and angle in radians.

Geometry

Area of a Circular Sector

A=\frac12 r^2\theta

Calculates sector area when the central angle is measured in radians.

Geometry

Area of an Ellipse

Calculates the area enclosed by an ellipse from its semi-major and semi-minor axes.

Algebra

Arithmetic Sequence Term

a_n=a_1+(n-1)d

Computes the nth term of an arithmetic sequence.

Graph Theory

Average Graph Degree

\bar d=\frac{2|E|}{|V|}

Computes the mean degree of a finite undirected graph.

Machine Learning

Binary Cross Entropy

L=-[y\log p+(1-y)\log(1-p)]

Measures probabilistic classification loss for a binary target.

Discrete Math

Binet Formula for Fibonacci Numbers

F_n=\frac{\varphi^n-(1-\varphi)^n}{\sqrt5}

Expresses the nth Fibonacci number using powers of the golden ratio.

Probability

Binomial Probability Mass Function

P(X=k)=\binom nkp^k(1-p)^{n-k}

Computes exact successes in independent Bernoulli trials.

Algebra

Binomial Theorem

(a+b)^n=\sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^k

Expands a nonnegative integer power of a binomial.

Physics

Boltzmann Entropy

S=k_B\ln\Omega

Relates thermodynamic entropy to multiplicity.

Combinatorics

Catalan Number

C_n=\frac{1}{n+1}\binom{2n}{n}

Counts many recursively structured objects such as balanced parentheses and polygon triangulations.

Linear Algebra

Cauchy–Schwarz Inequality

|\langle x,y\rangle|^2\le \langle x,x\rangle\langle y,y\rangle

Bounds the absolute inner product by the product of vector norms.

Numerical Methods

Central Difference Derivative

f\prime(x)\approx\frac{f(x+h)-f(x-h)}{2h}

Approximates a derivative with a symmetric second-order difference.

Numerical Methods

Central Difference First Derivative

f'(x)\approx\frac{f(x+h)-f(x-h)}{2h}

Approximates a first derivative using symmetric samples around x.

Numerical Methods

Central Difference Second Derivative

f''(x)\approx\frac{f(x+h)-2f(x)+f(x-h)}{h^2}

Approximates a second derivative using three equally spaced samples.

Statistics

Chi-Square Statistic

\chi^2=\sum\frac{(O-E)^2}{E}

Compares observed and expected categorical counts.

Number Theory

Chinese Remainder Reconstruction

x\equiv\sum_i a_iM_iN_i\pmod M

Reconstructs a simultaneous congruence solution under coprime moduli.

Geometry

Circular Arc Length

Calculates an arc length from a circle radius and a central angle measured in radians.

Geometry

Circular Arc Length

Computes circular arc length when the angle is measured in radians.

Physics

Classical Kinetic Energy

K=\frac12mv^2

Computes translational kinetic energy in classical mechanics.

Numerical Methods

Classical Runge–Kutta Method

y_{n+1}=y_n+\frac{h}{6}(k_1+2k_2+2k_3+k_4)

Combines four slope estimates to produce a fourth-order ODE step.

Statistics

Coefficient of Determination

R^2=1-\frac{SS_{res}}{SS_{tot}}

Measures the fraction of response variation explained by a regression model relative to a constant-mean baseline.

Statistics

Coefficient of Variation

CV=\frac{s}{\bar{x}}\times100\%

Expresses sample standard deviation relative to the sample mean.

Probability

Combination Formula

\binom{n}{r}=\frac{n!}{r!(n-r)!}

Counts unordered selections of r items from n distinct items.

Graph Theory

Complete Graph Edge Count

|E(K_n)|=\frac{n(n-1)}{2}

Counts the edges in a simple complete graph on n vertices.

Algebra

Complex Number Modulus

|z|=\sqrt{a^2+b^2},\quad z=a+bi

Calculates the distance of a complex number from the origin in the complex plane.

Algebra

Complex Number Polar Form

z=r(\cos\theta+i\sin\theta)=re^{i\theta}

Expresses a complex number by magnitude and argument.

Financial Mathematics

Compound Annual Growth Rate

CAGR=\left(\frac{V_f}{V_i}\right)^{1/n}-1

Finds the constant annual growth rate connecting an initial and final value.

Geometry

Cone Volume

V=\frac{1}{3}\pi r^2h

Calculates the volume of a right circular cone.

Applied Math

Continuous Compound Interest

Models continuously compounded growth of a principal amount.

Financial Mathematics

Continuous Compounding

Computes accumulated value under continuous compounding.

Transforms

Convolution Definition

(f*g)(t)=\int_{-\infty}^{\infty}f(\tau)g(t-\tau)\,d\tau

Combines functions through shifted overlap integration.

Trigonometry

Cosine Double-Angle Identity

\cos(2\theta)=\cos^2\theta-\sin^2\theta

Rewrites the cosine of a doubled angle using squared sine and cosine.

Trigonometry

Cosine Double-Angle Identity

\cos(2x)=\cos^2x-\sin^2x

Expresses the cosine of twice an angle in squared trigonometric terms.

Physics

Coulomb Law

F=k\frac{|q_1q_2|}{r^2}

Computes electrostatic force magnitude between point charges.

Physics

Coulomb's Law

F=k\frac{|q_1q_2|}{r^2}

Calculates the magnitude of the electrostatic force between two point charges.

Financial Mathematics

Coupon Bond Price

P=\sum_{t=1}^{n}\frac{C}{(1+y)^t}+\frac{F}{(1+y)^n}

Discounts coupon payments and face value using yield per period.

Statistics

Covariance

\operatorname{Cov}(X,Y)=E[(X-E[X])(Y-E[Y])]

Measures how two variables vary together.

Statistics

Covariance Definition

\operatorname{Cov}(X,Y)=\mathbb E[(X-\mu_X)(Y-\mu_Y)]

Measures joint linear variation of two random variables.

Statistics

Covariance Matrix

\Sigma=\mathbb E[(X-\mu)(X-\mu)^T]

Collects variances and pairwise covariances of a random vector.

Tensor Calculus

Covariant Derivative of a Vector

\nabla_\mu V^\nu=\partial_\mu V^\nu+\Gamma^\nu_{\mu\lambda}V^\lambda

Differentiates vector components while accounting for basis variation.

Statistics

Cramér–Rao Lower Bound

\operatorname{Var}(\hat\theta)\ge\frac{1}{I(\theta)}

Bounds the variance of an unbiased estimator using Fisher information.

Linear Algebra

Cramer's Rule for a 2×2 System

x=\frac{d_1b_2-b_1d_2}{a_1b_2-b_1a_2},\quad y=\frac{a_1d_2-d_1a_2}{a_1b_2-b_1a_2}

Solves a nonsingular two-equation linear system using determinant ratios.

Information Theory

Cross-Entropy

H(p,q)=-\sum_x p(x)\log q(x)

Measures the expected coding cost when q is used for data distributed as p.

Calculus

Curl

\nabla\times\mathbf F

Measures local rotation of a three-dimensional vector field.

Geometry

Cylinder Volume

Calculates the volume of a right circular cylinder.

Physics

de Broglie Wavelength

\lambda=\frac hp

Relates particle wavelength to momentum.

Logic

De Morgan Law for Logic

\neg(p\land q)\equiv(\neg p)\lor(\neg q)

Rewrites the negation of a conjunction as a disjunction of negations.

Set Theory

De Morgan Law for Sets

(A\cup B)^c=A^c\cap B^c

Relates the complement of a union to the intersection of complements.

Graph Theory

Degree from Adjacency Matrix

\deg(v_i)=\sum_j A_{ij}

Computes a vertex degree by summing its row in a simple undirected graph adjacency matrix.

Physics

Density Formula

\rho=\frac{m}{V}

Calculates mass per unit volume for a material or object.

Calculus

Derivative Chain Rule

\frac{d}{dx}f(g(x))=f\prime(g(x))g\prime(x)

Differentiates a composite function.

Calculus

Derivative of Cosine

\frac{d}{dx}\cos x=-\sin x

Gives the derivative of the cosine function.

Calculus

Derivative of Exponential

\frac{d}{dx}e^x=e^x

Shows that the natural exponential is its own derivative.

Calculus

Derivative of Natural Logarithm

\frac{d}{dx}\ln x=\frac{1}{x}

Differentiates the natural logarithm on its real domain.

Calculus

Derivative of Sine

\frac{d}{dx}\sin x=\cos x

Gives the derivative of the sine function.

Calculus

Derivative Power Rule

\frac{d}{dx}x^n=nx^{n-1}

Differentiates a power of x.

Calculus

Derivative Power Rule

\frac{d}{dx}x^n=nx^{n-1}

Differentiates a power of x.

Calculus

Derivative Product Rule

(fg)\prime=f\prime g+fg\prime

Differentiates the product of two differentiable functions.

Calculus

Derivative Quotient Rule

\left(\frac fg\right)\prime=\frac{f\prime g-fg\prime}{g^2}

Differentiates the quotient of two functions.

Linear Algebra

Determinant as Eigenvalue Product

\det(A)=\prod_{i=1}^{n}\lambda_i

Relates a square matrix determinant to its eigenvalues with multiplicity.

Probability

Discrete Expected Value

\mathbb E[X]=\sum_x xP(X=x)

Computes the probability-weighted average of a discrete random variable.

Geometry

Distance Between Two Points in 3D

d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}

Computes Euclidean distance between two points in three-dimensional space.

Calculus

Divergence

\nabla\cdot\mathbf F=\sum_i\frac{\partial F_i}{\partial x_i}

Measures local source or sink strength of a vector field.

Vector Calculus

Divergence Theorem

\iiint_V \nabla\cdot\mathbf F\,dV=\iint_{\partial V}\mathbf F\cdot\mathbf n\,dS

Relates volume divergence to outward flux across the boundary.

Financial Mathematics

Effective Annual Rate

EAR=\left(1+\frac{r}{m}\right)^m-1

Converts a nominal annual rate compounded m times into an effective annual rate.

Linear Algebra

Eigenvalue Equation

A\mathbf v=\lambda\mathbf v

Defines an eigenvector and its eigenvalue.

Tensor Calculus

Einstein Summation Convention

a_i b^i=\sum_i a_i b^i

Suppresses an explicit summation sign when an index appears once up and once down.

Physics

Electric Power

Calculates electrical power from voltage and current.

Physics

Energy Stored in a Capacitor

U=\frac12CV^2

Computes energy stored in an ideal capacitor.

Number Theory

Euclidean Algorithm

\gcd(a,b)=\gcd(b,a\bmod b)

Computes a greatest common divisor by repeated remainders.

Linear Algebra

Euclidean Vector Norm

\|\mathbf v\|_2=\sqrt{\sum_i v_i^2}

Calculates the Euclidean length of a vector.

Topology

Euler Characteristic for Convex Polyhedra

Relates vertices, edges, and faces of a convex polyhedron.

Graph Theory

Euler Formula for Planar Graphs

|V|-|E|+|F|=2

Relates vertices, edges, and faces in a connected planar embedding.

Algebra

Euler Identity

Connects the constants e, i, π, one, and zero in a single complex exponential identity.

Numerical Methods

Euler Method

y_{n+1}=y_n+h f(t_n,y_n)

Advances an ordinary differential equation solution using the current slope.

Number Theory

Euler Totient Formula for Prime Powers

\varphi(p^k)=p^k-p^{k-1}

Counts positive integers up to a prime power that are coprime to it.

Number Theory

Euler Totient Product

\varphi(n)=n\prod_{p\mid n}\left(1-\frac1p\right)

Computes Euler totient from the distinct prime divisors of n.

Algebra

Exponent Product Rule

a^m a^n=a^{m+n}

Combines powers with the same base by adding exponents.

Algebra

Exponent Quotient Rule

\frac{a^m}{a^n}=a^{m-n}

Divides powers with the same nonzero base by subtracting exponents.

Differential Equations

Exponential Decay

N(t)=N_0e^{-\lambda t}

Models decay at a constant proportional rate.

Differential Equations

Exponential Decay Model

y(t)=y_0e^{-kt}

Solves a proportional decay law y′=−ky with positive rate k.

Applied Math

Exponential Population Growth

N(t)=N_0e^{rt}

Models unrestricted growth or decay at a constant proportional rate.

Statistics

F Statistic for Two Variances

F=\frac{s_1^2}{s_2^2}

Compares two sample variances under an F-distribution model.

Algebra

Finite Geometric Series Sum

S_n=a_1\frac{1-r^n}{1-r}

Computes a finite geometric series when the ratio is not one.

Physics

First Law of Thermodynamics

Relates internal-energy change, heat, and work under a stated sign convention.

Differential Equations

First-Order Linear ODE Solution

y=e^{-\int P(x)dx}\left(\int Q(x)e^{\int P(x)dx}dx+C\right)

Gives the integrating-factor solution of y′+P(x)y=Q(x).

Statistics

Fisher Information

I(\theta)=\mathbb E\left[\left(\frac{\partial}{\partial\theta}\log p(X;\theta)\right)^2\right]

Measures how much an observable random variable tells us about an unknown parameter.

Numerical Methods

Forward Difference Derivative

f\prime(x)\approx\frac{f(x+h)-f(x)}{h}

Approximates a derivative with a first-order forward difference.

Analysis

Fourier Transform

\hat f(\omega)=\int_{-\infty}^{\infty}f(t)e^{-i\omega t}\,dt

Transforms a time- or space-domain function into frequency components.

Financial Mathematics

Future Value of Ordinary Annuity

FV=PMT\frac{(1+r)^n-1}{r}

Computes the future value of equal end-of-period payments.

Probability

Geometric Distribution PMF

P(X=k)=(1-p)^{k-1}p

Models the trial number of the first success.

Statistics

Geometric Mean

G=\left(\prod_{i=1}^{n}x_i\right)^{1/n}

Averages positive values multiplicatively.

Algebra

Geometric Sequence Term

a_n=a_1r^{n-1}

Computes the nth term of a geometric sequence.

Calculus

Gradient

\nabla f=\left(\frac{\partial f}{\partial x_1},\ldots,\frac{\partial f}{\partial x_n}\right)

Forms a vector of partial derivatives of a scalar field.

Optimization

Gradient Descent Update

x_{k+1}=x_k-\eta\nabla f(x_k)

Updates parameters in the direction opposite the objective gradient.

Optimization

Gradient Descent Update

\theta_{t+1}=\theta_t-\eta\nabla J(\theta_t)

Updates parameters in the negative gradient direction.

Physics

Gravitational Potential Energy

Approximates gravitational potential energy near Earth surface.

Vector Calculus

Green’s Theorem

\oint_C(P\,dx+Q\,dy)=\iint_D\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)dA

Relates a planar line integral to a double integral over its enclosed region.

Graph Theory

Handshake Lemma

\sum_{v\in V}\deg(v)=2|E|

Relates the sum of all vertex degrees in a finite undirected graph to twice its edge count.

Statistics

Harmonic Mean

H=\frac{n}{\sum_{i=1}^{n}1/x_i}

Averages positive rates or ratios.

Algebra

Harmonic Number

H_n=\sum_{k=1}^{n}\frac1k

Defines the nth harmonic number.

Differential Equations

Heat Equation

\frac{\partial u}{\partial t}=\alpha\nabla^2u

Models diffusion of heat or another conserved quantity through a medium.

Quantum Mechanics

Heisenberg Uncertainty Principle

\sigma_x\sigma_p\ge\frac{\hbar}{2}

Bounds simultaneous position and momentum uncertainty.

Quantum Mechanics

Heisenberg Uncertainty Relation

\sigma_x\sigma_p\ge\frac{\hbar}{2}

Bounds the product of position and momentum standard deviations.

Geometry

Heron Formula

A=\sqrt{s(s-a)(s-b)(s-c)},\quad s=\frac{a+b+c}{2}

Calculates triangle area from its three side lengths.

Physics

Ideal Gas Law

Relates pressure, volume, amount, and absolute temperature for an ideal gas.

Combinatorics

Inclusion–Exclusion for Two Sets

|A\cup B|=|A|+|B|-|A\cap B|

Counts elements in a union without counting the intersection twice.

Algebra

Infinite Geometric Series Sum

S=\frac{a_1}{1-r},\quad |r|<1

Computes the sum of a convergent infinite geometric series.

Calculus

Integral of Exponential

\int e^x\,dx=e^x+C

Integrates the natural exponential function.

Calculus

Integral Power Rule

\int x^n\,dx=\frac{x^{n+1}}{n+1}+C

Integrates a power of x when n is not −1.

Financial Mathematics

Internal Rate of Return Equation

0=\sum_{t=0}^{n}\frac{CF_t}{(1+IRR)^t}

Defines an internal rate of return as a discount rate that makes NPV zero.

Analysis

Inverse Fourier Transform

f(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\hat f(\omega)e^{i\omega t}\,d\omega

Reconstructs a function from its angular-frequency representation.

Linear Algebra

Inverse of a 2×2 Matrix

A^{-1}=\frac1{ad-bc}\begin{bmatrix}d&-b\\-c&a\end{bmatrix}

Computes the inverse of a nonsingular two-by-two matrix.

Optimization

Jensen’s Inequality

f(\mathbb E[X])\leq\mathbb E[f(X)]

Relates a convex function of an expectation to the expectation of the convex function.

Physics

Kinetic Energy

K=\frac{1}{2}mv^2

Calculates translational kinetic energy.

Optimization

KKT Stationarity

\nabla f(x)+\sum_i\lambda_i\nabla g_i(x)+\sum_j\nu_j\nabla h_j(x)=0

Gives the stationarity part of the Karush–Kuhn–Tucker conditions.

Information Theory

Kullback–Leibler Divergence

D_{\mathrm{KL}}(p\|q)=\sum_x p(x)\log\frac{p(x)}{q(x)}

Measures directed discrepancy from a reference distribution q to a target distribution p.

Optimization

Lagrange Multiplier Condition

\nabla f(x)=\lambda\nabla g(x)

States the first-order condition for an equality-constrained extremum with one regular constraint.

Differential Equations

Laplace Equation

Defines harmonic functions and steady-state potential fields.

Geometry

Law of Cosines

c^2=a^2+b^2-2ab\cos C

Relates the sides of any triangle to the cosine of one included angle.

Probability

Law of Total Probability

P(A)=\sum_i P(A\mid B_i)P(B_i)

Combines conditional probabilities across a partition.

Probability

Law of Total Probability

P(B)=\sum_iP(B\mid A_i)P(A_i)

Computes an event probability over a partition of cases.

Statistics

Least-Squares Normal Equation

\hat\beta=(X^TX)^{-1}X^Ty

Computes ordinary least-squares coefficients when XᵀX is invertible.

Linear Algebra

Least-Squares Normal Equations

X^{\mathsf T}X\hat\beta=X^{\mathsf T}y

Characterizes ordinary least-squares coefficients through normal equations.

Financial Mathematics

Level Loan Payment

PMT=P\frac{r(1+r)^n}{(1+r)^n-1}

Calculates the fixed payment for a fully amortizing loan.

Physics

Linear Momentum

\mathbf p=m\mathbf v

Defines linear momentum as mass times velocity.

Optimization

Linear Program Standard Form

\min_x c^Tx\quad\text{s.t.}\quad Ax=b,\;x\geq0

Expresses a linear objective with equality constraints and nonnegative variables.

Algebra

Logarithm Change of Base

\log_b x=\frac{\log_k x}{\log_k b}

Rewrites a logarithm using another convenient base.

Algebra

Logarithm Product Rule

\log_b(xy)=\log_b x+\log_b y

Converts a logarithm of a product into a sum of logarithms.

Differential Equations

Logistic Differential Equation

\frac{dP}{dt}=rP\left(1-\frac{P}{K}\right)

Models growth that slows as a population approaches carrying capacity.

Applied Math

Logistic Growth Function

P(t)=\frac{K}{1+Ae^{-rt}}

Models growth that begins nearly exponentially and then approaches a carrying capacity.

Differential Equations

Logistic Growth Solution

N(t)=\frac{K}{1+Ae^{-rt}}

Describes bounded growth toward a carrying capacity.

Statistics

Logistic Regression Probability

p(x)=\frac{1}{1+e^{-(\beta_0+\beta^Tx)}}

Maps a linear predictor to a probability between zero and one.

Statistics

Logistic Sigmoid

\sigma(z)=\frac{1}{1+e^{-z}}

Maps a real-valued score to a number between zero and one.

Physics

Mass Energy Equivalence

Relates rest energy to mass.

Linear Algebra

Matrix Multiplication

(AB)_{ij}=\sum_k A_{ik}B_{kj}

Defines each entry of a matrix product.

Linear Algebra

Matrix Trace

\operatorname{tr}(A)=\sum_{i=1}^{n}a_{ii}

Adds the entries on the main diagonal of a square matrix.

Statistics

Mean Absolute Error

MAE=\frac{1}{n}\sum_{i=1}^{n}|y_i-\hat y_i|

Averages absolute prediction errors.

Statistics

Mean Confidence Interval

\bar x\pm z_{\alpha/2}\frac{\sigma}{\sqrt n}

Estimates a population mean when population standard deviation is known.

Statistics

Mean Confidence Interval

\bar{x}\pm t_{\alpha/2,n-1}\frac{s}{\sqrt n}

Gives a t-based confidence interval for a population mean.

Statistics

Mean Squared Error

MSE=\frac{1}{n}\sum_{i=1}^{n}(y_i-\hat y_i)^2

Averages squared prediction errors.

Geometry

Midpoint Formula

M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)

Finds the point halfway between two coordinate points.

Chemistry

Molarity Formula

M=\frac{n}{V}

Calculates solution concentration as moles of solute per litre of solution.

Analysis

Multivariable Taylor Expansion

f(x+h)\approx f(x)+\nabla f(x)^Th+\frac12 h^T H_f(x)h

Approximates a smooth multivariable function near a reference point.

Information Theory

Mutual Information

I(X;Y)=\sum_{x,y}p(x,y)\log\frac{p(x,y)}{p(x)p(y)}

Measures how much knowing one random variable reduces uncertainty about another.

Financial Mathematics

Net Present Value

NPV=\sum_{t=0}^{n}\frac{CF_t}{(1+r)^t}

Sums discounted cash flows including the initial investment.

Numerical Methods

Newton Method Iteration

x_{n+1}=x_n-\frac{f(x_n)}{f\prime(x_n)}

Iteratively approximates a root of a differentiable function.

Optimization

Newton Optimization Step

x_{k+1}=x_k-[\nabla^2f(x_k)]^{-1}\nabla f(x_k)

Uses gradient and Hessian information to form a local quadratic step.

Physics

Newton Second Law

\mathbf F_{net}=m\mathbf a

Relates net force, mass, and acceleration.

Numerical Methods

Newton–Raphson Iteration

x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}

Iteratively approximates a root using the tangent line at the current estimate.

Probability

Normal Distribution Density

f(x)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}

Defines the bell-shaped normal probability density.

Physics

Ohm Law

Relates voltage, current, and resistance for an ohmic element.

Statistics

One-Sample t Statistic

t=\frac{\bar x-\mu_0}{s/\sqrt n}

Standardizes a sample mean when population standard deviation is unknown.

Quantum Mechanics

Operator Commutator

Measures the failure of two operators to commute.

Geometry

Parallelogram Area

Calculates area from a base and perpendicular height.

Statistics

PCA Eigenvalue Problem

\Sigma v_k=\lambda_k v_k

Finds principal directions as covariance-matrix eigenvectors.

Statistics

Pearson Correlation

r=\frac{\sum (x_i-\bar x)(y_i-\bar y)}{\sqrt{\sum(x_i-\bar x)^2\sum(y_i-\bar y)^2}}

Measures linear association between two variables.

Probability

Permutation Formula

P(n,r)=\frac{n!}{(n-r)!}

Counts ordered selections of r items from n distinct items.

Physics

Photon Energy

Relates photon energy to frequency through Planck constant.

Combinatorics

Pigeonhole Principle Bound

\max_i n_i\ge\left\lceil\frac{N}{m}\right\rceil

Guarantees that at least one of m containers holds at least the ceiling of N divided by m objects.

Quantum Mechanics

Planck Relation

E=\hbar\omega

Relates energy to angular frequency.

Algebra

Point-Slope Form

y-y_1=m(x-x_1)

Defines a line from one point and its slope.

Probability

Poisson Probability Mass Function

P(X=k)=\frac{e^{-\lambda}\lambda^k}{k!}

Models counts of independent events in a fixed interval.

Statistics

Population Variance

\sigma^2=\frac{1}{N}\sum_{i=1}^{N}(x_i-\mu)^2

Measures average squared deviation from the population mean.

Financial Mathematics

Present Value

PV=\frac{FV}{(1+r)^n}

Discounts a future amount back n compound periods.

Financial Mathematics

Present Value of Ordinary Annuity

PV=PMT\frac{1-(1+r)^{-n}}{r}

Discounts equal end-of-period payments to a present value.

Physics

Pressure Formula

P=\frac{F}{A}

Calculates average pressure as perpendicular force divided by the area receiving that force.

Probability

Probability Addition Rule

P(A\cup B)=P(A)+P(B)-P(A\cap B)

Finds the probability of at least one of two events.

Probability

Probability Complement Rule

P(A^c)=1-P(A)

Finds the probability that an event does not occur.

Trigonometry

Pythagorean Trigonometric Identity

\sin^2\theta+\cos^2\theta=1

Relates sine and cosine for the same angle.

Algebra

Quadratic Discriminant

\Delta=b^2-4ac

Determines the number and type of roots of a quadratic equation.

Linear Algebra

Quadratic Form

Defines a scalar quadratic expression associated with a square matrix.

Algebra

Quadratic Vertex Form

Expresses a parabola using its vertex and vertical scale.

Algebra

Quadratic Vertex Formula

x_v=-\frac{b}{2a},\quad y_v=f(x_v)

Finds the vertex coordinates of a quadratic function.

Calculus

Quotient Rule

\left(\frac{f}{g}\right)\prime=\frac{f\prime g-fg\prime}{g^2}

Differentiates a quotient of two functions.

Geometry

Rectangle Area

Calculates the area of a rectangle.

Geometry

Rectangle Perimeter

Calculates the perimeter of a rectangle.

Geometry

Rectangular Prism Volume

Calculates volume from length, width, and height.

Statistics

Root Mean Square

x_{\mathrm{rms}}=\sqrt{\frac1n\sum_{i=1}^{n}x_i^2}

Measures the quadratic mean of a set of values.

Statistics

Root Mean Squared Error

RMSE=\sqrt{\frac{1}{n}\sum_{i=1}^{n}(y_i-\hat y_i)^2}

Returns the square root of mean squared error in the response variable’s units.

Statistics

Shannon Entropy

H(X)=-\sum_i p_i\log_2 p_i

Measures uncertainty in a discrete probability distribution in bits.

Physics

Simple Harmonic Motion

x(t)=A\cos(\omega t+\varphi)

Describes ideal periodic motion whose restoring force is proportional to displacement.

Differential Equations

Simple Harmonic Oscillator Equation

m\ddot x+kx=0

Describes an ideal mass-spring oscillator without damping or forcing.

Financial Mathematics

Simple Interest

Computes accumulated value when interest is applied only to the original principal.

Statistics

Simple Regression Intercept

b_0=\bar y-b_1\bar x

Computes the intercept of a fitted simple regression line.

Statistics

Simple Regression Slope

b_1=\frac{\sum(x_i-\bar x)(y_i-\bar y)}{\sum(x_i-\bar x)^2}

Computes the least-squares slope in simple linear regression.

Numerical Methods

Simpson’s Rule

\int_a^b f(x)dx\approx\frac{h}{3}\left[f(x_0)+4\sum_{i\ odd}f(x_i)+2\sum_{i\ even}f(x_i)+f(x_n)\right]

Approximates a definite integral with piecewise quadratic interpolation.

Trigonometry

Sine Double-Angle Identity

\sin(2\theta)=2\sin\theta\cos\theta

Rewrites the sine of a doubled angle as a product.

Trigonometry

Sine Double-Angle Identity

\sin(2x)=2\sin x\cos x

Expresses the sine of twice an angle using sine and cosine.

Trigonometry

Sine Half-Angle Identity

\sin^2\frac{x}{2}=\frac{1-\cos x}{2}

Relates a half-angle sine square to the cosine of the full angle.

Optimization

Single-Constraint Lagrange Condition

\nabla f(x)=\lambda\nabla g(x)

Gives a first-order condition for a constrained extremum with one equality constraint.

Statistics

Softmax Function

p_i=\frac{e^{z_i}}{\sum_j e^{z_j}}

Converts a vector of real scores into positive values that sum to one.

Statistics

Softmax Normalization Formula

\operatorname{softmax}(z)_i=\frac{e^{z_i}}{\sum_j e^{z_j}}

Converts a vector of scores into positive values summing to one.

Geometry

Sphere Surface Area

Calculates the surface area of a sphere.

Vector Calculus

Stokes’ Theorem

\iint_S(\nabla\times\mathbf F)\cdot\mathbf n\,dS=\oint_{\partial S}\mathbf F\cdot d\mathbf r

Relates surface curl to circulation around the oriented boundary.

Algebra

Sum of the First Cubes

\sum_{k=1}^{n}k^3=\left[\frac{n(n+1)}2\right]^2

Computes the sum of the first n cube numbers.

Algebra

Sum of the First Integers

\sum_{k=1}^{n}k=\frac{n(n+1)}2

Computes the sum of the first n positive integers.

Discrete Math

Sum of the First n Cubes

\sum_{k=1}^{n}k^3=\left(\frac{n(n+1)}{2}\right)^2

Calculates the sum of the cubes from one through n.

Discrete Math

Sum of the First n Positive Integers

\sum_{k=1}^{n}k=\frac{n(n+1)}{2}

Calculates the total of the integers from one through n.

Discrete Math

Sum of the First n Squares

\sum_{k=1}^{n}k^2=\frac{n(n+1)(2n+1)}{6}

Calculates the sum of the squared integers from one through n.

Algebra

Sum of the First Squares

\sum_{k=1}^{n}k^2=\frac{n(n+1)(2n+1)}6

Computes the sum of the first n square numbers.

Calculus

Taylor Series

f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^n

Represents a sufficiently smooth function as a power series around a point.

Linear Algebra

Three by Three Determinant

\det A=a(ei-fh)-b(di-fg)+c(dh-eg)

Expands the determinant of a 3×3 matrix along the first row.

Quantum Mechanics

Time-Dependent Schrödinger Equation

i\hbar\frac{\partial}{\partial t}|\psi(t)\rangle=\hat H|\psi(t)\rangle

Describes the time evolution of a quantum state under a Hamiltonian.

Quantum Mechanics

Time-Dependent Schrödinger Equation

i\hbar\frac{\partial}{\partial t}\Psi=\hat H\Psi

Describes unitary time evolution of a quantum state.

Geometry

Total Surface Area of a Cylinder

S=2\pi r^2+2\pi rh

Adds the areas of two circular bases and the curved lateral surface.

Geometry

Trapezoid Area

A=\frac{1}{2}(b_1+b_2)h

Calculates area from parallel bases and height.

Numerical Methods

Trapezoidal Rule

\int_a^b f(x)dx\approx\frac{h}{2}\left[f(x_0)+2\sum_{i=1}^{n-1}f(x_i)+f(x_n)\right]

Approximates a definite integral with piecewise linear segments.

Graph Theory

Tree Edge Count

States the edge count of a finite connected acyclic graph.

Linear Algebra

Triangle Inequality for Norms

\|x+y\|\le \|x\|+\|y\|

States that the direct displacement is no longer than a path through an intermediate point.

Linear Algebra

Two by Two Matrix Inverse

A^{-1}=\frac{1}{ad-bc}\begin{bmatrix}d&-b\\-c&a\end{bmatrix}

Finds the inverse of a nonsingular 2×2 matrix.

Discrete Mathematics

Two-Set Inclusion–Exclusion Formula

|A\cup B|=|A|+|B|-|A\cap B|

Counts a union while correcting for elements counted in both sets.

Probability

Variance Identity

\operatorname{Var}(X)=\mathbb E[X^2]-\mathbb E[X]^2

Expresses variance using first and second moments.

Linear Algebra

Vector Projection

\operatorname{proj}_{\mathbf b}\mathbf a=\frac{\mathbf a\cdot\mathbf b}{\mathbf b\cdot\mathbf b}\mathbf b

Projects one vector onto the direction of another.

Linear Algebra

Vector Projection

\operatorname{proj}_{u}v=\frac{v\cdot u}{u\cdot u}u

Projects one vector onto the direction of another.

Differential Equations

Wave Equation

\frac{\partial^2u}{\partial t^2}=c^2\nabla^2u

Models ideal wave propagation with constant speed c.

Physics

Wave Speed

Relates wave speed, frequency, and wavelength.

Statistics

Weighted Mean

\bar x_w=\frac{\sum_i w_ix_i}{\sum_i w_i}

Calculates an average where observations have different weights.