The Ultimate Maths Cheat Sheet

- Mathematics

Last revision:

Girl sitting on a small deserted island at sunrise reading a magical book under a brain-shaped tree

The following is a compilation of pretty much every single mathematical topic that I learned throughout my life, covering topics from all levels of education, along with external links for each of them for quick reference.

I have compiled this list after extracting all of the relevant information from my personal notes and book collection, ensuring that every single external reference contained the same information as them or better, and ordered it in structural order as much as I reasonably could.

Note that this cheat sheet is not a proper substitute for a standard mathematical education, but rather a complement to it for easy reference. If you are looking for such an education, I recommend looking at Khan Academy's free math courses.

Logic

Math symbols

Axioms

Propositional calculus

Zermelo-Fraenkel set theory

Mathematical proof methods

Numbers

Decimal system

Bases

Relative quantities

Common number sets

Basic arithmetic

Integer addition and subtraction

Method of complements

Integer multiplication and division

Quotients and reminders

Arithmetic tables

Field axioms

Integer factorization

Fractions

Reduced fractions

Alternate fraction representations

Fraction addition and subtraction

Fraction multiplication and division

Geometry

Basic geometry vocabulary

Measurements, and their common formulas

Euclidean geometry, and its most popular theorems

Common shapes, and their most interesting properties

Cartesian coordinates

Isometries and similarities

Distances, and their common formulas

Pi and Tau

Angles

Algebra over a field

Functions

Function graphs

Intervals

Domains and ranges

Increasing and decreasing functions

Extremums

Function transformations

Commun functions, and their graphs

Equations, and solving them

Permutations and combinations

Polynomials

Polynomial addition and subtraction

Polynomial multiplication and division

Binomial theorem

Polynomial factorization

Quadratic formula

Inequalities, and their most interesting properties

Optimization

Parametric equations

Advanced arithmetic, and their algebraic identities

Imaginary unit

Complex conjugate

Absolute value

Powers and roots

Exponent laws

Rationalization

Euler's number

Logarithms

Logarithm identities

Trigonometry

Trigonometric identities

Laws of sines and cosines

Euler formula

Linear algebra

Scalars and vectors

Vector addition and subtraction

Multiplication with vectors

Vector bases

Matrices

Matrix addition and subtraction

Multiplication with matrices

Square matrices

Simple square matrices

Common matrix to scalar functions

Common matrix transformations

Most interesting properties of matrix operations

Special square matrices

Linear transformations

Solving systems of linear equations

Eigenvalues and eigenvectors

Eigendecomposition

Calculus

Limits, and their most interesting properties

Squeeze theorem

Derivatives and indefinite integrals, and their common formulas

L'Hôpital's rule

Finding extremums

Sums and series

Convergence tests

Common sums and series

Riemann sums and definite integrals

Fundamental theorems of calculus

Differential equations

Integral transforms

Vector calculus, and their common theorems

Randomness

Probability theory, and its most interesting properties

Expected value

Mean and variance, their most interesting properties, and their formulas for common distribution functions

Statistics

Estimators and biases

Median, quartiles and percentiles

Common charts

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