The Ultimate Maths Cheat Sheet

- Mathematics

Last revision:

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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.


Math symbols


Propositional calculus

Zermelo-Fraenkel set theory

Mathematical proof methods


Decimal system


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


Reduced fractions

Alternate fraction representations

Fraction addition and subtraction

Fraction multiplication and division


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


Algebra over a field


Function graphs


Domains and ranges

Increasing and decreasing functions


Function transformations

Commun functions, and their graphs

Equations, and solving them

Permutations and combinations


Polynomial addition and subtraction

Polynomial multiplication and division

Binomial theorem

Polynomial factorization

Quadratic formula



Parametric equations

Advanced arithmetic, and their algebraic identities

Absolute value

Powers and roots

Exponent laws


Euler's number


Logarithm identities


Trigonometric identities

Laws of sines and cosines

Euler formula

Linear algebra


Vector addition and subtraction

Multiplication with vectors

Vector bases


Matrix addition and subtraction

Multiplication with matrices

Square matrices

Matrix trace


Matrix inverse

Matrix transpose

Most interesting properties of matrix operations

Orthogonal matrices

Linear transformations

Solving systems of linear equations

Eigenvalues and eigenvectors



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


Probability theory, and its most interesting properties

Expected value

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


Estimators and biases

Median, quartiles and percentiles

Common charts

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