The Ultimate Maths Cheat Sheet

- Mathematics

Latest 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.


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

Inequalities, and their most interesting properties


Parametric equations

Advanced arithmetic, and their algebraic identities

Imaginary unit

Complex conjugate

Absolute value

Powers and roots

Exponent laws


Euler's number


Logarithm identities


Trigonometric identities

Laws of sines and cosines

Euler formula

Linear algebra

Scalars and vectors

Vector addition and subtraction

Multiplication with vectors

Vector bases


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



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 formulas for common distribution functions


Estimators and biases

Median, quartiles and percentiles

Common charts

Related content I wrote

Dice stacked in a triangle shape, with their face numbers matching their row position

I Designed the Perfect Gambling Game, But...

- Mathematics, Business, Game Design

Back in 2006-07-08, during the 13th Canadian Undergraduate Mathematics Conference at McGill University, I presented a gambling game I designed with the novel property of being both advantageous to players and the house, and that despite this proprety, that pretty much nobody in their right mind…

Stream of zeros and ones in space

Minifying JSON Text Beyond Whitespace

- Programming, Mathematics

JSON is a common data serialization format to transmit information over the Internet. However, as I mentioned in a previous article, it's far from optimal. Nevertheless, due to business requirements, producing data in this format may be necessary. I won't go into the details as to how one could…

Medusa in fiery scenery

Deep Learning in Python with PyTorch - Tutorial and Demo

- Programming, Mathematics

As I am continuing my personal journey into deep learning research and development, I wanted to try out PyTorch, a machine learning framework with GPU acceleration primarily designed for the Python programming language. However, I couldn't find any good introductory resource online for it. So I read…

Lady Justice

Reasonable Doubt as a Game Mechanic

- Game Design, Mathematics

Detective fiction, and particularly whodunits, have been really good at being engaging people in attempting to solving the mystery presented before the final reveal. Video games allows such stories to thrive with a level of interactivity that can directly engage the player in this process as an…

City numerically connected to computers

Free Quantum Programming on the IBM Q Experience

- Programming, Mathematics

You can now watch yours truly explain the basics of quantum programming and how to execute a custom quantum program on the IBM Q Experience in less than 10 minutes! The following video summarizes my very first blog post Quantum Programming 101 in a dynamic format, and appends to it a real-time…

See all of my articles