The Ultimate Maths Cheat Sheet
- Mathematics
Last revision:

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
- https://mathworld.wolfram.com/Set.html
- https://en.wikipedia.org/wiki/Algebra_of_sets
- https://en.wikipedia.org/wiki/Set_theory
- https://en.wikipedia.org/wiki/Zermelo–Fraenkel_set_theory
Mathematical proof methods
Numbers
Decimal system
Bases
Relative quantities
- https://mathworld.wolfram.com/Positive.html
- https://mathworld.wolfram.com/Negative.html
- https://mathworld.wolfram.com/Nonpositive.html
- https://mathworld.wolfram.com/Nonnegative.html
Common number sets
- https://mathworld.wolfram.com/NaturalNumber.html
- https://mathworld.wolfram.com/Integer.html
- https://mathworld.wolfram.com/RationalNumber.html
- https://mathworld.wolfram.com/RealNumber.html
- https://mathworld.wolfram.com/ComplexNumber.html
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
- https://mathworld.wolfram.com/DecimalExpansion.html
- https://mathworld.wolfram.com/RepeatingDecimal.html
- https://mathworld.wolfram.com/Percent.html
- https://en.wikipedia.org/wiki/Scientific_notation
Fraction addition and subtraction
- https://www.mathsisfun.com/fractions_addition.html
- https://www.mathsisfun.com/fractions_subtraction.html
Fraction multiplication and division
- https://www.mathsisfun.com/fractions_multiplication.html
- https://www.mathsisfun.com/fractions_division.html
Geometry
Basic geometry vocabulary
Measurements, and their common formulas
- https://mathworld.wolfram.com/Perimeter.html
- https://mathworld.wolfram.com/Area.html
- https://mathworld.wolfram.com/SurfaceArea.html
- https://mathworld.wolfram.com/Volume.html
Euclidean geometry, and its most popular theorems
Common shapes, and their most interesting properties
- https://mathworld.wolfram.com/Line.html
- https://mathworld.wolfram.com/Plane.html
- https://mathworld.wolfram.com/Triangle.html
- https://mathworld.wolfram.com/Parallelogram.html
- https://mathworld.wolfram.com/Parabola.html
- https://mathworld.wolfram.com/Hyperbola.html
- https://mathworld.wolfram.com/Circle.html
- https://mathworld.wolfram.com/Ellipse.html
- https://mathworld.wolfram.com/Cylinder.html
- https://mathworld.wolfram.com/Sphere.html
- https://mathworld.wolfram.com/Cone.html
Cartesian coordinates
Isometries and similarities
Distances, and their common formulas
- https://mathworld.wolfram.com/Distance.html
- https://en.wikipedia.org/wiki/Distance_from_a_point_to_a_line
- https://en.wikipedia.org/wiki/Distance_from_a_point_to_a_plane
- https://en.wikipedia.org/wiki/Distance_between_two_parallel_lines
- https://en.wikipedia.org/wiki/Skew_lines
Pi and Tau
Angles
Algebra over a field
Functions
Function graphs
Intervals
Domains and ranges
Increasing and decreasing functions
- https://mathworld.wolfram.com/IncreasingFunction.html
- https://mathworld.wolfram.com/DecreasingFunction.html
Extremums
Function transformations
Commun functions, and their graphs
Equations, and solving them
Permutations and combinations
Polynomials
Polynomial addition and subtraction
Polynomial multiplication and division
- https://www.mathsisfun.com/algebra/polynomials-multiplication-long.html
- https://www.mathsisfun.com/algebra/polynomials-division-long.html
Binomial theorem
Polynomial factorization
Quadratic formula
Inequalities
Optimization
- https://en.wikipedia.org/wiki/Mathematical_optimization
- https://mathworld.wolfram.com/GlobalOptimization.html
Parametric equations
- https://mathworld.wolfram.com/ParametricEquations.html
- https://mathworld.wolfram.com/PolarCoordinates.html
Advanced arithmetic, and their algebraic identities
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
Vectors
Vector addition and subtraction
Multiplication with vectors
- https://mathworld.wolfram.com/ScalarMultiplication.html
- https://mathworld.wolfram.com/DotProduct.html
- https://mathworld.wolfram.com/CrossProduct.html
Vector bases
Matrices
Matrix addition and subtraction
Multiplication with matrices
- https://en.wikipedia.org/wiki/Scalar_multiplication
- https://mathworld.wolfram.com/MatrixMultiplication.html
- https://mathworld.wolfram.com/KroneckerProduct.html
Square matrices
- https://mathworld.wolfram.com/SquareMatrix.html
- https://mathworld.wolfram.com/TriangularMatrix.html
- https://mathworld.wolfram.com/DiagonalMatrix.html
- https://mathworld.wolfram.com/IdentityMatrix.html
Matrix trace
Determinants
Matrix inverse
Matrix transpose
Most interesting properties of matrix operations
Orthogonal matrices
Linear transformations
- https://mathworld.wolfram.com/LinearTransformation.html
- https://en.wikipedia.org/wiki/Transformation_matrix
Solving systems of linear equations
Eigenvalues and eigenvectors
- https://mathworld.wolfram.com/Eigenvalue.html
- https://mathworld.wolfram.com/RightEigenvector.html
- https://mathworld.wolfram.com/LeftEigenvector.html
Eigendecomposition
Calculus
Limits, and their most interesting properties
Squeeze theorem
Derivatives and indefinite integrals, and their common formulas
L'Hôpital's rule
Finding extremums
- https://mathworld.wolfram.com/FirstDerivativeTest.html
- https://mathworld.wolfram.com/SecondDerivativeTest.html
- https://mathworld.wolfram.com/ExtremumTest.html
Sums and series
Convergence tests
Common sums and series
- https://mathworld.wolfram.com/ArithmeticSeries.html
- https://mathworld.wolfram.com/GeometricSeries.html
- https://mathworld.wolfram.com/PowerSum.html
- https://mathworld.wolfram.com/PowerSeries.html
- https://mathworld.wolfram.com/HarmonicSeries.html
- https://mathworld.wolfram.com/TaylorSeries.html
- https://mathworld.wolfram.com/BinomialSeries.html
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
- https://en.wikipedia.org/wiki/Probability
- https://en.wikipedia.org/wiki/Probability_theory
- https://en.wikipedia.org/wiki/Probability_axioms
- https://en.wikipedia.org/wiki/Probability_distribution
Expected value
Mean and variance, their most interesting properties, and their common distribution functions
Statistics
Estimators and biases
Median, quartiles and percentiles
Common charts
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