The Ultimate Maths Cheat Sheet

2023-03-27 - Mathematics

Latest revision: 2024-08-29

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

• https://www.rapidtables.com/math/symbols/Basic_Math_Symbols.html

Axioms

• https://en.wikipedia.org/wiki/Axiom

Propositional calculus

• https://mathworld.wolfram.com/PropositionalCalculus.html

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

• https://en.wikipedia.org/wiki/Mathematical_proof

Numbers

Decimal system

• https://www.mathsisfun.com/numbers/counting-names-1000.html

Bases

• https://mathworld.wolfram.com/Base.html

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

• https://mathworld.wolfram.com/Addition.html
• https://mathworld.wolfram.com/Subtraction.html

Method of complements

• https://en.wikipedia.org/wiki/Method_of_complements

Integer multiplication and division

• https://mathworld.wolfram.com/Multiplication.html
• https://mathworld.wolfram.com/Division.html

Quotients and reminders

• https://en.wikipedia.org/wiki/Modular_arithmetic
• https://en.wikipedia.org/wiki/Modulo

Arithmetic tables

• https://www.mathsisfun.com/numbers/addition-table.html
• https://www.mathsisfun.com/tables.html

Field axioms

• https://mathworld.wolfram.com/FieldAxioms.html

Integer factorization

• https://en.wikipedia.org/wiki/Integer_factorization

Fractions

• https://mathworld.wolfram.com/Fraction.html

Reduced fractions

• https://www.mathsisfun.com/simplifying-fractions.html

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

• https://www.vocabulary.com/lists/149639

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

• https://en.wikipedia.org/wiki/Euclidean_geometry

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

• https://mathworld.wolfram.com/CartesianCoordinates.html

Isometries and similarities

• https://mathworld.wolfram.com/Isometry.html
• https://mathworld.wolfram.com/Similarity.html

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

• https://mathworld.wolfram.com/Pi.html
• https://tauday.com/tau-manifesto

Angles

• https://en.wikipedia.org/wiki/Angle

Algebra over a field

Functions

• https://mathworld.wolfram.com/Function.html

Function graphs

• https://mathworld.wolfram.com/FunctionGraph.html

Intervals

• https://mathworld.wolfram.com/Interval.html

Domains and ranges

• https://mathworld.wolfram.com/Domain.html
• https://mathworld.wolfram.com/Range.html

Increasing and decreasing functions

• https://mathworld.wolfram.com/IncreasingFunction.html
• https://mathworld.wolfram.com/DecreasingFunction.html

Extremums

• https://mathworld.wolfram.com/Extremum.html

Function transformations

• https://www.mathsisfun.com/sets/function-transformations.html

Commun functions, and their graphs

• https://www.mathsisfun.com/sets/functions-common.html

Equations, and solving them

• https://www.mathsisfun.com/algebra/equations-solving.html

Permutations and combinations

• https://en.wikipedia.org/wiki/Factorial
• https://en.wikipedia.org/wiki/Binomial_coefficient

Polynomials

• https://mathworld.wolfram.com/Polynomial.html

Polynomial addition and subtraction

• https://www.mathsisfun.com/algebra/polynomials-adding-subtracting.html

Polynomial multiplication and division

• https://www.mathsisfun.com/algebra/polynomials-multiplication-long.html
• https://www.mathsisfun.com/algebra/polynomials-division-long.html

Binomial theorem

• https://mathworld.wolfram.com/BinomialTheorem.html

Polynomial factorization

• https://en.wikipedia.org/wiki/Factorization_of_polynomials

Quadratic formula

• https://mathworld.wolfram.com/QuadraticFormula.html

Inequalities, and their most interesting properties

• https://mathworld.wolfram.com/Inequality.html
• https://en.wikipedia.org/wiki/Inequality_(mathematics)

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

Imaginary unit

• https://mathworld.wolfram.com/ImaginaryUnit.html

Complex conjugate

• https://en.wikipedia.org/wiki/Complex_conjugate

Absolute value

• https://mathworld.wolfram.com/AbsoluteValue.html

Powers and roots

• https://mathworld.wolfram.com/Power.html
• https://mathworld.wolfram.com/nthRoot.html

Exponent laws

• https://mathworld.wolfram.com/ExponentLaws.html

Rationalization

• https://www.mathsisfun.com/algebra/rationalize-denominator.html

Euler's number

• https://mathworld.wolfram.com/e.html

Logarithms

• https://mathworld.wolfram.com/Logarithm.html

Logarithm identities

• https://en.wikipedia.org/wiki/List_of_logarithmic_identities

Trigonometry

• https://mathworld.wolfram.com/Trigonometry.html

Trigonometric identities

• https://en.wikipedia.org/wiki/List_of_trigonometric_identities

Laws of sines and cosines

• https://en.wikipedia.org/wiki/Law_of_sines
• https://en.wikipedia.org/wiki/Law_of_cosines

Euler formula

• https://mathworld.wolfram.com/EulerFormula.html

Linear algebra

Scalars and vectors

• https://dictionary.cambridge.org/dictionary/english/scalar
• https://mathworld.wolfram.com/Vector.html

Vector addition and subtraction

• https://mathworld.wolfram.com/VectorAddition.html

Multiplication with vectors

• https://mathworld.wolfram.com/ScalarMultiplication.html
• https://mathworld.wolfram.com/DotProduct.html
• https://mathworld.wolfram.com/CrossProduct.html

Vector bases

• https://mathworld.wolfram.com/UnitVector.html
• https://mathworld.wolfram.com/VectorBasis.html

Matrices

• https://mathworld.wolfram.com/Matrix.html

Matrix addition and subtraction

• https://en.wikipedia.org/wiki/Matrix_addition

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

Simple square matrices

• https://mathworld.wolfram.com/TriangularMatrix.html
• https://mathworld.wolfram.com/DiagonalMatrix.html
• https://mathworld.wolfram.com/IdentityMatrix.html

Common matrix to scalar functions

• https://mathworld.wolfram.com/MatrixTrace.html
• https://mathworld.wolfram.com/Determinant.html

Common matrix transformations

• https://mathworld.wolfram.com/MatrixInverse.html
• https://mathworld.wolfram.com/Transpose.html
• https://mathworld.wolfram.com/ConjugateMatrix.html
• https://mathworld.wolfram.com/ConjugateTranspose.html

Most interesting properties of matrix operations

• https://math.mit.edu/~dyatlov/54summer10/matalg.pdf

Special square matrices

• https://mathworld.wolfram.com/OrthogonalMatrix.html
• https://mathworld.wolfram.com/HermitianMatrix.html
• https://mathworld.wolfram.com/UnitaryMatrix.html
• https://mathworld.wolfram.com/NormalMatrix.html
• https://mathworld.wolfram.com/SimilarMatrices.html

Linear transformations

• https://mathworld.wolfram.com/LinearTransformation.html
• https://en.wikipedia.org/wiki/Transformation_matrix

Solving systems of linear equations

• https://en.wikipedia.org/wiki/System_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

• https://mathworld.wolfram.com/EigenDecomposition.html

Calculus

Limits, and their most interesting properties

• https://mathworld.wolfram.com/Limit.html
• https://en.wikipedia.org/wiki/Limit_of_a_function

Squeeze theorem

• https://en.wikipedia.org/wiki/Squeeze_theorem

Derivatives and indefinite integrals, and their common formulas

• https://mathworld.wolfram.com/Derivative.html
• https://mathworld.wolfram.com/IndefiniteIntegral.html

L'Hôpital's rule

• https://en.wikipedia.org/wiki/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

• https://mathworld.wolfram.com/Sum.html
• https://mathworld.wolfram.com/Series.html

Convergence tests

• https://en.wikipedia.org/wiki/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

• https://mathworld.wolfram.com/RiemannSum.html
• https://mathworld.wolfram.com/DefiniteIntegral.html

Fundamental theorems of calculus

• https://mathworld.wolfram.com/FundamentalTheoremsofCalculus.html

Differential equations

• https://www.mathsisfun.com/calculus/differential-equations.html

Integral transforms

• https://en.wikipedia.org/wiki/Integral_transform

Vector calculus, and their common theorems

• https://en.wikipedia.org/wiki/Vector_calculus
• https://en.wikipedia.org/wiki/Gradient

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

• https://mathworld.wolfram.com/ExpectationValue.html

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

• https://en.wikipedia.org/wiki/Arithmetic_mean
• https://en.wikipedia.org/wiki/Variance

Statistics

• https://en.wikipedia.org/wiki/Statistic
• https://en.wikipedia.org/wiki/Statistics

Estimators and biases

• https://en.wikipedia.org/wiki/Estimator

Median, quartiles and percentiles

• https://en.wikipedia.org/wiki/Percentile

Common charts

• https://en.wikipedia.org/wiki/Chart

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