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05 Math

01 Encyclopedia

Algebra

binomial coefficient

degree of a polynomial

greatest common divisor

Arithmetic

Calculus

definite integral

l'Hopital's rule

Code Theory

Hamming distance

minimum Hamming distance

Combinatorics

Complex Numbers

complex conjugate

Foundations

axiomatic system

mathematical object

mathematical property

primitive notion

Functions

equality as functions

function of a real variable

inverse function

real-valued function

Fundamentals

Geometry

Coordinate

Cartesian coordinate system

coordinate system

polar coordinate system

Models

three-point geometry

Systems

incidence geometry

Groups

general linear group

linearly ordered group

order in a group

order of a group

Homogeneous Relations

antisymmetric relation

asymmetric relation

connected relation

equivalence relation

irreflexive relation

reflexive relation

strongly connected relation

symmetric relation

transitive relation

trichotomous relation

Linear Algebra

augmented matrix

identity matrix

invertible matrix

linear combination

linear independence

Logic

Logical Operations

alternative denial

converse implication

converse nonimplication

exclusive disjunction

logical connective

material implication

material nonimplication

Logical Quantifiers

logical quantifier

formal language

Measure

measurable function

measurable space

pushforward measure

Metric

metric space axioms

Morphism

Numbers

Number Systems

algebraic number

imaginary number

irrational number

rational number

transcendental number

Sequences

triangular number

Operations

binary operation

Cartesian product

Order

Orderings

strict partial order

strict total order

strict weak order

strict well-order

order-convex set

partially ordered set

well-ordering principle

Probability

Distribution

Examples

Bernoulli distribution

binomial distribution

continuous uniform distribution

exponential distribution

normal distribution

Poisson distribution

cumulative distribution function

probability density function

probability distribution

probability mass function

Experiment

Bernoulli trial

binomial experiment

collectively exhaustive events

mutually exclusive events

Structure

probability axioms

probability measure

probability space

conditional probability

random variable

Proof

reductio ad absurdum

Propositional Logic

Real Analysis

continuous function

law of trichotomy

Relations

Domain

codomain of definition

domain of definition

Properties

Totality

surjective relation

Uniqueness

functional relation

injective relation

bijective relation

binary relation

complementary relation

composition of relations

converse relation

equivalence class

finitary relation

Rings

endomorphism ring

Sequences

arithmetic sequence

geometric sequence

Series

geometric series

Sets

Intervals

nested intervals

partition of an interval

Set Operations

set intersection

De Morgan's laws

equality as sets

identity element

inclusion-exclusion principle

indicator function

inverse element

partition of a set

Shapes

Polygons

Quadrilaterals

Triangles

regular polygon

Statistics

correlation coefficient

standard deviation

Structures

algebraic structure

integral domain

linear subspace

Trigonometry

trigonometric function

Vectors

inner product space

Euclid's postulates

Euler's product formula

Euler's totient function

Pascal's identity

Pascal's triangle

pigeonhole principle

Russel's paradox

02 Distinctions

comparable and incomparable elements

convergent and divergent series

degree wrt polynomial

discrete and continuous random variables

discrete and continuous variables

finite and infinite

greatest and least elements

homogeneous and heterogeneous relations

independent and dependent events

independent and dependent random variables

independent and dependent variables

maximal and minimal elements

subset and superset

supremum and infimum

upper and lower bounds

03 Theorems

Abstract Algebra

A group element of finite order generates a finite cyclic subgroup of that order.

A group element of infinite order generates an infinite cyclic subgroup.

Cyclic subgroups are equivalent if one power is expressible as the gcd of the other with the order of the base.

Every permutation is the product of disjoint cycles.

Every power of a group element with infinite order is unique.

Every subgroup of a cyclic group is itself cyclic.

For each factor of the order of a cyclic group, the number of elements in the group with the order of that factor is equal to the number of totatives of the factor.

Fundamental Theory of Cyclic Groups

Greatest Common Divisor is expressible as a linear combination of its arguments

Group Cancellation Laws

If a group is finite, then every element has finite order.

Powers distribute over an operation if and only if the group is abelian.

The group center is a subgroup.

The identity element of a group is unique.

The inverse of an element in a group is unique.

The order of a group element divides the difference in exponent of equivalent powers of that element.

The order of a group element to some power is equal to the order of the generator, divided by the gcd of the order of the generator with the power of the element.

The order of any cyclic subgroup divides the order of any cyclic group which contains it.

The product of two elements with finite order which commute in a group has finite order which divides the product of the orders of the elements.

There is exactly one subgroup of the order of each divisor of the order of its container cyclic group.

Algebra

Fundamental Theorem of Algebra

Calculus

decomposition of definite integral

Fundamental Theorem of Calculus

swapping limits of integration in definite integral

Code Theory

An error correcting code has disjoint Hamming balls.

Hamming distance for binary codes can be calculated using the binomial coefficient.

Sphere Packing Bound

The cardinality of a Hamming ball is the total number of elements that are any distance up to and including its radius.

Fallacy

Division by Zero Fallacy

Fundamentals

principle of mathematical induction

Geometry

triangle inequality

Linear Algebra

A regular Markov chain has a stationary distribution and it is unique.

Fundamental Theorem of Regular Markov Chains

Markov chain contraction mapping

Number Theory

An odd number squared is odd.

The square root of two is irrational.

Probability

Bayes's theorem

law of total probability

multiplication rule for conditional probability

Real Analysis

Archimedean property

Cantor's theorem

density theorem

Every number is between two natural numbers

nested intervals property

The infimum of the set of reciprocals of the natural numbers is zero.

Statistics

central limit theorem

Triangular Numbers

The sum of two consecutive triangular numbers is a square number.

Bernoulli's inequality

binomial theorem

Division Algorithm

Euler's formula

Euler's theorem

Euler's totient function is multiplicative for coprime values.

Pythagorean theorem

04 Explainers

converting between cartesian and polar coordinates

error detection and correction

horizontal transformation of functions

introduction to probability

synthetic division

15 LaTeX

01 Showcase

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Euler's totient function is multiplicative for coprime values.

See Euler's totient function

If $m, n$ are coprime, then $φ (m) φ (n) = φ (m n)$ .

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Euler's totient function

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