Mathematics

Part: General · Chapter: Mathematics · Edition: 2026-07-12
Keywords: number, algebra, analysis, geometry, proof, probability
Glossary: set, function, proof, group, limit — see GLOSSARY

At a glance

Mathematics studies structure, quantity, change, and space via definitions and proof.
From number systems to algebra, analysis, geometry, and probability.
Notation is a technology: clarity of definition precedes calculation.

Online references (live)

Offline snapshots (link-rot shield)

Offline snapshot omitted in this web edition. Offline snapshot · Mathematics · https://en.wikipedia.org/wiki/Mathematics · retrieved 2026-07-12

Mathematics

Offline snapshot omitted in this web edition. Offline snapshot · Set theory · https://en.wikipedia.org/wiki/Set_theory · retrieved 2026-07-12

Set theory

Offline snapshot omitted in this web edition. Offline snapshot · Proof · https://en.wikipedia.org/wiki/Mathematical_proof · retrieved 2026-07-12

Mathematical proof

Core formal map (scientific summary)

Mathematics is organised by objects, structures, and proofs.

1. Language of mathematics

Sets collect objects; membership is primitive.
Functions $f: A \to B$ assign to each $a \in A$ a unique $f(a) \in B$ .
Relations generalise functions (order, equivalence).
Proof is a finite derivation from axioms and inference rules.

2. Algebraic structures (sketch)

Structure	Operations	Example
Group	associative binary op, identity, inverses	$(\mathbb{Z}, +)$
Ring	two ops $+,\cdot$ with distributivity	$\mathbb{Z}$
Field	ring with multiplicative inverses (≠0)	$\mathbb{R}, \mathbb{C}$
Vector space	scalars from a field + vectors	$\mathbb{R}^n$

3. Analysis (sketch)

Limits formalise approximation: $\lim_{n\to\infty} a_n = L$ means that for every $\varepsilon > 0$ there exists $N$ such that $n > N \Rightarrow |a_n - L| < \varepsilon$ . Continuity, derivatives, and integrals build on this language.

4. Probability (sketch)

A probability space $(\Omega, \mathcal{F}, P)$ assigns measures to events. Random variables are measurable functions; expectation is integration against $P$.

5. How to use this chapter

The archive outline below is a curriculum map. For each numbered item: write a one-line definition, one example, one theorem name, one Wikipedia snapshot from the chapter card.

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Summary

Mathematics: A Comprehensive Guide

Part I: Foundations of Mathematics

Numbers and Arithmetic

1.1 Natural Numbers

1.2 Integers

1.3 Rational Numbers

1.4 Irrational Numbers

1.5 Real Numbers

1.6 Complex Numbers

Algebra

2.1 Variables and Expressions

2.2 Equations and Inequalities

2.3 Polynomials

2.4 Factoring

2.5 Functions and Relations

Geometry

3.1 Euclidean Geometry

3.2 Coordinate Geometry

3.3 Trigonometry

3.4 Analytic Geometry

3.5 Non-Euclidean Geometry

Set Theory

4.1 Sets and Elements

4.2 Set Operations

4.3 Venn Diagrams

4.4 Set Cardinality

4.5 Axiomatic Set Theory

Logic and Proof

5.1 Propositions and Truth Values

5.2 Logical Connectives

5.3 Truth Tables

5.4 Proof Techniques

5.5 Mathematical Induction

Part II: Advanced Topics

Calculus

6.1 Limits and Continuity

6.2 Differentiation

6.3 Integration

6.4 Applications of Derivatives

6.5 Techniques of Integration

6.6 Sequences and Series

Linear Algebra

7.1 Matrices and Vectors

7.2 Systems of Linear Equations

7.3 Eigenvalues and Eigenvectors

7.4 Linear Transformations

7.5 Inner Product Spaces

Probability and Statistics

8.1 Probability Theory

8.2 Random Variables

8.3 Probability Distributions

8.4 Statistical Inference

8.5 Hypothesis Testing

8.6 Regression Analysis

Number Theory

9.1 Prime Numbers

9.2 Divisibility and Congruence

9.3 Diophantine Equations

9.4 Cryptography

9.5 Continued Fractions

Abstract Algebra

10.1 Groups

10.2 Rings

10.3 Fields

10.4 Galois Theory

10.5 Algebraic Structures

Topology

11.1 Open and Closed Sets

11.2 Continuity and Homeomorphism

11.3 Compactness

11.4 Connectedness

11.5 Metric Spaces

Real Analysis

12.1 Limits of Functions

12.2 Sequences and Series of Functions

12.3 Continuity and Differentiability

12.4 Riemann Integration

12.5 Measure Theory

12.6 Lebesgue Integration

Part III: Advanced Pure Mathematics

Complex Analysis

13.1 Complex Functions

13.2 Contour Integration

13.3 Analytic Continuation

13.4 Conformal Mapping

Differential Equations

14.1 Ordinary Differential Equations

14.2 Partial Differential Equations

14.3 Existence and Uniqueness

14.4 Sturm-Liouville Theory

14.5 Green’s Functions

Algebraic Geometry

15.1 Algebraic Varieties

15.2 Projective Geometry

15.3 Bezout’s Theorem

15.4 Sheaf Theory

15.5 Schemes

Functional Analysis

16.1 Normed Vector Spaces

16.2 Banach Spaces

16.3 Hilbert Spaces

16.4 Operator Theory

16.5 Spectral Theory

Algebraic Number Theory

17.1 Dedekind Domains

17.2 Class Field Theory

17.3 Cyclotomic Fields

17.4 Quadratic Forms

17.5 Zeta Functions

Differential Topology

18.1 Smooth Manifolds

18.2 Tangent Spaces

18.3 Vector Bundles

18.4 Differential Forms

18.5 Morse Theory

Part IV: Cutting-Edge Mathematics

Algebraic Topology

19.1 Homotopy Theory

19.2 Homology and Cohomology

19.3 Singular Homology

19.4 Fiber Bundles

Representation Theory

20.1 Group Representations

20.2 Lie Algebras

20.3 Character Theory

20.4 Module Theory

20.5 Tensor Categories

Number Theory and Cryptography

21.1 Elliptic Curves

21.2 Modular Forms

21.3 L-Functions

21.4 Public Key Cryptosystems

21.5 Elliptic Curve Cryptography

Advanced Functional Analysis

22.1 C*-Algebras

22.2 von Neumann Algebras

22.3 Operator Algebras

22.4 Quantum Mechanics

22.5 Quantum Information Theory

Advanced Differential Geometry

23.1 Riemannian Manifolds

23.2 Curvature

23.3 Einstein’s Field Equations

23.4 Differential Forms on Manifolds

23.5 Geometric Topology

Glossary of Terms

6174

An Exceptionally Simple Theory of Everything

Sure! Here’s a structured index of fundamental topics in number theory, along with short descriptions:

Offline snapshot omitted in this web edition.

Fundamentals of Number Theory

1. Introduction to Number Theory

The study of integers, their properties, and relationships.

2. Divisibility and Prime Numbers

Divisibility: Understanding factors, multiples, and divisibility rules.
Prime Numbers: Numbers greater than 1 with exactly two divisors: 1 and itself.
Greatest Common Divisor (GCD) and Least Common Multiple (LCM): Fundamental tools for divisibility analysis.

3. Euclidean Algorithm

An efficient algorithm for computing the GCD of two integers.

4. Fundamental Theorem of Arithmetic

Every integer greater than 1 can be uniquely factored into prime numbers.

5. Congruences and Modular Arithmetic

Modular arithmetic: Remainder operations in division, useful in cryptography.
Linear congruences: Solving equations of the form $ax \equiv b \mod m$ .

6. Diophantine Equations

Equations where integer solutions are sought, such as Pell’s equation and Fermat’s Last Theorem.

7. Euler’s Theorem and Fermat’s Little Theorem

Fermat’s Little Theorem: If ( p ) is prime, then $a^{p-1} \equiv 1 \mod p$ .
Euler’s Theorem: A generalization of Fermat’s theorem using Euler’s totient function.

8. Chinese Remainder Theorem

A method for solving systems of congruences with different moduli.

9. Quadratic Residues and Reciprocity

Legendre and Jacobi symbols: Tools for determining whether a number is a square modulo another number.
Quadratic reciprocity law: Describes when an integer is a quadratic residue.

10. Perfect Numbers and Mersenne Primes

Perfect numbers: Numbers equal to the sum of their proper divisors.
Mersenne primes: Primes of the form ( 2^p - 1 ).

11. Continued Fractions and Approximation

Representing numbers as fractions for better approximations, important in Diophantine approximation.

12. Cryptographic Applications

RSA Encryption: Uses modular exponentiation and Euler’s theorem.
Elliptic Curve Cryptography: Advanced applications of number theory.

13. Additive Number Theory

Studying how numbers can be expressed as sums of others, including Goldbach’s Conjecture.

14. Analytical Number Theory

Involves the use of analysis to study prime distribution, including Riemann Zeta Function.

15. Computational Number Theory

Algorithms for primality testing, integer factorization, and fast modular exponentiation.

Definitions and references

This section does not replace the curriculum outline above. It supplies formal anchors and full references.

Core objects

Sets, functions, relations — language of modern mathematics.
Groups / rings / fields / vector spaces — algebraic structure.
Limits & continuity — $\varepsilon$ -(N) / $\varepsilon$ - $\delta$ analysis.
Probability spaces $(\Omega,\mathcal F,P)$ — measure-theoretic probability (intro level).

Full papers & books

Work	Role
Euclid — Elements	Axiomatic geometry heritage
Hardy — A Course of Pure Mathematics	Analysis foundations
Rudin — Principles of Mathematical Analysis	Rigorous undergrad analysis
Artin — Algebra	Modern algebra
Tao — Analysis I	Contemporary teaching text
Wikipedia hubs	Mathematics, Proof

Part index

See General index.