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Mathematics (Greek mathema: science, learning; mathematikos: fond of learning) is the study of patterns that involves the abstract properties of quantity, structure, change and space. In the modern view, it is the investigation of axiomatically defined abstract structures using logic as the common framework. The specific structures investigated often have their origin in the natural sciences, most commonly in physics, but mathematicians also define and investigate structures for reasons purely internal to mathematics, for instance because they realize that the structure provides a unifying generalization for several subfields or a helpful tool in common calculations.

Historically, mathematics arose out of the need to do calculations in commerce, to measure land and to predict astronomical events. These three needs can be roughly related to the broad subdivision of mathematics into the study of structure, space and change.

The study of structure starts with numbers, initially the familiar natural numbers and integers. The rules governing arithmetical operations are recorded in elementary algebra, and the deeper properties of whole numbers are studied in number theory. The investigation of methods to solve equations leads to the field of abstract algebra, which, among other things, studies rings and fields, structures that generalize the properties possessed by the familiar numbers. The physically important concept of vector, generalized to vector spaces and studied in linear algebra, belongs to the two branches of structure and space.

The study of space originates with geometry, first the Euclidean geometry and trigonometry of familiar three-dimensional space, but later also generalized to non-Euclidean geometries which play a central role in general relativity. The modern fields of differential geometry and algebraic geometry generalize geometry in different directions: differential geometry emphasizes the concepts of coordinate system, smoothness and direction, while in algebraic geometry geometrical objects are described as solution sets of equations. Group theory investigates the concept of symmetry abstractly and provides a link between the studies of space and structure. Topology connects the study of space and the study of change by focusing on the concept of continuity.

Understanding and describing the change in measurable variables is the central topic of the natural sciences, and calculus was developed as a most useful tool for doing just this. The central concept used to describe a changing variable is that of a function. Many problems lead quite naturally to relations between a quantity and its rate of change, and the methods to solve these are studied in the field of differential equations. The numbers used to represent continuous quantities are the real numbers, and the detailed study of their properties and the properties of real-valued functions is known as real analysis. For mathematical reasons, it is convenient to introduce the complex numbers which are studied in complex analysis. Functional analysis focuses attention on (typically infinite-dimensional) spaces of functions, laying the groundwork for quantum mechanics among many other things.

In order to clarify and investigate the foundations of mathematics, the fields of set theory, mathematical logic and model theory were developed.

When computers were first conceived, several surrounding theoretical questions were tackled by mathematicians, leading to the fields of computability theory, computational complexity theory, information theory and algorithmic information theory. Many of these questions are now investigated in theoretical computer science.

An important field in applied mathematics is probability and statistics, which allows the description, analysis and prediction of random phenomena and is used in all sciences. Numerical analysis investigates the methods for performing calculations on computers and discrete mathematics is the common name for those fields of mathematics useful in computer science.

This following list of subfields and topics reflects one organizational view of mathematics:

Quantity

Numbers -- Natural numbers -- Integers -- Rational numbers -- Real numbers -- Complex numbers -- Quaternions -- Octonions -- Sedenions -- Hyperreal numbers -- Surreal numbers -- Ordinal numbers -- Cardinal numbers -- p-adic numbers -- Integer sequences -- Mathematical constants -- Number names -- Infinity

Change

Calculus -- Vector calculus -- Analysis -- Differential equations -- Dynamical systems and chaos theory -- List of functions

Structure

Abstract algebra -- Number theory -- Algebraic geometry -- Group theory -- Monoids -- Analysis -- Topology -- Linear algebra -- Graph theory -- Universal algebra -- Category theory

Space

Topology -- Geometry -- Trigonometry -- Algebraic geometry -- Differential geometry -- Differential topology -- Algebraic topology -- Linear Algebra

Finite Mathematics

Combinatorics -- Basic set theory -- Probability and statistics -- Theory of computation -- Discrete mathematics -- Cryptography -- Graph theory -- Game theory

Applied Mathematics

Mechanics -- Numerical analysis -- Optimization -- Probability and statistics

Famous Theorems and Conjectures

Fermat's last theorem -- Riemann hypothesis -- Continuum hypothesis -- P=NP -- Goldbach's conjecture -- Twin Prime Conjecture -- Gödel's incompleteness theorems -- Poincaré conjecture -- Cantor's diagonal argument -- Pythagorean theorem -- Central limit theorem -- Fundamental theorem of calculus -- Fundamental theorem of algebra -- Four color theorem -- Zorn's lemma -- "The most remarkable formula in the world"

Foundations and Methods

Philosophy of mathematics -- Mathematical intuitionism -- Mathematical constructivism -- Foundations of mathematics -- Set theory -- Symbolic logic -- Model theory -- Category theory -- Theorem-proving

History and the World of Mathematicians

History of mathematics -- Mathematicians -- Fields medal -- Millennium Prize Problems (Clay Math Prize) -- International Mathematical Union -- Mathematics competitions

What are our priorities for writing in this area? To help develop a list of the most basic topics in Mathematics, please see Mathematics basic topics.

Further Reading:

• Davis, Philip J.; Hersh, Reuben: The Mathematical Experience. Birkhäuser, Boston, Mass., 1980. A gentle introduction to the world of mathematics.
• Rusin, Dave: The Mathematical Atlas, http://www.math-atlas.org. A tour through the various branches of modern mathematics.
• Weisstein, Eric: World of Mathematics, http://www.mathworld.com. An encyclopedia of mathematics.
• Planet Math, http://planetmath.org. A free math encyclopedia under construction.
• Mathematical Society of Japan: Encyclopedic Dictionary of Mathematics, 2nd ed.. MIT Press, Cambridge, Mass., 1993. Definitions, theorems and references.