Areas of mathematics


encompasses a growing variety and depth of subjects over history, and comprehension requires a system to categorize and organize the many subjects into more general areas of mathematics. A number of different classification schemes have arisen, and though they share some similarities, there are differences due in part to the different purposes they serve. In addition, as mathematics continues to be developed, these classification schemes must change as well to account for newly created areas or newly discovered links between different areas. Classification is made more difficult by some subjects, often the most active, which straddle the boundary between different areas.
A traditional division of mathematics is into pure mathematics, mathematics studied for its intrinsic interest, and applied mathematics, mathematics that can be directly applied to real world problems.
This division is not always clear and many subjects have been developed as pure mathematics to find unexpected applications later on. Broad divisions, such as discrete mathematics and computational mathematics, have emerged more recently.
An ideal system of classification permits adding new areas into the organization of previous knowledge, and fitting surprising discoveries and unexpected interactions into the outline.
For example, the Langlands program has found unexpected connections between areas previously thought unconnected, at least Galois groups, Riemann surfaces and number theory.

Classification systems

Pure mathematics

Foundations">Foundations of mathematics">Foundations

;including set theory and mathematical logic
Mathematicians have always worked with logic and symbols, but for centuries the underlying laws of logic were taken for granted, and never expressed symbolically. Mathematical logic, also known as symbolic logic, was developed when people finally realized that the tools of mathematics can be used to study the structure of logic itself. Areas of research in this field have expanded rapidly, and are usually subdivided into several distinct subfields.
;History and biography
The history of mathematics is inextricably intertwined with the subject itself. This is perfectly natural: mathematics has an internal organic structure, deriving new theorems from those that have come before. As each new generation of mathematicians builds upon the achievements of their ancestors, the subject itself expands and grows new layers, like an onion.
;Recreational mathematics
From magic squares to the Mandelbrot set, numbers have been a source of amusement and delight for millions of people throughout the ages. Many important branches of "serious" mathematics have their roots in what was once a mere puzzle and/or game.

[Number Theory]

is the study of numbers and the properties of operations between them. Number theory is traditionally concerned with the properties of integers, but more recently, it has come to be concerned with wider classes of problems that have arisen naturally from the study of integers.

Algebra">Abstract algebra">Algebra

The study of structure begins with numbers, first the familiar natural numbers and integers and their arithmetical operations, which are recorded in elementary algebra. The deeper properties of these 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 everyday numbers. Long standing questions about compass and straightedge constructions were finally settled by Galois theory. The physically important concept of vectors, generalized to vector spaces, is studied in linear algebra. Themes common to all kinds of algebraic structures are studied in universal algebra.

[Combinatorics]

Combinatorics is the study of finite or discrete collections of objects that satisfy specified criteria. In particular, it is concerned with "counting" the objects in those collections and with deciding whether certain "optimal" objects exist. It includes graph theory, used to describe inter-connected objects. See also the list of combinatorics topics, list of graph theory topics and glossary of graph theory. A combinatorial flavour is present in many parts of problem-solving.

[Geometry]

Geometry deals with spatial relationships, using fundamental qualities or axioms. Such axioms can be used in conjunction with mathematical definitions for points, straight lines, curves, surfaces, and solids to draw logical conclusions. See also List of geometry topics.

[Topology]

Deals with the properties of a figure that do not change when the figure is continuously deformed. The main areas are point set topology, algebraic topology, and the topology of manifolds, defined below.

[Mathematical Analysis]

Within the world of mathematics, analysis is the branch that focuses on change: rates of change, accumulated change, and multiple things changing relative to one another.
Modern analysis is a vast and rapidly expanding branch of mathematics that touches almost every other subdivision of the discipline, finding direct and indirect applications in topics as diverse as number theory, cryptography, and abstract algebra. It is also the language of science itself and is used across chemistry, biology, and physics, from astrophysics to X-ray crystallography.

Applied mathematics

Probability and statistics

;Numerical analysis
;Computer algebra

Physical sciences

; Mechanics
; Mechanics of structures
; Mechanics of deformable solids
; Fluid mechanics
; Particle mechanics

Other applied mathematics