This page reflects one organizational view of mathematics. /Schemes offers other views. Following the list of subfields, a narrative broadly outlines their origins and connections.
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.
Mathematics (Greek mathema: science, learning; mathematikos: fond of learning) studies 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 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 ground work for quantum mechanics among many other things.
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.