Mathematics

Overview of fields of mathematics

The major disciplines within mathematics first 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 (i.e. algebra, geometry and analysis). In addition to these three main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to logic and other simpler systems (foundations) and to the empirical systems of the various sciences (applied mathematics).

Related Topics:
Astronomical - Algebra - Geometry - Analysis

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

The study of structure starts with numbers, first the familiar natural numbers and integers and their arithmetical operations, which are characterized in elementary algebra. 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 everyday numbers. Long-standing questions about ruler-and-compass constructions were finally settled by Galois theory. The physically important concept of vectors, generalized to vector spaces and studied in linear algebra, belongs to the two branches of structure and space.

Related Topics:
Number - Natural number - Integer - Arithmetic - Elementary algebra - Number theory - Abstract algebra - Rings - Field - Ruler-and-compass construction - Galois theory - Vector - Vector space - Linear algebra

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

The study of space originates with geometry, first the Euclidean geometry and trigonometry of familiar three-dimensional space (also applying to both more and fewer dimensions), 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 functions, fiber bundles, derivatives, smoothness, and direction, while in algebraic geometry geometrical objects are described as solution sets of polynomial equations. Group theory investigates the concept of symmetry abstractly; topology, the greatest growth area in the twentieth century, has a focus on the concept of continuity. Both the group theory of Lie groups and topology reveal the intimate connections of space, structure and change.

Related Topics:
Geometry - Euclidean geometry - Trigonometry - Dimension - Non-Euclidean geometries - General relativity - Differential geometry - Algebraic geometry - Fiber bundle - Derivative - Smoothness - Solution set - Polynomial - Group theory - Symmetry - Topology - Twentieth century - Continuity - Lie group

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Understanding and describing change in measurable quantities is the common theme of the natural sciences, and calculus was developed as a most useful tool for that. 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 several reasons, it is convenient to generalize to 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. Many phenomena in nature can be described by dynamical systems; chaos theory makes precise the ways in which many of these systems exhibit unpredictable yet still deterministic behavior.

Related Topics:
Calculus - Function - Differential equations - Real numbers - Real analysis - Complex number - Complex analysis - Functional analysis - Quantum mechanics - Dynamical system - Chaos theory - Deterministic

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

In order to clarify the foundations of mathematics, the fields first of mathematical logic and then set theory were developed. Mathematical logic, which divides into recursion theory, model theory and proof theory, is now closely linked to computer science. When electronic computers were first conceived, several essential theoretical concepts were shaped by mathematicians, leading to the fields of computability theory, computational complexity theory, and information theory. Many of those topics are now investigated in theoretical computer science. Discrete mathematics is the common name for the fields of mathematics most generally useful in computer science.

Related Topics:
Foundations of mathematics - Mathematical logic - Set theory - Recursion theory - Model theory - Proof theory - Computer science - Electronic computer - Computability theory - Computational complexity theory - Information theory - Theoretical computer science - Discrete mathematics

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

An important field in applied mathematics is statistics, which uses probability theory as a tool and allows the description, analysis and prediction of phenomena where chance plays a part. It is used in all sciences. Numerical analysis investigates methods for efficiently solving a broad range of mathematical problems numerically on computers, beyond human capacities, and taking rounding errors and other sources of error into account to obtain credible answers.

Related Topics:
Applied mathematics - Statistics - Probability theory - Numerical analysis - Rounding error

~ ~ ~ ~ ~ ~ ~ ~ ~ ~