Difference between revisions of "Mathematics"

Revision as of 18:50, July 1, 2009

Mathematics is the study and practice of rigorous inference about abstract structures. It includes many practical results concerning quantity and measure, such as calculations involving numbers, financial accounting, geometric construction of building, astronomical calculations, calendar dating, telling time, engineering, physics, chemistry, etc. but also more abstract issues such as establishing the conditions under which certain kinds of equations and formulas have solutions.

Symbols, Equations, and Theories

Pure and Applied Mathematics

Applied mathematics concerns the use of mathematical methods for practical purposes. Pure mathematics involves reasoning about abstract structures.

Applied mathematics

Applied mathematics has its emphasis in applications, and is used extensively in the sciences such as physics, chemistry, medicine, and biology, as well as engineering, mechanics and technology. Economics and information theory also uses applied mathematics. Mathmeticians involved in research can and do create new theories, mathematical ideas, and new areas of study simply from their use of applied mathematics to solve various problems.

Pure mathematics

Branches of Mathematics

Arithmetic

Algebra

Broadly, speaking algebra concerns 'addition' and 'multiplication', but in the widest possible sense. The objects that are being added or multiplied can be numbers, as in number theory, but they can also can be more general structures such as matrices, functions, polynomials, vectors or many others. Concentrating on addition and multiplication does not exclude subtraction or division, since subtraction is formally considered to be addition of an additive inverse and division is considered to be multiplication by a multiplicative inverse. That is, subtracting 3 from 2 is rigorously defined to adding the number -2 to 3. Minus two is called the additive inverse of +2. Similarly, dividing 3 by 2 is formally defined in terms of multiplying 3 by (1/2), where 1/2 is the multiplicative inverse of 2. Abstract algebra is the study of algebraic structures such as groups, rings, and fields.

Analysis

Analysis is concerned with limits and other infinite processes. This subject includes the theory of limits of sequences and series and all forms of statistics, and calculus, including the calculus of several variables, vector calculus and tensor calculus. Also included is numerical analysis, the study of error propagation in algorithms carried out to finite precision. Additional topics in analysis include real analysis and complex analysis.

Chaos Theory

Geometry

Geometry was defined by Felix Klein as the study of invariants under groups of transformations. For example, the Euclidean transformations are translation, rotation and reflection. The quantities that are not altered by these transformations are things like angles and distances, so these are the subjects of interest in Euclidean geometry. Other types of transformations, such as the affine transformations, define other types of geometry. Topology is concerned with the connectedness of objects, rather than the distance between them. It is sometimes called 'rubber sheet geometry', as it concerns properties (that is, arcs between nodes in networks) of objects that would be preserved even if a diagram of them were to be stretched or shrunk.

Logic and set theory

All of mathematics can be expressed in terms of sets. Sets are defined by a collection of axioms called the Zermelo-Fraenkel axioms. One of the axioms, the Axiom of Choice, has been the subject of much discussion

Probability and Statistics

Topology

Topology began with Leonard Euler's consideration of the Königsberg Bridges Problem, which also introduced Graph Theory. Beck's map of the London Underground in 1933 used a topological distortion of the locations of the subway stations in order to produce a more useful and artistic map. Differential geometry is a specialized field of its own.

Triginometry