Mathematics
From Encyclopædia
Mathematics is the study of
numbers, sets of points, and various
abstract elements, together with relations between them and operations performed on them. Mathematics deals with size, order, shape, and other relationships among quantities. For the history of this subject, see
history of mathematics.ASPECTS OF MATHEMATICSMathematics is variously considered a
language, an art, a
science, a tool, and a game.A LanguageA
language is an agreed-upon set of symbols or sounds; mathematics may be considered as the
language used to express size and order. Equations and statements of inequality are mathematical sentences.
constants correspond to nouns, and variables are the counterparts of pronouns.An ArtMathematical ideas fit together in a harmonious manner. A beauty exists in the patterns, relationships, and symmetry in arithmetic and geometry. Developing new mathematical theories, concepts, and systems is aesthetically satisfying. The study of mathematics can be a rewarding endeavor in much the same sense as can the study of history,
literature, or
music.A ScienceMathematics is the
science of logical reasoning, in which valid conclusions are arrived at from a set of AXIOMS (see
logic). It involves a search for truth. It is rigorous and precise. Although some theories discovered 2,000 years ago are still valid, mathematics continues to change and develop.A ToolMathematics is a tool in that it contains the skills for PROBLEM SOLVING; organizing, simplifying, and interpreting data; and performing calculations that are necessary in subjects such as
science, business, and industry. The development of modern computers and calculators has enabled mathematicians to solve problems that previously were extremely difficult or impossible to solve. Some areas of matematics were developed specifically to solve certain types of problems. One goal of mathematics is to solve a problem in a systematic way so that similar problems can be solved more easily in the same way.A GameThe individual can create a set of consistent rules and regulations (axioms) and proceed by logical reasoning to invent and play the game (see
games, MATHEMATICAL). Those who consider mathematics a game enjoy the challenge of developing new mathematics and of solving previously unsolved problems.BRANCHES OF MATHEMATICSSome branches of mathematics were developed in order to solve certain physical problems or to explain physical phenomena. For example, in his study of astronomy, Johannes KEPLER found it necessary to develop new mathematics. On the other
hand, mathematical calculations sometimes
lead to the discovery of new physical phenomena. Deviations in the motions of Neptune from the predictions of the mathematical theory led to the conclusion that an unknown planet existed. Exhaustive calculations pinpointed the position of this body and led to the discovery of the planet Pluto (1931).Mathematics today can be subdivided into pure mathematics and applied mathematics. Applied mathematics deals with solutions to practical problems in areas such as
physics,
economics, business,
navigation, and astronomy. Pure mathematics deals with the study of the
abstract properties of mathematical quantities and systems without regard to application. When a new application is found, a topic that was originally classified as pure mathematics may then become a part of applied mathematics. Computer
science,
probability, STATISTICS, and
operations research are often considered part of applied mathematics, whereas
abstract algebra,
number theory, and topology are usually considered part of pure mathematics.Mathematics may also be divided into various branches, depending on the elements and axioms used. Some major branches can be further subdivided into several subbranches. Each branch usually consists of definitions, undefined terms, elements, axioms, operations, relations, and theorems.ARITHMETIC deals with
numbers and the fundamental operations (addition, subtraction, multiplication, and division) and the extensions of these (raising to powers and extracting roots). Arithmetic is sometimes called the art of computation.ALGEBRA involves the operations of arithmetic. Unknown
numbers are represented by symbols called variables. Open mathematical sentences (equations and inequalities involving variables) are solved for the "unknowns." Systems of equations are used to solve practical problems. Solution of systems of the LINEAR EQUATION
leads to the study of LINEAR ALGEBRA, in which the elements are matrices and vectors (see MATRIX).
abstract algebra is the study of systems that satisfy certain sets of axioms. These structures may be called a FIELD, a RING, a group, a domain (see GROUP THEORY). The elements used in
abstract algebra may be
numbers, vectors, or even geometric transformations.GEOMETRY is the branch of mathematics that deals with sets of points in a plane or in space. The study of plane curves, angles, polygons, and lines is called plane geometry. The study of space curves (in three-dimensional space) such as spheres, cones, cylinders, and po?yhedra is called solid geometry. In about 300 BC, EUCLID established a set of axioms for geometry. In
Euclidean geometry, all of these axioms are obeyed. NON-
Euclidean geometry has been developed by denying the validity of the famous fifth postulate (parallel postulate), which was equivalent to the statement that "given a line and a point not on the line, one and only one coplanar line can be drawn through the point parallel to the given line."ANALYTIC GEOMETRY is the study of geometry through algebraic methods. DIFFERENTIAL GEOMETRY applies techniques of
calculus to geometry and studies such local properties as
tangents and curvature. TOPOLOGY, which has been developed in the 20th century, is a study of generalized geometric elements and such properties as connectedness and compactness. Fractal geometry (see
fractal geometry), which, like topology, includes structures that are non-Euclidean, has been applied in
Chaos Theory.
trigonometry is the branch of mathematics used in computing, rather than directly measuring, distances. The trigonometric functions (
sine, cosine,
tangent, cotangent,
secant, and cosecant) can be defined as the ratios of lengths of sides of right triangles or in terms of coordinates of terminal points of arcs on the unit circle (circular functions).Analysis is the name given to the branches of mathematics that use the concept of a LIMIT.
calculus is considered a branch of analysis; so are subjects that depend on the concepts of
calculus, such as DIFFERENTIAL EQUATIONS,
vector analysis, real analysis, and complex analysis. DIFFERENTIAL
calculus involves derivatives and deals with such topics as MAXIMA AND MINIMA and rates of change of functions. The definite integral--a quantity studied in INTEGRAL
calculus--can be used to find areas and volumes of irregular figures, to find lengths of curves, and to determine convergence or divergence of infinite series of
numbers.
number THEORY is one of the oldest branches of pure mathematics. The elements used are the integers, and the topics investigated include prime
numbers (see
PRIME NUMBER), factorization, congruences, and DIOPHANTINE EQUATIONS (equations with integral solutions).Joe K. SmithBibliography: Barrow, John D., Pi in the Sky: Counting, Thinking, and Being (1992);
beavers,
Mary, Essential Mathematics (1983); Courant, Richard, and Robbins, Herbert, What Is Mathematics? (1978); Dunham,
William, Journey through Genius (1990); Kline, Morris, Mathematics and the Search for Knowledge (1986) and Mathematics for Non-Mathematicians (1985); McLeish, John,
number (1992); Peterson, Ivars, The Mathematical Tourist (1988); Wheeler, Ruric, Modern Mathematics, 7th ed. (1988).
