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Mathematics, field of thought concerned with relationships involving concepts of quantity, space, and symbolism. Over the past several centuries mathematics developed to include axiomatic-deductive reasoning. This aspect of mathematics is credited to the classical Greeks and is traced back to Euclid, who formalized it in 300 B.C. with his work Elements. Axiomatic-deductive systems are based on elementary ideas assumed self-evident (axioms) and formal rules governing the mathematical system. Consequences (theorems) can then be deduced systematically and logically from axioms. All mathematical systems have this quality. Familiar examples include algebra, where relationships between known and unknown quantities are represented symbolically; arithmetic, the science of quantity and space concerned with numbers and rules for manipulating them such as addition or multiplication; calculus, dealing with relationships involving rates of change; and geometry, concerned with spatial relationships. Mathematics is pursued to solve practical problems as well as to enhance its logical and often abstract nature. Consequently it is often categorized as applied and pure. The applied mathematician uses or develops mathematics as a tool, solving problems or relationships in other fields. Physicists and engineers often apply calculus to questions of motion, economists apply concepts of linear algebra to determine cost effective solutions, and statistics and probability are frequently used by psychologists. The scholar of pure mathematics investigates logical relationships of abstract quantities or objects. Questions of the completeness and consistency within given mathematical constructs are addressed in pure mathematics.

Major contributions in the development of western mathematics came from Egypt (3000–1600 B.C.), Babylonia (1700–300 B.C.), Greece (600–200 B.C.), and the Hindu and Arab world (600 B.C.A.D. 1450). Major achievements before modern times have also occurred in China, Japan, and Incan and Aztec empires.

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