Mathematics is the study of numbers, sets of points, and various abstract elements, together with relations between them and operations performed on them. Originally mathematics was concerned with the properties of numbers and space, as the science of quantity, whether of magnitudes, as in geometry, or of numbers, as in arithmetic, or the generalization of these two fields, as in algebra.

Toward the middle of the 19th century, mathematics increasingly came to be regarded as the science of relations, more generally concerned with deductions made in abstract systems. Science/theory.gif This latter view encompasses mathematical or symbolic logic, the science of using symbols to provide an exact theory of logical deduction and inference based on definitions, axioms, postulates, and rules for combining and transforming primitive elements into more complex relations and theorems.

Mathematics is the science of logical reasoning, in which valid conclusions are deduced from a set of basic assumptions, or axioms, It involves a search for truth. It is rigorous and precise.

Mathematics is a tool for problem solving; organizing, simplifying, and interpreting data; and performing calculations that are necessary in subjects such as science, commerce, and industry. The development of modern computers and electronic calculators has enabled mathematicians to solve problems that previously were extremely difficult or impossible to solve.

Mathematics is usually divided into pure mathematics - abstract reasoning based on axioms and rules for maing deductions from them - and applied mathematics, in which mathematical methods are applied to 'real world' problems in engineering, physics, economics, business, navigation, astronomy, chemistry, electronics, computer science, etc. Applied mathematics mathematics concerns itself with the application of mathematical knowledge to other domains.

Some branches of mathematics were developed in order to solve certain physical problems or to explain physical phenomena. In his study of astronomy and astrophysics, Johannes Kepler found it necessary to develop new mathematics.

Mathematical calculations sometimes lead to the discovery of new physical phenomena. Deviations in the motions of Neptune from the predictions of mathematical theory led to the conclusion that an unknown planet existed. Calculations pinpointed the position of this body and led to the discovery of the planet Pluto (1931).

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Mathematics is commonly defined as the study of patterns of structure, change, and space. In the modern formalist view, it is the investigation of axiomatically defined abstract structures using logic and mathematical notation. Mathematics is often abbreviated to math in North America and maths in other English-speaking countries.

These 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, because the structures may provide, for instance, a unifying generalization for several subfields, or a helpful tool for common calculations. Finally, many mathematicians study the areas they do for purely aesthetic reasons, viewing mathematics as an art form rather than as a practical or applied science.