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.
Mathematics is the study of patterns of quantity, structure, change and space. In the modern view, it is the investigation of axiomatically defined abstract structures using formal logic as the common framework. The 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, for instance because they realize that the structure provides a unifying generalization for several subfields or a helpful tool in common calculations.
Finally, many mathematicians study in the areas that they do for aesthetic reasons – simply because they find the structures they investigate beautiful in and of themselves. 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. 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 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 making 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 dwarf planet Pluto (1931).