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Mathematics is a deductive study of numbers, geometry and various abstract constructs, or structures.

Mathematics is very broadly divided into foundations, algebra, analysis, geometry, and applied mathematics. The term foundations is used to refer to the formulation and analysis of the language, axioms, and logical methods on which all of mathematics rests. The scope and complexity of modern mathematics requires a very fine analysis of the formal language in which meaningful mathematical statements may be formulated and perhaps be proved true or false. Most apparent mathematical contradictions have been shown to derive from an imprecise and inconsistent use of language. A basic task is to furnish a set of axioms effectively free of contradictions and at the same time rich enough to constitute a deductive source for all of modern mathematics.

The general case of several equations of high degree leads to algebraic geometry, so named because the sets of solutions of such systems are often studied by geometric methods. Some of the concepts of modern algebra have found their way into elementary mathematics education in the so-called new mathematics. Some important abstractions recently introduced in algebra are the notions of category and functor, which grew out of so-called homological algebra. Arithmetic and number theory are also a part of algebra. The essential ingredient of analysis is the use of infinite processes. For example, the area of a circle may be computed as the limiting value of the areas of inscribed regular polygons as the number of sides of the polygons increases indefinitely. The basic branch of analysis is the calculus. The general problem of measuring lengths, areas, volumes, and other quantities as limits by means of approximating polygonal figures leads to the integral calculus. The differential calculus arises similarly from the problem of finding the tangent line to a curve at a point. Other branches of analysis result from the application of the concepts and methods of the calculus to various mathematical entities. For example, vector analy¬sis is the calculus of functions whose variables are vectors. Here various types of derivatives and integrals may be introduced. They lead, among other things, to the theory of differential and integral equations, in which the unknowns are functions rather than numbers, as in algebraic equations. Differential equations are often the most natural way in which to express the laws governing the behaviour of various physical systems. Calculus is one of the most powerful tools of mathematics. Its applications both in pure mathematics and in virtually every scientific domain are manifold.

The 20th century has seen an enormous development of topology which is the study of very general geometric objects, called topological spaces, with respect to relations that are much weaker than congruence and similarity. Other branches of geometry include algebraic geometry, which is now in a vigorous state of development, and differential geometry in which the methods of analysis are brought to bear on geometric problems. Applied mathematics is a term loosely designating a wide range of studies with significant current use in the empirical sciences. It includes numerical methods and computer science, which seeks concrete solutions, sometimes approximate, to explicit mathematical problems. In addition, probability theory and mathematical statistics are often considered parts of applied mathematics.

Mathematics is very broadly divided into foundations, algebra, analysis, geometry, and applied mathematics. The term foundations is used to refer to the formulation and analysis of the language, axioms, and logical methods on which all of mathematics rests. The scope and complexity of modern mathematics requires a very fine analysis of the formal language in which meaningful mathematical statements may be formulated and perhaps be proved true or false. Most apparent mathematical contradictions have been shown to derive from an imprecise and inconsistent use of language. A basic task is to furnish a set of axioms effectively free of contradictions and at the same time rich enough to constitute a deductive source for all of modern mathematics.

The general case of several equations of high degree leads to algebraic geometry, so named because the sets of solutions of such systems are often studied by geometric methods. Some of the concepts of modern algebra have found their way into elementary mathematics education in the so-called new mathematics. Some important abstractions recently introduced in algebra are the notions of category and functor, which grew out of so-called homological algebra. Arithmetic and number theory are also a part of algebra. The essential ingredient of analysis is the use of infinite processes. For example, the area of a circle may be computed as the limiting value of the areas of inscribed regular polygons as the number of sides of the polygons increases indefinitely. The basic branch of analysis is the calculus. The general problem of measuring lengths, areas, volumes, and other quantities as limits by means of approximating polygonal figures leads to the integral calculus. The differential calculus arises similarly from the problem of finding the tangent line to a curve at a point. Other branches of analysis result from the application of the concepts and methods of the calculus to various mathematical entities. For example, vector analy¬sis is the calculus of functions whose variables are vectors. Here various types of derivatives and integrals may be introduced. They lead, among other things, to the theory of differential and integral equations, in which the unknowns are functions rather than numbers, as in algebraic equations. Differential equations are often the most natural way in which to express the laws governing the behaviour of various physical systems. Calculus is one of the most powerful tools of mathematics. Its applications both in pure mathematics and in virtually every scientific domain are manifold.

The 20th century has seen an enormous development of topology which is the study of very general geometric objects, called topological spaces, with respect to relations that are much weaker than congruence and similarity. Other branches of geometry include algebraic geometry, which is now in a vigorous state of development, and differential geometry in which the methods of analysis are brought to bear on geometric problems. Applied mathematics is a term loosely designating a wide range of studies with significant current use in the empirical sciences. It includes numerical methods and computer science, which seeks concrete solutions, sometimes approximate, to explicit mathematical problems. In addition, probability theory and mathematical statistics are often considered parts of applied mathematics.

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