Mathematics

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Pure Mathematics

Pure mathematics is more abstract than applied mathematics. It emphasizes rigorous proof, manipulates symbols rather than numbers, and seeks to obtain the most general results possible with the fewest possible assumptions. British mathematician G. H. Hardy, one of the foremost spokesmen for pure mathematics, represents this approach in his classic book A Mathematician's Apology (1941).

Pure mathematics began to come into its own during the 1800s when rigorous proof and detailed analysis became more common. The beauty of the mathematical proof—that is, its simplicity and its brevity—became just as important as the result, more important even than the specific application that inspired it. British mathematician and logician Bertrand Russell wrote in 1910, “Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.” German mathematician Hermann Weyl remarked in the early 1900s, only partly in jest, “My work has always tried to unite the true with the beautiful and when I had to choose one or the other, I usually chose the beautiful.”

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Applied Mathematics

Applied mathematics, while just as concerned with rigorous mathematical methods, emphasizes applications. Applied mathematics has had close ties with the sciences and engineering throughout its history. Applied mathematicians believe that new mathematical ideas and areas of study can come from using mathematics to solve problems in physics, chemistry, biology, medicine, engineering, and technology. Much of the current research in applied mathematics takes place outside traditional mathematics departments.

Subject areas in applied mathematics often overlap areas associated with other fields, including economics, physics, mechanics, and information theory. For example, mathematicians who study the structure of matter and the behavior of subatomic particles overlap in their area of research with physicists. Some areas of applied mathematics depend heavily on pure mathematics. Numerical analysis, which studies computational methods for solving mathematical problems, relies on the pure mathematical areas of partial differential equations and variational methods. Other areas, such as computer science, are as broad as the entire field of mathematics.

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Applied mathematics is older than pure mathematics because it was used in areas that formed the core of early physics research, such as mechanics, optics, and fluid dynamics. As mathematical tools became more powerful, these areas of physics became more mathematically based.

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Historical Differentiation

One force that tends to differentiate mathematical disciplines is research funding. In the United States, pure mathematics has been funded by the National Science Foundation (NSF) and carried out in universities as well as at a few industrial research and development facilities, such as IBM Research Labs and AT&T Bell Laboratories. Within the government, the National Security Agency (NSA) has been a big supporter of pure mathematics areas related to cryptography, such as algebra, graph theory, and number theory.

Since the end of World War II in 1945, government laboratories have employed thousands of pure and applied mathematicians. These facilities include Los Alamos National Laboratory, Oak Ridge National Laboratory, Brookhaven National Laboratory, and Lawrence Berkeley National Laboratory and Lawrence Livermore National Laboratory. The military, through the Office of Naval Research, the Air Force Office of Scientific Research, and the Army Research Office, has also supported both pure and applied mathematics in the United States. The applied mathematics community has received large amounts of research funding from the energy, computing, and communications industries.

The distinction between pure and applied mathematics is not rigid, however. Many mathematicians receive their training in pure mathematics, then become interested in applying their expertise to other areas. Other mathematicians seek to build links between different areas of mathematics. Interdisciplinary mathematics, in which methods from more than one area of mathematics are used, is one of the fastest growing areas of mathematics.

Trends also influence the direction of mathematics. A research area can become popular, as catastrophe theory did in the 1970s and 1980s, then virtually disappear. Mathematical research has traditionally been tied to individuals, schools, and even countries. In the 1800s new mathematical areas were started by individuals in the great schools of Europe and the United States. These universities included Göttingen (Germany), Moscow (Russia), Paris (France), Cambridge (Britain), and Princeton University and the University of Chicago in the United States. As pioneering researchers trained students in mathematics, their universities became associated with particular areas of mathematics. Because many of these universities were the leading institutions in their countries, the countries themselves became associated with different mathematical interests.

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Branches of Mathematics

The American Mathematical Society, a professional organization of mathematicians, classifies subject areas of mathematics as either pure mathematics, applied mathematics, or borderline areas. Out of the many subject areas of mathematics, this article describes ten significant areas.

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Arithmetic

Arithmetic, one of the oldest branches of mathematics, arises from the most fundamental of mathematical operations: counting. The arithmetic operations—addition, subtraction, multiplication, division, and placeholding—form the basis of the mathematics that we use regularly. In many countries arithmetic is the primary area of mathematical study during the first six years of school.

Although arithmetic itself is not an area of mathematics research, research on how best to teach arithmetic is crucial to the field of mathematics education. Models of learning and mastering the basics of arithmetic are often used in cognitive science—the study of the processes of acquiring, storing, and using knowledge. Cognitive sciences encompass a range of activities, including the design of computer-aided instructional systems and the study of artificial intelligence. Arithmetic and logic also form the basis for all computer software—the instructions that tell computers what to do.