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a somewhat vaguely defined part of mathematics. Elementary mathematics embraces the branches, problems, and methods of mathematics in which the general concepts of, for example, variable, function, and limit are not used. In other words, it uses the general mathematical concepts (abstractions) that took shape before the appearance of mathematical analysis. Within the framework of these concepts it has continued to develop and has even obtained new results (seeMATHEMATICS: History of mathematics up to the 19th century: Period of elementary mathematics).
Elementary mathematics encompasses primarily arithmetic, elementary number theory, elementary algebra, elementary geometry, and trigonometry. It may be concisely characterized as the mathematics of constants. This description, however, is not entirely accurate, since elementary mathematics deals not only with constant quantities but also with geometric figures, whose “magnitude”—for example, position—may not be pertinent to the problem at hand. Moreover, the purview of elementary mathematics extends to such variable quantities as trigonometric functions.
The variable quantities dealt with in elementary mathematics are defined in a special manner. The determination of the circumference of a circle provides an illustration of this idea. In essence, such a determination makes use of the concept of limit, but not in a general form; a specially defined sequence is considered—the sequence of perimeters of inscribed and circumscribed polygons. The general concepts of function and limit fall beyond the scope of elementary mathematics, as do the general concepts of curve and surface; indeed, the only figures dealt with in elementary mathematics are those given by a special construction.
Number theory often makes use of “elementary” proofs, in which the methods of mathematical analysis are not employed. This elementary number theory, it should be noted, is not at all elementary in the sense of being characterized by simplicity.
The term “elementary mathematics” is also applied to the mathematical disciplines studied in secondary general-education schools. In this sense elementary mathematics is distinguished from higher mathematics.