CHAPTER 1

NUMBERS AN D EXPRESSION S

SECTION 1. REA L NUMBERS

SECTION 2 . NUMBER S AN D SETS

SECTION 3 . INTEGRA L EXPRESSION S

SECTION 4. DIVISIO N OF INTEGRAL AND FRACTIONAL EXPRESSION S

The concept of real numbers can be understood intuitively by relating them to the

points on the number line. Thi s concept received a fonnal mathematica l defmition onl y in

the second half of the nineteenth century. Th e concept of real numbers provides a basic

demonstration of the efficiency o f the concept of a set. "Th e continuity of real numbers"

arises both explicitly and implicitly in various areas of science, and the practical usefulnes s

of complex numbers, introduced in Chapter 2, sterns from th e continuity of real numbers.

Extending th e set of numbers to encompass real numbers by adding irrational numbers to

rational numbers has other uses besides the mere calculation of square roots.

An algebraic symbolic system that represents both unknown and known quantities as

letters was first introduced by Vieta (1540-1603) and perfected by Descartes (1596-1650).

This approach systematized algebra and laid a foundation fo r the extensive development of

algebra up to the present day.

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http://dx.doi.org/10.1090/mawrld/008/01