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
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