chapter  4
13 Pages

Real Numbers

WithGarrett Birkhoff, Saunders Mac Lane

This chapter describes the real numbers by a brief set of postulates. Although "modern" algebra properly stresses the wealth of properties holding in general fields and integral domains, the real and complex fields are indispensable for describing quantitatively the world in which we live. A completely geometric approach to real numbers was used by the Greeks. For them, a number was simply a ratio (a :b) between two line segments a and b. They gave direct geometric constructions for equality between ratios and for addition, multiplication, subtraction, and division of ratios. The postulates stating that the real numbers form an ordered field appeared to the Greeks as a series of geometric theorems, to be proved from postulates for plane geometry (including the parallel postulate). The concept of real numbers as least upper bounds of sets of rationals is directly involved in the familiar representation of real numbers by unlimited decimals.