Introduction

In Chapter 1, as you will recall, the inequality a > b was defined in terms of the set P of positive numbers. You may also recall that for the validity of several of the results of Chapter 2, such as Theorem 2.5 concerning the multiplication of inequalities, it was necessary to specify that certain of the numbers involved should be positive. Again, in many instances the fractional powers of numbers that appear in Theorem 2.7 would not even be real if the numbers themselves were negative; consider, for instance, a1/2 with a = –9. Many of the fundamental inequalities, which will be derived in Chapter 4, involve just such fractional powers of numbers. It is natural, then, that we should often restrict our attention to positive numbers or to nonnegative numbers (positive numbers and zero) in this study.

In applied problems involving inequalities, we often deal with weights, volumes, etc., and with the magnitudes, or absolute values, of certain mathematical objects such as real numbers, complex numbers, vectors. The magnitudes of all these are measured by nonnegative numbers. Thus, even though you may choose to denote gains by positive numbers and losses by negative numbers, a loss of $3 is still a loss of greater magnitude than a loss of $2; the absolute value of –3 is greater than the absolute value of –2.