We deal only with real numbers in this chapter.

10.1 ELEMENTARY PROPERTIES

If a and b are real numbers, a < b (read "a is less that b") is defined to mean that b - a is positive. This definition and the following three properties can be used in proving elementary properties of inequalities:

(1) Closure of the positive numbers. If a and b are positive numbers, then a + b and ab are positive numbers.

(2) Trichotomy. One and only one of the following is true for a given real number a:
(a) a is zero; (b) a is positive; (c) -a is positive.

(3) Roots. If p is a positive number and n is a positive integer, then there is exactly one positive number r such that rⁿ = p. (This number r is called the positive nth root of p or the principal nth root of p.)

We can write the statement a < b in the form b > a (read "b is greater than a"). The notation  (read "a is less than or equal to b") means that either a < b or a = b, andis defined analogously. The notation x < y < z or z > y > x means that x < y and y < z are true simultaneously.

As mentioned is Section 8.4, the absolute value of a real number x is written as |x| and is defined as follows: If  then |x| = x; if x < 0, then |x| = -x.

Example 1. Show that if x < y and y < z, that is x < y < z, then x < z.

Solution: If x < y then y - x = p, a positive number, and similarly y < z implies that

z - y = q where q is positive. Hence (z - y) + (y - x) = q + p, or
 

Since z - x is the sum of the positive numbers q and p, it is positive by the closure property and thus x < z by the definition.
 
 
 

Monday, June 22, 1998