Why Read This Book? - Index of

56 Chapter 2 Properties of Real Numbers
EXERCISE 2.3.10 For all real numbers x and y, � �|x| − |y| � � ≤ |x − y|. 14
Exercises 2.3.8–2.3.10 reveal how addition and subtraction relate to absolute
value. The final exercise for this section reveals how multiplication and division
relate to absolute value.
EXERCISE 2.3.11 Suppose x and y are real numbers. Then the following are
true.
(a) |xy| = |x||y|.
(b) If y �= 0, then � �y−1 � � = |y| −1 . 15
(c) If y �= 0, then |x/y| = |x|/|y|.
2.4 The Division Algorithm
This section is devoted to one very important theorem about the integers. Even
though it is called the division algorithm, it is not an algorithm in the sense of
computer science, where we think of an algorithm as a step-by-step process for
performing a task. Instead, the division algorithm is merely a unique existence
theorem, but a very useful one.
One way to assign meaning to the words even integer and odd integer is the
following.
Definition 2.4.1 If n is an integer, we say that n is even provided there exists an
integer k such that n = 2k. We say that n is odd provided there exists an integer k
such that n = 2k + 1.
Let’s prove some easy results about even and odd integers. Notice how the
proof of the first one appeals to the definition of even integer in both directions.
First, when we are given that a particular integer n is even, then we can claim the
existence of some integer k1 such that n = 2k1. Then, when we need to demonstrate
that some integer p is even, we must be able to find some integer k2 such that
p = 2k2.
We present two slightly different proofs of the next theorem to illustrate subtly
different tastes for the flow of a proof.
Theorem 2.4.2 Let m and n be integers, at least one of which is even. Then mn
is even.
Proof 1. Pick integers m and n, and suppose (without loss of generality) that
n is even. Then there exists integer k1 such that n = 2k1. Thus we have that
14 Apply Corollary 2.3.5 by writing the expression in Exercise 2.3.9 in two ways, once switching the roles
of x and y.
15 Use a technique like the proof of Theorem 2.1.18.