r · r' = 1 = r' · r. To prove that an if--then statement (p
q) is true, we assume that p is
true and use that to show that q must also be true.4. This question can be rewritten as an if--then statement. If
r is a non-zero rational number, then there exists another rational number r', such that
r · r' = 1 = r' · r. To prove that an if--then statement (p q) is true, we assume that p is
true and use that to show that q must also be true.
Proof Suppose that r is
a non-zero rational number; hence r = a/b for some integers a and b, where a 
0 and
b 0.
Let r' = b/a. Then r' is a rational number, since a
and b are both integers and a 0. Furthermore,
| (a/b) · (b/a) | = | (ab)/(ab) |
| = | 1. | |
| By the same reasoning (b/a) · (a/b) | = | 1. |
Therefore for any non-zero rational number r = a/b, a multiplicative inverse exists, namely the rational number b/a.