Chapter 5.4, Problem 42HP

To determine

The solution of the statement of an indirect proof.

Explanation of Solution

Given information:

If $x$ is a rational number, then it can be repersented by the quotient $\frac{a}{b}$ for some integer $a$ and $b$ , if $b\ne 0$ . An irrational number cannot be represented by the quotient of two integers. Write an indirect proof to show that the product of a nonzero rational number is an irrational number.

Calculation:

Given: $x$ rational, $x\ne 0;y$ irrational

Prove: $xy$ is irrational

Let assume that $xy$ is rational. Then there are two integers $a,b,b\ne 0$ with no common multiplies, so that can be write:

  $xy=\frac{a}{b}$

Also $x$ is rational and $\ne 0$ . Then there are two integers $c,d,d\ne 0$ , $c\ne 0$ , with no common multiplies so that can be write:

  $x=\frac{c}{d}$

Replace the expression of $x$ in the equation for $xy$ :

  $\begin{array}{l}\frac{c}{d}\cdot y=\frac{a}{b}\\ \frac{cy}{d}=\frac{a}{b}\\ bcy=ad\\ y=\frac{ad}{bc}\end{array}$

This means that $y$ can be written as a quotient of two integers $ad$ and $bc$ ,where $bc\ne 0$ .

Therefore, the assumption is false.