Question

Use a direct proof to show that the product of two rational numbers is rational.

Short answer

The multiplication of two rational numbers is rational.

Step-by-step solution

Introduction

The purpose is to show that the multiplication of two rational numbers is a rational number.

Defining two rational numbers

Let m and n be two rational numbers.

Therefore, m and n can be expressed as follows:

\(m = \frac{a}{b}\)and\(n = \frac{c}{d}\), where a, b, c, d are integers and \(b \ne 0,d \ne 0\)

Finding their product for the result

Consider the product \(m.n\)as shown below:

\(m.n = \frac{a}{b}.\frac{c}{d}\)

\( = \frac{{ac}}{{bd}}\)

As\(b \ne 0,d \ne 0\)therefore\(bd \ne 0\)

Therefore, the product is rational as\(\frac{{ac}}{{bd}}\)is rational.

Hence, the multiplication of two rational numbers is rational.