Let and be rational numbers. Show that is a rational number.

Question

Homi · Accepted Answer

Understand the Question

A rational number is defined as any number that can be written in the form $\dfrac{p}{q}$ , where $p$ and $q$ are integers and $q \neq 0$ .
If $x$ and $y$ are rational numbers, we can represent them as fractions of integers with non-zero denominators.
By multiplying the two fractions, we show that the resulting numerator and denominator are also integers (with a non-zero denominator), proving that the product $xy$ is rational.

Step 1 · Express $x$ and $y$ in Rational Form

Since $x$ and $y$ are rational numbers, they can be written as:

$x = \dfrac{a}{b}, \quad \text{where } a, b \text{ are integers and } b \neq 0$

$y = \dfrac{c}{d}, \quad \text{where } c, d \text{ are integers and } d \neq 0$

Step 2 · Multiply the Rational Numbers

Finding the product $xy$

$xy = \dfrac{a}{b} \times \dfrac{c}{d} = \dfrac{a \times c}{b \times d}$

Let $P = a \times c$ and $Q = b \times d$ .

Since the product of two integers is always an integer, both $P$ and $Q$ are integers.
Since $b \neq 0$ and $d \neq 0$ , their product $Q = b \times d \neq 0$ .

Therefore,

$xy = \dfrac{P}{Q} = \dfrac{\text{integer}}{\text{non-zero integer}}$

This matches the definition of a rational number.

Answer

Hence, $xy$ is a rational number.

Common Mistakes

Omitting the Non-Zero Condition: Forgetting to state that $b \neq 0$ and $d \neq 0$ , which is essential to guarantee that the denominator $bd \neq 0$ .
Missing Integer Closure: Not explicitly stating that the product of integers is also an integer, which is a required condition for the ratio to be rational.

Question 4