Question 4

A rational number can be written as a fraction.

Step 1 — Define rational numbers

Let's start with our numbers.

We are given $x$ and $y$.

Both $x$ and $y$ are rational numbers.

This means we can write $x$ as a fraction.

Let $x = \frac{\mathbf{a}}{\mathbf{b}}$.

Here, $\mathbf{a}$ and $\mathbf{b}$ are integers.

Also, $\mathbf{b}$ cannot be zero.

Similarly, we can write $y$ as a fraction.

Let $y = \frac{\mathbf{c}}{\mathbf{d}}$.

Here, $\mathbf{c}$ and $\mathbf{d}$ are integers.

Also, $\mathbf{d}$ cannot be zero.

Step 2 — Multiply the numbers

Now, let's find the product $xy$.

We will substitute the fractions.

$xy = \frac{\mathbf{a}}{\mathbf{b}} \times \frac{\mathbf{c}}{\mathbf{d}}$

To multiply fractions, we multiply numerators.

We also multiply denominators.

$xy = \frac{\mathbf{a} \times \mathbf{c}}{\mathbf{b} \times \mathbf{d}}$

Let's call the new numerator $\mathbf{P}$.

So, $\mathbf{P} = \mathbf{a} \times \mathbf{c}$.

Let's call the new denominator $\mathbf{Q}$.

So, $\mathbf{Q} = \mathbf{b} \times \mathbf{d}$.

Since $\mathbf{a}, \mathbf{b}, \mathbf{c}, \mathbf{d}$ are integers, $\mathbf{P}$ must be an integer.

Also, $\mathbf{Q}$ must be an integer.

Remember that $\mathbf{b}$ is not zero.

Also, $\mathbf{d}$ is not zero.

This means their product $\mathbf{Q}$ cannot be zero.

So, $xy$ is in the form $\frac{\mathbf{P}}{\mathbf{Q}}$.

$\boxed{xy = \frac{\text{integer}}{\text{non-zero integer}}}$

This matches the definition of a rational number.

Answer

(i) The product $xy$ is a rational number.