Given that the product of two rational numbers is rational, and suppose and are rationals, what can you conclude about ?

Question

Homi · Accepted Answer

Understand the Question

A rational number can be expressed in the form $\dfrac{p}{q}$ , where $p, q \in \mathbb{Z}$ and $q \neq 0$ .
We are given that the product of any two rational numbers is always rational.
Since $a$ and $b$ are both rational numbers, applying this property directly tells us about the nature of their product $ab$ .

Step 1 · Apply the Property of Rational Numbers

Given that $a$ and $b$ are rational numbers.

Since the product of any two rational numbers is rational, their product $ab$ must also be a rational number.

Answer

$ab$ is rational

Common Mistakes

Confusing Rationals with Irrationals: The product of two rational numbers is always rational, whereas the product of two irrational numbers is not necessarily irrational (e.g., $\sqrt{2} \times \sqrt{2} = 2$ , which is rational).
Overcomplicating the Proof: When a universal statement is already given as a premise ("the product of two rational numbers is rational"), you can apply it directly as a deductive step without reproving closure from scratch.

Question 2