Product of Two Irrationals: Counterexample

When we encounter a "prove or disprove" HSC question about irrational numbers and multiplication, it's often testing whether we really understand what it means to be irrational. This problem sits in the proofs and number systems area of the syllabus, and working through it gives you a powerful technique: disproving a universal claim with a single well-chosen counterexample. The insight you'll gain is that the irrational numbers are not closed under multiplication—a fact that can surprise students who assume operations automatically stay within the same number set.

Problem Statement

Prove or disprove: If $x$ and $y$ are irrational numbers with $x \neq y$, then $xy$ is irrational.

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Hints

The statement is false. Find a counterexample using multiples of $\sqrt{2}$.

Consider $x = \sqrt{2}$ and $y = k\sqrt{2}$ for some rational $k \neq 1$. What is $xy$? Is it rational or irrational?

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Solutions

The statement is false. We disprove by counterexample.

A classic approach is to use multiples of $\sqrt{2}$ because the square of $\sqrt{2}$ is rational, which can produce a rational product when multiplied. So we choose two distinct multiples that give a rational outcome.

Counterexample:

Let $x = \sqrt{2}$ and $y = 2\sqrt{2}$.

• Check $x$ is irrational: $\sqrt{2}$ is irrational (well-known).
• Check $y$ is irrational: $2\sqrt{2}$ is the product of rational $2$ and irrational $\sqrt{2}$, so it's irrational.
• Check $x \neq y$: Clearly $\sqrt{2} \neq 2\sqrt{2}$ since $1 \neq 2$.
• Compute $xy$: Now multiply them together and simplify using the fact that $(\sqrt{2})^2 = 2$.

$$xy = (\sqrt{2})(2\sqrt{2}) = 2 \cdot (\sqrt{2})^2 = 2 \cdot 2 = 4$$

• Check rationality of $xy$: $4 = \frac{4}{1}$ is rational.

Since we found irrational numbers $x$ and $y$ with $x \neq y$ such that $xy$ is rational, the original statement is disproven. $\blacksquare$

Note: This shows that the irrationals are not closed under multiplication. The rationals are closed under multiplication, but the irrationals are not.

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Takeaways

• To disprove a universal statement about numbers, you only need one valid counterexample—choose it cleverly.
• The irrational numbers are not closed under multiplication; the product of two irrationals can be rational.
• Using known irrationals like $\sqrt{2}$ and scaling by rational factors often yields a rational product because the irrational part cancels.

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Further Readings

If you found this proof interesting, be sure to check out these relevant HSC booklets to sharpen your reasoning skills: HSC Proofs, HSC Induction, HSC Vectors