Question

Can the product of two irrational numbers be rational? Explain your answer and support with an example.

Answer 1

Checking if the product of two irrational numbers is rational:

Definition of rational and irrational numbers

Rational numbers are those which can be expressed in the form of $\frac{\mathrm{p}}{\mathrm{q}}$ where $\mathrm{p},\mathrm{q}\ne 0$.

While irrational numbers are those which cannot be expressed in the form of a simple fraction.

Considering the case of two irrational numbers $\sqrt{8}$ and $\sqrt{2}$.

We see that the product of these two irrational numbers give,

$$\begin{gathered} \sqrt{8}\times \sqrt{2} \\ =\sqrt{16} \\ =4 \end{gathered}$$

which is a rational number.

The best general example for this is if you take the square root of a non-perfect square and square it, or multiply it by itself. This "undoes" the squaring, so you get a whole number.

Hence, we can say that the product of two irrational numbers can be a rational number.