Question

Give an example of two irrational numbers to show that their product is not always an irrational number.

Answer 1

Irrational numbers:

Irrational numbers are the number having non-terminating and non- recurring decimals.

Let us assume the two irrational numbers be

$3+\sqrt{5}$ and $3-\sqrt{5}$

On multiplying these numbers we get

$$\begin{gathered} (3+\sqrt{5})\times (3-\sqrt{5})=9-3\sqrt{5}+3\sqrt{5}-5 \\ (3+\sqrt{5})\times (3-\sqrt{5})=4 \end{gathered}$$

Where, $4$ is not an irrational number its a rational number.

Therefore, the two irrational numbers having their product not an irrational number are $3+\sqrt{5}$ and $3-\sqrt{5}$