Question

Is the product of two irrationals always irrational? Justify your answer.

Answer 1

Hint:
Before we judge whether two irrationals are always irrational or not, first we need to know what is irrational? Irrational number is a number which cannot be expressed as a fraction that is in the form of ${\displaystyle \frac{p}{q}}$ where $p$ and $q$ are integers. Take some suitable examples and perform multiplication operations between those two numbers and we get to know if the product is irrational or not.

Complete Step by Step Solution:
First we need to understand rational and irrational numbers.
Rational numbers are those numbers which can be expressed as ${\displaystyle \frac{p}{q}}$ form, whereas irrational numbers are those which cannot be expressed in the form of ${\displaystyle \frac{p}{q}}$.
Depending on the product of two numbers we need to decide whether the product is always irrational or not.
Now, we take some examples to understand concept

Example:1
Let, the two numbers be $\sqrt{5}\times \sqrt{5}$
If we multiply $\sqrt{5}\times \sqrt{5}$ we get the answer as $5$, which is a rational number rather than irrational.

Example:2
If we take an another example, that is $\sqrt{5}\times \sqrt{3}$
In this case if we multiply $\sqrt{5}\times \sqrt{3}$ we get the answer as $\sqrt{15}$ or $3.87298335$ which is an irrational number.
So, from the above two examples we can say that the product of two irrational numbers can be rational sometimes and irrational sometimes.
Therefore, for the given question we can say that the product of two irrational numbers are not always irrational.

Note:
While taking examples to prove whether the product of two irrationals is irrational or not try to take as many as you can because the result will be depending on the numbers we take for multiplication. The result may be rational or irrational but always it is not irrational.