Question

Give an example of two irrational numbers, whose difference is a rational number.

• A. $4+\sqrt{2},2+\sqrt{2}$
• B. $4-\sqrt{2},2+\sqrt{2}$
• C. $4+\sqrt{2},2-\sqrt{2}$
• D. $4+\sqrt{2},-2-\sqrt{2}$

Answer 1

The correct option is A $4+\sqrt{2},2+\sqrt{2}$

A rational number is a number that can be written as a ratio. That means it can be written as a fraction, in which both the numerator and the denominator are whole numbers.

On the other hand, all numbers that are not rational are considered irrational. An irrational number can be written as a decimal, but not as a fraction.

So, let us take two irrational numbers $a=4+\sqrt{2}$ and $b=2+\sqrt{2}$ and we now calculate $a-b$ as follows:

$a-b=(4+\sqrt{2})-(2+\sqrt{2})=4+\sqrt{2}-2-\sqrt{2}=2=\frac{2}{1}$

Therefore, the difference of two irrational numbers $a$ and $b$ is $2$ and from the above defination of rational numbers, we get that $2$ is a rational number.

Hence, the difference of two irrational numbers $4+\sqrt{2}$ and $2+\sqrt{2}$ is a rational number.