Question

Classify the following numbers as rational or irrational:

(i) $2-\sqrt{5}$ (ii) $(3+\sqrt{23})-\sqrt{23}$ (iii) $\frac{2\sqrt{7}}{7\sqrt{7}}$

(iv) $\frac{1}{\sqrt{2}}$ (v) $2\pi$

Answer 1

We know that the sum of rational and irrational will be an irrational number.

The difference of a rational and an irrational will be an irrational.

Product of rational and irrational number is always irrational if rational number is non zero.

Division of a non-zero rational and irrational is always irrational.

(i) $2-\sqrt{5}$

$2$ is a rational number and $\sqrt{5}$ is an irrational number, so, the given number is a difference of rational and irrational. Hence it is an irrational number.

(ii)$(3+\sqrt{23})-\sqrt{23}=3=\frac{3}{1}$

As it can be represented in $\frac{p}{q}$ form, therefore, it is a rational number.

(iii) $\frac{2\sqrt{7}}{7\sqrt{7}}=\frac{2}{7}$

As it can be represented in $\frac{p}{q}$ form, therefore, it is a rational number.

(iv) $\frac{1}{\sqrt{2}}$

$1$ is a rational number and $\sqrt{2}$ is an irrational number, so the given number is an example of division of a non-zero rational number by an irrational number. Hence, it is an irrational number.

(v) $2\pi$

$2$ is a rational number and $\pi$ is an irrational number, so the given number is an example of product of a non-zero rational number by an irrational number. Hence, it is an irrational number.