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

Only the square roots of the square numbers, i.e. the square roots of the perfect squares are the rational numbers.

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

$2$ is a rational number and $\sqrt{5}$ is an irrational number, then $2-\sqrt{5}$ is a irrational number. $[\because$ difference of a rational and an irrational number is irrational $]$

(ii) In $(3+\sqrt{23})-\sqrt{23}=3+\sqrt{23}-\sqrt{23}=3$ is a rational number.

(iii) $\frac{2\sqrt{7}}{7\sqrt{7}}=\frac{2}{7}$ is a rational number.

(iv) In $\frac{1}{\sqrt{2}}$, the number $\sqrt{2}$ is not a rational number, then $\frac{1}{\sqrt{2}}$ is irrational number.

(v) $\pi$ is an irrational number, hence $2\pi$ is an irrational number.