Question

Classify the following number 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

$\left(i\right)$ $2-\sqrt{5}$:

Since $\sqrt{5}$ is an irrational number, $2-\sqrt{5}$ is also an irrational number.

$\left(ii\right)$ $(3+\sqrt{23})-\sqrt{23}$:

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

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

$\frac{2\sqrt{7}}{7\sqrt{7}}=\frac{2}{7}$, which is a rational number.

$\left(iv\right)$ $\frac{1}{\sqrt{2}}$:

Since $\sqrt{2}$ is an irrational number, $\frac{1}{\sqrt{2}}$, which is also an irrational number.

$\left(v\right)$ $2\pi$:

Since $\pi$ is an irrational number, $2\pi$is also an irrational number