Question

Which of the following is an irrational number?

(i) $\sqrt{3}$

(ii) $0.143\overline{32}$

(iii) $5.\overline{46}$

(iv) $\sqrt{5}$

• A. (i) and (iv)
• B. Only (iv)
• C. (ii) and (iii)
• D. Only (iii)

Answer 1

The correct option is A (i) and (iv)

i) We know that $\sqrt{3}$ cannot be expressed in the form of $\frac{p}{q}$ where $p$ and $q$ are integers and $q\ne 0$. Therefore, $\sqrt{3}=\mathrm{1.7321428......}$ is an irrational number.

ii) $0.143\overline{32}$ is a recurring decimal as 32 repeates over and over. Recurring decimals are rational numbers. Hence $0.143\overline{32}$ is a rational number.

iii) $5.\overline{46}$ is again a recurring decimal as 46 repeates over and over. Recurring decimals are rational numbers. Hence $5.\overline{46}$ is a rational number.

iv) We know that $\sqrt{5}$ cannot be represented in the form of $\frac{p}{q}$ where $p$ and $q$ are integers and $q\ne 0$.
Its value is $\mathrm{2.23606797749979...}$ which is a non-terminating and non-recurring decimal. Irrational numbers are non-terminating and non-repeating.
Hence $\sqrt{5}$ is an irrational number.