Which of the following numbers are irrational?
Question:

Which of the following numbers are irrational?

(a) $\sqrt{2}$

(b) $\sqrt[3]{6}$

(c) $3.142857$

(d) $2 . \overline{3}$

(e) $\pi$

(f) $\frac{22}{7}$

(g) $0.232332333 \ldots$

(h) $5.27 \overline{41}$

Solution:

(a) $\sqrt{2}$ is irrational ( $\because$ if $p$ is prime, then $\sqrt{p}$ is irrational).

(b) $\sqrt[3]{6}=\sqrt[3]{2} \times \sqrt[3]{3}$ is irrational.

(c) $3.142857$ is rational because it is a terminating decimal.

(d) $2 . \overline{3}$ is rational because it is a non-terminating, repeating decimal.

(e) $\pi$ is irrational because it is a non-repeating, non-terminating decimal.

(f) $\frac{22}{7}$ is rational because it is in the form of $\frac{p}{q}, q \neq 0$.

(g) $0.232332333 \ldots$ is irrational because it is a non-terminating, non-repeating decimal.

(h) $5.2741$ is rational because it is a non-terminating, repeating decimal.