Question

Which of the following numbers are irrational?

(a) $\sqrt{2}$
(b) $\sqrt[3]{6}$
(c) 3.142857
(d) $2.\overline{3}$
(e) π
(f) $\frac{22}{7}$
(g) 0.232332333...
(h) $5.27\overline{41}$

Answer 1

(a) $\sqrt{2}$ is irrational (∵ 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) 2.3 is rational because it is a non-terminating, repeating decimal.

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

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

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

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