Question

Which of the following are irrational numbers?

i) $2\sqrt{3}$

ii) $0.143\overline{32}$

iii) $5.\overline{46}$

iv) $\sqrt{5}$

Answer 1

i) $2\sqrt{3}$ is product of $2$ and $\sqrt{3}$. We know that $\sqrt{3}$ is an irrational number and cannot be expressed in the form of $\frac{p}{q}$ where $p$ and $q$ are integers and $q\ne 0$. When we multiply a rational number with an irrational number, the result is an irrational number. Therefore, $2\sqrt{3}$ is an irrational number.

ii) $0.143\overline{32}$ is a recurring decimal.
In this case we see that $143$ does not repeat itself but the block $32$ repeats itself. Since two digits are repeating, it is a rational number.

iii) $5.\overline{46}$ is again a recurring decimal. Here the number 46 repeats after the decimal and hence it is a rational number.

iv) $\sqrt{5}$ We know that $\sqrt{5}$ is an irrational number since it 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 which is the very definition of an irrational number.