Question

Classify the following numbers as rational or irrational:
(1) $\sqrt{23}$
(2) $\sqrt{225}$
(3) $0.3796$
(4) $7.478478----$
(5) $1.101001000100001---$

Answer 1

Hint: Here, we have to classify the numbers as rational and irrational numbers. Basically, a number in the form of ${\displaystyle \frac{p}{q}}$ where $p$ and $q$ are integers and $q\ne 0$ is said to be a rational number otherwise irrational numbers. We can identify the number as rational or irrational as:
If the given number is in decimal form then we have to check whether it is terminating or non-terminating or repeating if the decimal number is terminating or repeating then it is a rational number otherwise irrational number.
If the given number is in the form $\sqrt{x}$ then it is a rational number if and only if $x$ is a perfect square otherwise irrational number.

Complete step-by-step answer:
(1) $\sqrt{23}$
Here, $23$ is not a perfect square number so, $\sqrt{23}$ can’t be written in the form of ${\displaystyle \frac{p}{q}}$ because the decimal is neither terminating nor repeating. So, $\sqrt{23}$ is an irrational number.
(2) $\sqrt{225}$
Here, $225$ is a perfect square number so, $\sqrt{225}={\displaystyle \frac{15}{1}}$ which is in the form of ${\displaystyle \frac{p}{q}}$ . So, $\sqrt{225}$ is a rational number.
(3) $0.3796$
This is a terminating decimal, so this can be written in the form of ${\displaystyle \frac{p}{q}}$. So, $0.3796$ is a rational number.
(4) $7.478478----=7.\overline{478}$
Here, three digits after decimal that is $478$ repeats regularly so it is a repeating decimal and it can be written in the form of ${\displaystyle \frac{p}{q}}$. So, $7.478478----$ is a rational number.
(5) $1.101001000100001---$
This decimal number is neither terminating non-repeating so, this can’t be written in the form of ${\displaystyle \frac{p}{q}}$. So, $1.101001000100001---$ is an irrational number.

Note:
$\sqrt{x}$ is a rational number if $x$ is perfect square if $x$ is not a perfect square then $\sqrt{x}$ can not be written in the form of ${\displaystyle \frac{p}{q}}$ because the decimal expression of $\sqrt{x}$ is neither terminating nor repeating.