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 $\dfrac{p}{q}$ 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 $\dfrac{p}{q}$ 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, $\sqrt {225} = \dfrac{{15}}{1}$ which is in the form of $\dfrac{p}{q}$ . So, $\sqrt {225} $ is a rational number.
(3) $0.3796$
This is a terminating decimal, so this can be written in the form of $\dfrac{p}{q}$. 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 $\dfrac{p}{q}$. 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 $\dfrac{p}{q}$. 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 $\dfrac{p}{q}$ because the decimal expression of $\sqrt x $ is neither terminating nor repeating.