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

Last updated date: 06th Sep 2024
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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.

(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.
$\sqrt x$ 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.