Question

Classify the following numbers as rational or irrational:
i). $\sqrt {23} $
ii) $\sqrt {225} $
iii). $0.3796$
iv). $7.478478.....$
v). $1.101001000100001.................$

Answer 1

Hint- To solve this question, you need to understand the concept of rational numbers and irrational numbers. The rational numbers are those numbers which can be written in the form of $\dfrac{p}{q}$ where p and q are both integers and q is never equal to zero. Irrational numbers are those which cannot be written in terms of fractions. We will use these basic definitions to reach the answer.

Complete step-by-step solution -
Now we will check each option
i). $\sqrt {23} $
The number $\sqrt {23} $ is not a perfect square of any integer. By finding the square root of the number 23, we see that the number is 4.79583152331.. , the number is not repeating and cannot be expressed in terms of $\dfrac{p}{q}$ Hence, the number is an irrational number.
ii). $\sqrt {225} $
 The number $\sqrt {225} $ is a perfect square of 15 and it can be expressed in the form of $\dfrac{p}{q}$
As $\dfrac{{15}}{1}.$ hence, the number is a rational number.
iii). $0.3796$
The number is a terminating decimal, means that it can be expressed in the $\dfrac{p}{q}$ form such as
$\dfrac{{3796}}{{10000}} = \dfrac{{949}}{{2500}}$ Hence, the number is a rational number.
iv). $7.478478.....$
The number $7.478478.....$ is a number with its decimal part repeating. It is a non- terminating number but its decimal part is repeating. Hence, the given number is a rational number.
v). $1.101001000100001.................$
The number $1.101001000100001.................$ is a number with its decimal part is non – terminating and non- repeating and therefore it cannot be represented in the $\dfrac{p}{q}$ form. Hence, the given number is an irrational number.



Note- In order to solve these types of questions, remember the basic definitions of rational, irrational, integers, whole numbers and more. In order to differentiate between decimal numbers whether they are rational or irrational remember that terminating numbers are always rational numbers while non- terminating and non- repeating are irrational numbers. The one exception is that the non-terminating but repeating numbers are rational numbers.