Question

Examine whether the following number is rational or irrational: 7.478478…..

Answer 1

Hint: We will refer to the basic definition of rational number and irrational number to check whether 7.478478….. is a rational number or irrational number. If the number is rational then it can be represented in a ratio of fraction form but if it is an irrational number, then we cannot represent it in ratio.

Complete step-by-step answer:
It is given in the question that we have to check whether the number 7.478478….. is a rational number or irrational number.
Now, we look over the basic definition of rational number and irrational number. So, the rational number is basically the ratio between two integers which are represented in the fraction form $\dfrac{p}{q}$, the condition for q is that $q\ne 0$.
Also, the number which is not a rational number is called irrational number or we can say that the number which is present in decimal form but it cannot be represented in ratio or any fraction form is called irrational number.
Some of the examples of rational numbers are \[2,\text{ }4,\text{ }6,\text{ }8,\ \sqrt{4},\ 0\ etc.\] and some of the example of irrational numbers are $\sqrt{8},\ \sqrt{1.6},\ -\sqrt{11},\ \pi ,\ etc.$
Difference between rational numbers and irrational numbers:

Rational Numbers	Irrational Numbers
(1) It can be expressed in $\dfrac{p}{q}$ form where both p and q are the whole number and $q\ne 0$.	(1) It cannot be expressed in $\dfrac{p}{q}$ form.
(2) The decimal expansion for rational numbers executes finite or recurring decimal.Ex- 0.7777 is a recurring decimal.	(2) The non-finite and non-recurring decimals are executed.Ex- $\pi $, 3.14151 is a non-recurring decimal.

In the question the given number is 7.478478…. is a non-recurring and non- terminating decimal.
Also, it is impossible to represent non-recurring and non-terminating decimals in fraction or any ratio form.

Note: We have to keep in mind the basic definition of rational number and irrational number, that rational number can be expressed in fraction form but it is impossible to represent irrational number in ratio. Also, if a number is a perfect square then its root will be a rational number, but if the number is not a perfect square then the root of the number is an irrational number. For example: $\sqrt{4}$ is a rational number as 4 is a perfect square and $\sqrt{7}$ is an irrational number as it is not a perfect square.