Question

Classify the following number as a rational / irrational number.
7.484848….

Answer 1

- Hint:- Check if the number can be represented in the form of \[{}^{p}/{}_{q}\], where p and q are integers and \[q\ne 0\]. Thus check if rational / irrational. Then find if the number is terminating or non-terminating.

Complete step-by-step answer: -
Rational numbers are numbers that can be represented in the form of \[{}^{p}/{}_{q}\], where p and q are integers and \[q\ne 0\]. In the case of irrational numbers they cannot be expressed in the form of fractions or \[{}^{p}/{}_{q}\].
The rational numbers can be expressed in the form of decimal fraction. When rational numbers are converted into decimal fraction it can be both terminating and non-terminating decimals. We have been given the number 7.484848….
Now this term can be expressed in the form of \[{}^{p}/{}_{q}\] so 7.484848…. is a rational number. Now we need to find if this number is terminating or non-terminating.
Terminating decimals are those numbers which come to an end after a few repetitions of decimal points. For example- 0.25, 0.016, 7.135 etc.
Non-terminating decimals are those decimals that keep on continuing after the decimal point, i.e. they go on forever. They don’t come to an end. For example- \[\pi =3.141592653.....\]
Thus the number given to us 7.484848…..is non-terminating as it keeps on continuing.
\[\therefore \]7.484848…. is a rational non-terminating number.

Note:-If a rational number can be expressed in the form \[\left( \dfrac{p}{{{2}^{n}}\times {{5}^{m}}} \right)\], then the rational number will be terminating decimal. Otherwise the rational number will be non-terminating, recurring decimal. For us it has been closely given as a non-terminating number.