Question

Is 0.6 a rational number ?

Answer 1

Hint:Here the number is given in the form of decimal. To determine if the given number is rational or not firstly we multiply both the numerator and denominator by 10. Then on multiplying we obtain a number in the form of fraction. On simplification if we get in the form of \[\dfrac{p}{q}\].

Complete step by step answer:
In mathematics we have different forms of numbers.
Natural numbers - Contain all counting numbers which start from 1.
Example: All numbers such as 1, 2, 3, 4, 5, 6, …
Whole Numbers - Collection of zero and natural numbers.
Example: All numbers including 0 such as 0, 1, 2, 3, 4, 5, 6, …
Integers- The collective result of whole numbers and negative of all-natural numbers.
Example: \[ - \infty , \cdot \cdot \cdot 0,1,2,3, \cdot \cdot \cdot + \infty \]
Rational Numbers- Numbers that can be written in the form of \[\dfrac{p}{q}\] where \[q \ne 0\]
Example: 3, -7, -100, \[\dfrac{1}{2}\], \[\dfrac{5}{3}\], 0.16, 0.4666 etc
Irrational Numbers- All the numbers which are not rational and cannot be written in the form of \[\dfrac{p}{q}\]
Example: \[\sqrt 2 \], \[\pi \], \[\sqrt 3 \], \[2\sqrt 2 \] and \[ - \sqrt {45} \] etc
Real numbers: Real numbers can be defined as the union of both the rational and irrational numbers. They can be both positive or negative and are denoted by the symbol “R”.

Now on considering the given question. the given number is 0.6
On multiplying and dividing by 10.
\[ \Rightarrow 0.6 \times \dfrac{{10}}{{10}}\]
\[ \Rightarrow \dfrac{6}{{10}}\]
On simplifying we get
\[ \Rightarrow \dfrac{3}{5}\]
The simplified form is in fraction and it is in the form of \[\dfrac{p}{q}\]

Therefore, 0.6 is a rational number.

Note:As per definition, the rational numbers include all the integers, fractions and repeating decimals, so while deciding keep the definitions in mind. Also, remember that the decimal expansion for rational numbers that execute finite or recurring decimals.