Question

Express the following in the form of $ \dfrac{p}{q} $ , where p and q are integer.
i. $ 0.\overline 6 $

Answer 1

Hint: In this question it is needed to express $ 0.\overline 6 $ , the bar over the digit 6, represents the recurring of digit 6 infinite times. So to solve this we need to assume some variable as $ 0.\overline 6 $ , then multiply both sides with 10 and subtract one from another and proceed accordingly.

Complete step-by-step answer:
Given, the number is $ 0.\overline 6 $ , we have to represent this number in the form of $ \dfrac{p}{q} $ .
Consider the number $ 0.\overline 6 $ , the bar over the digit 6 represents the repetition of the 6 for infinite times, i.e., it is recurring.
Let $ x = 0.\overline 6 $
So,
 $ x = 0.6666 $ ----(i)
Now, multiply with 10 on both the sides.
 $ 10x = 6.6666 $ ----(ii)
Now, subtract (i) from (ii)
 $
\Rightarrow 10x - x = 6.6666 - 0.6666 \\
  9x = 6 \\
  $
Now, solve further
 $
\Rightarrow 9x = 6 \\
\Rightarrow x = \dfrac{6}{9} \\
\Rightarrow x = \dfrac{2}{3} \\
  $
Therefore, $ \dfrac{2}{3} = 0.666666 = 0.\overline 6 $ .

Note: Numbers that can be represented as a fraction and even as positive numbers, negative numbers, and zero are rational numbers. Where q is not equal to zero, it can be written as $ \dfrac{p}{q} $ .The term logical is derived from the term 'ratio,' which means a comparison of two or more values or integer numbers and is referred to as a fraction. It is the ratio of two integers in plain terms.
Numbers that are not rational are referred to as irrational numbers. Now, let's elaborate, irrational numbers could be expressed in decimals, but not in the form of fractions, meaning that the ratio of two integers should not be expressed. Following the decimal mark, irrational numbers have infinite non-repeating digits.