Question

Write recurring decimal expressions for the rational numbers $ \dfrac{1}{{21}} $ and $ \dfrac{1}{{14}} $ and hence write two irrational numbers between the numbers $ \dfrac{1}{{21}} $ and $ \dfrac{1}{{14}} $ .

Answer 1

Hint: A rational number is the number which can be expressed as the ratio of two numbers or which can be expressed as the p/q form or as the quotient or the fraction with non-zero denominator whereas, the numbers which are not represented as the rational are known as the irrational number. Recurring decimal expressions are the expressions or the numbers which are repeated again and again following some pattern like a group of three numbers are repeated again and again and likewise.

Complete step-by-step answer:
Take the given rational number:
First convert the given fraction in the form of the decimal.
 $ \dfrac{1}{{21}} = 0.047619047619047619 $
We can observe that five digits after the decimal point are recurring again and again.
Similarly for the second given fraction –
 $ \dfrac{1}{{14}} = 0.0714285714285 $
We can observe that five digits after the decimal point are recurring again and again.
As we know that the irrational numbers are all the non-terminating and the non-repeating numbers.
Irrational numbers between the two given fraction are –
 $ 0.051000002,{\text{ 0}}{\text{.5100000008,}}.... $
It can be any number between the two with any number of digits. Hence we have an infinite number of irrational numbers between the two irrational numbers.
So, the correct answer is $ 0.051000002,{\text{ 0}}{\text{.5100000008,}}.... $ ”.

Note: Always remember that between any two given numbers there are infinite rational and irrational numbers irrespective of how small or large the difference between the two may be. In irrational numbers the decimal form is in the non-repeating and non-terminating numbers.