Question

Find an irrational number between $ \dfrac{5}{7} $ and $ \dfrac{7}{9} $ . How many more there may be?

Answer 1

Hint: We are asked to find the irrational number between the two fractions. Fractions are the part of the whole. Generally it represents any number of equal parts and it describes the part from the certain size. Here first we will convert fraction in the decimal point and then will find the numbers between the two.

Complete step-by-step answer:
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. Remember zero is the rational number.
First convert the given fraction in the form of the decimal.
 $ \Rightarrow \dfrac{5}{7} = 0.71 $ (Decimal point up to two digits)
Similarly for the second given fraction –
 $ \Rightarrow \dfrac{7}{9} = 0.77 $ (Decimal point up to two digits)
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 –
 $ \Rightarrow 0.71000002,{\text{ 0}}{\text{.7100000008,}}.... $
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 “we have an infinite number of irrational numbers between the two irrational numbers.”.

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.