Question

Find three different irrational numbers between the rational numbers $\dfrac{5}{7}$and$\dfrac{9}{{11}}$.

Answer 1

Hint: In order to find three irrationals between them, we need to write three irrational numbers. An irrational number is something which cannot be expressed as fraction. In our case an irrational number between the given numbers $\dfrac{5}{7}$and$\dfrac{9}{{11}}$. So we just have to pick any non-terminating number between the given numbers.

Complete step-by-step answer:
We have to find three irrational numbers between the rational numbers $\dfrac{5}{7}$and$\dfrac{9}{{11}}$.
Irrational number is a number that cannot be expressed as a fraction for any integers and irrational numbers have decimal expansions that neither terminate nor become periodic. Every transcendental number is irrational.

The value of the given number - $\dfrac{5}{7}$= 0.7142857143…
And - $\dfrac{9}{{11}}$= 0.81818181….

So our goal is to pick an irrational number between the values of 0.7142857143 and 0.81818181,
So we pick three random irrational numbers in these limits:
Three different irrational is
$
   \Rightarrow 0.71231234... \\
   \Rightarrow 0.72232232.... \\
   \Rightarrow 0.7542112111... \\
 $

Note: In order to solve this type of problems the key is to have to write irrational numbers i.e. numbers should not be repeating or recurring. We can't write it in a fraction and we can write infinite irrational numbers between two rational numbers. Irrational numbers are normally characterized by non-ending decimal values.