Question

Find an irrational number between two numbers $\frac{1}{7}$ and $\frac{2}{7}$ and justify your answer.

Answer 1

$\frac{1}{7}=\mathrm{0.142857142857...}=0.\overline{142857}$

and $\frac{2}{7}=\mathrm{0.28571428571428...}=0.\overline{285714}$.

Here, the two decimal expansions are non-terminating recurring.

Hence, $\frac{1}{7}$ and $\frac{2}{7}$ are two rational numbers.

We know, between any two rational numbers, there are infinitely many irrational numbers.

An irrational number has non-terminating non-recurring decimal expansions.

Then an irrational number between $\frac{1}{7}$ and $\frac{2}{7}$ is

$\mathrm{0.15015001500015...}$.

Similarly, $\mathrm{0.21020020002...}$ is another irrational number between $\frac{1}{7}$ and $\frac{2}{7}$.

NOTE:

This is some of the various possible answers. It might have different solution.