Question

Insert any two irrational numbers between $\frac{4}{7}$ and $\frac{5}{7}$.

Answer 1

$\frac{5}{7}=\mathrm{0.714285714285...}=0.\overline{714285}$

and $\frac{4}{7}=\mathrm{0.571428571428...}=0.\overline{571428}$.

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

Hence, $\frac{5}{7}$ and $\frac{4}{7}$ 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 two different irrational number between $\frac{5}{7}$ and $\frac{4}{7}$ can be:

$\mathrm{0.5918264201001...}$ and $\mathrm{0.601001000100001...}$

NOTE:

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