Question

Find 3 irrational numbers between $\frac{2}{3}$ and $\frac{3}{4}$ using their decimal expansion.

Answer 1

The decimal expansions of $\frac{2}{3}$ and $\frac{3}{4}$ are as follows:

Here, the one decimal expansions are non-terminating recurring, while the other is terminating.

Hence, $\frac{2}{3}$ and $\frac{3}{4}$ 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 $3$ different irrational number between $\frac{2}{3}$ and $\frac{3}{4}$ can be:

$a=0.6707007000$

$b=0.6808008000$

$c=0.6909009000$

Here we observe that $\frac{2}{3}<a<b<c<\frac{3}{4}$.

NOTE:

This is $3$ of the various possible answers. It might have different solution.