Question

# Find two rational and two irrational number between $\sqrt 2$ and $\sqrt 3$

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Hint: As we know that $\sqrt 2$ and $\sqrt {\text{3}}$ are irrational numbers so their approximate values are $1.414$ and ${\text{1}}{\text{.732}}$. Now , we need to calculate rational an irrational number between $1.414$ and ${\text{1}}{\text{.732}}$

Given Irrational Numbers $\sqrt 2$ and $\sqrt {\text{3}}$
Calculating rational numbers between $1.4$and $1.7$.
So, they are $\Rightarrow \dfrac{{1.4 + 1.7}}{2} = 1.55$ and can also be integers as $1.5,1.6...$
$\Rightarrow \dfrac{{\sqrt 2 + \sqrt 3 }}{2} = 1.572$
So the numbers between $\sqrt 2$ and $\sqrt {\text{3}}$ which are non-terminating and cannot be expressed in $\dfrac{{\text{p}}}{q}$ form, so it can be $1.665\overline 7 ,1.543\overline 9 ,....$
Hence, $1.5,1.6$ are 2 rational numbers and $1.665\overline 7 ,1.543\overline 9$ are irrational terms between $\sqrt 2$ and $\sqrt {\text{3}}$.