Question
Write two rational numbers between $\sqrt{2}$ and $\sqrt{3}.$
✓
Answer
We want rational numbers $\mathrm{a} / \mathrm{b}$ and $\mathrm{c} / \mathrm{d}$ such that
$: \sqrt{2}<\frac{a}{b}<\frac{c}{d}<\sqrt{3}$ Consider any two rational numbers between $2$ and $3$ such that they are perfect squares.
Let us take $2.25$ and $2.56$ as $\sqrt{2.25 } =1.5$ and $\sqrt{ 2.56} =1.6$
Thus we have,
$\sqrt{ 2}<\sqrt{ 2.25} <\sqrt{ 2.56} <\sqrt{3 } $
$ \Rightarrow \sqrt{ 2} <1.5<1.6<\sqrt{ 3} $
$\Rightarrow \sqrt{ 2}<\frac{15}{10}<\frac{16}{10}<\sqrt{3 } $
$ \Rightarrow \sqrt{ 2} <\frac{3}{2}<\frac{8}{5}<\sqrt{3 } $
Therefore any two rational numbers between $\sqrt{2 } $ and $\sqrt{3 } $ are :
$\frac{3}{2}$ and $\frac{8}{5}$
$: \sqrt{2}<\frac{a}{b}<\frac{c}{d}<\sqrt{3}$ Consider any two rational numbers between $2$ and $3$ such that they are perfect squares.
Let us take $2.25$ and $2.56$ as $\sqrt{2.25 } =1.5$ and $\sqrt{ 2.56} =1.6$
Thus we have,
$\sqrt{ 2}<\sqrt{ 2.25} <\sqrt{ 2.56} <\sqrt{3 } $
$ \Rightarrow \sqrt{ 2} <1.5<1.6<\sqrt{ 3} $
$\Rightarrow \sqrt{ 2}<\frac{15}{10}<\frac{16}{10}<\sqrt{3 } $
$ \Rightarrow \sqrt{ 2} <\frac{3}{2}<\frac{8}{5}<\sqrt{3 } $
Therefore any two rational numbers between $\sqrt{2 } $ and $\sqrt{3 } $ are :
$\frac{3}{2}$ and $\frac{8}{5}$
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