Question
Write three rational numbers between $\sqrt{3}$ and $\sqrt{5}.$
✓
Answer
Consider some rational numbers between $3$ and $5$ such that they are perfect squares.
Let us take, $3.24,3.61,4,4.41$ and $4.84$ as $\sqrt{3.24}$
$=1.8, \sqrt{3.61} =1.9, \sqrt{4}=2, \sqrt{4.41} =2.1$ and $\sqrt{4.84} =2.2$
Thus we have,
$\sqrt{3} <\sqrt{3.24 } <\sqrt{ 3.61} <\sqrt{ 4} <\sqrt{ 4.41}<\sqrt{4.84 } <\sqrt{ 5} $
$ \Rightarrow \sqrt{ 3} <1.8<1.9<2<2.1<2.2<\sqrt{ 5} $
$ \Rightarrow \sqrt{3 } <\frac{18}{10}<\frac{19}{10}<2<\frac{21}{10}<\frac{22}{10}<\sqrt{ 5}$
$ \Rightarrow \sqrt{3} <\frac{9}{5}<\frac{19}{10}<2<\frac{21}{10}<\frac{11}{5}<\sqrt{ 5}$
Therefore, any three rational numbers between $\sqrt{3 } $ and $\sqrt{5 } $ are :
$\frac{9}{5}, \frac{19}{10}$ and $\frac{21}{10}$
Let us take, $3.24,3.61,4,4.41$ and $4.84$ as $\sqrt{3.24}$
$=1.8, \sqrt{3.61} =1.9, \sqrt{4}=2, \sqrt{4.41} =2.1$ and $\sqrt{4.84} =2.2$
Thus we have,
$\sqrt{3} <\sqrt{3.24 } <\sqrt{ 3.61} <\sqrt{ 4} <\sqrt{ 4.41}<\sqrt{4.84 } <\sqrt{ 5} $
$ \Rightarrow \sqrt{ 3} <1.8<1.9<2<2.1<2.2<\sqrt{ 5} $
$ \Rightarrow \sqrt{3 } <\frac{18}{10}<\frac{19}{10}<2<\frac{21}{10}<\frac{22}{10}<\sqrt{ 5}$
$ \Rightarrow \sqrt{3} <\frac{9}{5}<\frac{19}{10}<2<\frac{21}{10}<\frac{11}{5}<\sqrt{ 5}$
Therefore, any three rational numbers between $\sqrt{3 } $ and $\sqrt{5 } $ are :
$\frac{9}{5}, \frac{19}{10}$ and $\frac{21}{10}$
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