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'Rationals Between...' printed from https://nrich.maths.org/
Charlie and Alison are exploring fractions and surds.
They are looking for fractions with different denominators that lie between $\sqrt{65}$ and $\sqrt{67}$.
Can you find some fractions that lie between $\sqrt{65}$ and $\sqrt{67}$?
Charlie and Alison found that for some denominators, there is no fraction between $\sqrt{65}$ and $\sqrt{67}$. Click to reveal their thoughts.
Charlie said:
$\sqrt{65}$ is approximately $8.06$, and $\sqrt{67}$ is approximately $8.18$.
Fractions with a denominator of $4$ end in $0$ or $.25$ or $.5$ or $.75$ so there is no fraction with a denominator of $4$ between $\sqrt{65}$ and $\sqrt{67}$.
Alison agreed with Charlie but thought about it in a slightly different way:
I'm looking for a fraction $\frac{p}{q}$ where $\sqrt{65}<\frac{p}{q}<\sqrt{67}$.
This means that $65<\frac{p^2}{q^2}<67$,
or $65q^2<{p^2}<67q^2$.
Suppose $q=4$.
$65\times16<{p^2}<67\times16$
$1040<{p^2}<1072$
$32^2=1024$, and $33^2=1089$, so there is no perfect square between $1040$ and $1072$.
Therefore, $q\neq4$, so there is no fraction with a denominator of $4$ between $\sqrt{65}$ and $\sqrt{67}$.
Can you find other denominators where there is no fraction in the interval?
How will you know when you have found them all?