How to represent the square root of 90 as a fraction?

Question

How to represent the square root of 90 as a fraction?

I can usually do this with a calculator but it's not wanting to play nice. I need it to find the distance between two points but, unfortunately I am unable to do so because it asks for it in the form of a fraction. The decimal form is $9.486932981$ on my calculator.

Answer 1

Since sqrt(90) is an irrational number, by definition it cannot be represented as a ratio of two integers.

Edit: wrote "numbers" instead of "integers"...d'oh

Answer 2

There is no fraction of integer numbers $p,q\in\mathbb Z$ such that $\dfrac pq=\sqrt{90}$, however it can be approximated. One method which yields a rapid aproximation are continued fractions in which: $$\sqrt{90}=a_0+\dfrac{1}{a_1+\dfrac{1}{a_2+\dfrac{1}{a_3+\dfrac{1}{a_4+\dfrac{1}{a_5+\dots}}}}}$$ where you can approximate: $$C_5=a_0+\dfrac{1}{a_1+\dfrac{1}{a_2+\dfrac{1}{a_3+\dfrac{1}{a_4+\dfrac{1}{a_5}}}}}$$ a rational number.

For finding the appropiate $a_i$ (and $C_i=\dfrac{p_i}{q_i}$) we must first set $a_0=\lfloor\sqrt{90}\rfloor=9$, now: \begin{align} \sqrt{90} &= 9 + (\sqrt{90}-9) \\ &= 9 + \frac1{\frac1{\sqrt{90}-9}} \\ &= 9 + \frac1{\frac{\sqrt{90}+9}{(\sqrt{90}-9)(\sqrt{90}+9)}} \\ &= 9 + \frac1{\frac{\sqrt{90}+9}{90-81}} = \frac1{\frac{\sqrt{90}+9}9}\\ &= 9 + \frac1{\frac19\sqrt{90}+1} \\ \end{align} Where $1<\frac19\sqrt{90}<2$ and, therefor $\lfloor\frac19\sqrt{90}+1\rfloor=2$ so: \begin{align} \sqrt{90} &= 9 + \frac1{\frac19\sqrt{90}+1} \\ &= 9 + \frac1{2+(\frac19\sqrt{90}-1)} \\ &= 9 + \dfrac1{2+\dfrac1{\frac1{\frac19\sqrt{90}-1}}} \\ &= 9 + \dfrac1{2+\dfrac1{\frac9{\sqrt{90}-9}}} \\ &= 9 + \dfrac1{2+\dfrac1{\frac{9(\sqrt{90}+9)}{(\sqrt{90}-9)(\sqrt{90}+9)}}} \\ &= 9 + \dfrac1{2+\dfrac1{\frac{9(\sqrt{90}+9)}9}} \\ &= 9 + \dfrac1{2+\dfrac1{\sqrt{90}+9}} \\ &= 9 + \dfrac1{2+\dfrac1{18+(\sqrt{90}-9)}}. \\ \end{align}

Here you can probably see the pattern that: $$\sqrt{90}=9+\dfrac{1}{2+\dfrac{1}{18+\dfrac{1}{2+\dfrac{1}{18+\dfrac{1}{2+\dots}}}}}.$$

So, the successive approximations will have: \begin{align} \sqrt{90} &\simeq 9 \\ &\simeq 9+\dfrac12 = \frac{19}{2} = 9.5 \\ &\simeq 9+\dfrac1{2+\dfrac1{18}} = \frac{351}{37} \simeq9.4864864865 \\ &\simeq 9+\dfrac1{2+\dfrac1{18+\dfrac12}} = \frac{721}{76} \simeq9.4868421053 \\ &\simeq 9+\dfrac1{2+\dfrac1{18+\dfrac1{2+\dfrac1{18}}}} = \frac{13329}{1405} \simeq9.4868327402 \\ &\simeq \frac{27379}{2886}\simeq9.4868329868 \\ &\simeq \frac{506151}{53353}\simeq9.4868329803 \\ &\simeq \frac{1039681}{109592}\simeq9.4868329805 \end{align}

But neither is an exact result. $\sqrt{90}$ does not have an exact result in either a fraction, or a finite decimal representation. (A finite decimal representation is a fraction: $9.486832981=\dfrac{9486832981}{1000000000}$.)