# number between 17 and 18, and has a rational square root

"number between 17 and 18, and has a rational square root" Is there even one? They all keep coming up irrational for me

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It would be more intruiging to know how you found the irrational ones. –  Tom Collinge Apr 28 '14 at 12:28

Hint: Can you think of a rational number between $\sqrt{17}$ and $\sqrt{18}$?

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Perhaps to make it a little easier to see how to construct it: you want to form a number $\ \frac{p^2}{q^2} \$ , $\ p \$ and $\ q \$ being integers and with the numerator and denominator relatively prime, that is between $\ 17^2 \ = \ 289 \$ and $\ 18^2 \ = \ 324 \$ . –  RecklessReckoner Apr 27 '14 at 19:24
We can also use continued fractions. $\sqrt{17}=[4;\overline{8}]$ while $\sqrt{18}=[4,\overline{4,8}]$. Hence the "simplest" answer is $[4;5]=4\frac{1}{5}=4.2$. –  vadim123 Apr 27 '14 at 19:27
You can be as simple-minded as you like. Try $4.1$, $4.2$, $4.3$, etc., squaring each one, till you find one of the squares to lie between $17$ and $18$. In case you jumped over the interval $\langle17,18\rangle$, you could work on varying the second decimal place. –  Lubin Apr 27 '14 at 19:58
@vadim123: Wouldn't the simplest answer be $[4,\overline{6,8}] = 2\sqrt{13/3}$? –  Eric Towers Apr 27 '14 at 20:16
@EricTowers, that is indeed simple, but alas not rational. –  vadim123 Apr 27 '14 at 21:40

I like the positive root of $x^2 - x - \frac{217745}{16384}$.

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Any particular reason? –  vonbrand Apr 28 '14 at 10:45
It's as good as any other? –  orion Apr 28 '14 at 11:29
It is an irrational liking. –  Eric Towers Apr 28 '14 at 13:04

"Is there even one?" Yes – there are infinitely many. (Exercise: Why?)

Hint: Guess the square root, not the number between 17 and 18.

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