Find the continued fraction of the square root of a given integer How to find the continued fraction of $\sqrt{n}$, for an integer $n$?
I saw a site where they explained it, but it required a calculator. Is it possible to do it without a calculator?  

The easiest way to find a continued fraction for $\sqrt{n}$ is to use a generalized continued fraction (continued fractions where the numerator is something besides 1).
Starting from $\sqrt{n}$, partition ${n}$ as the sum of the perfect square which is less than or equal to ${n}$, ${s^{2}}$, and a remainder, ${r}$, like this: $\sqrt{s^{2} + r}$.
The generalized continued fraction of this answer will always be:
$$
\sqrt{s^{2} + r} = s + \frac{r}{2s+\frac{r}{2s+\frac{r}{2s+\frac{r}{2s+\frac{r}{2s+\ddots}}}}}
$$
...or, in Gauss' Kettenbruch notation:
$$
\sqrt{s^{2} + r} = s + \underset{i=1}{\overset{\infty}{\LARGE\mathrm K}}\frac{r}{2s}
$$
For example, what's the generalized continued fraction for $\sqrt{21}$? $\sqrt{21} \ = \ \sqrt{16 + 5} \ = \ \sqrt{4^{2} + 5}$, so the generalized continued fraction is:
$$
\sqrt{21} = 4 + \frac{5}{8+\frac{5}{8+\frac{5}{8+\frac{5}{8+\frac{5}{8+\ddots}}}}}
$$
..or..
$$
\sqrt{21} = 4 + \underset{i=1}{\overset{\infty}{\LARGE\mathrm K}}\frac{5}{8}
$$
Here's how to to verify this in Wolfram|Alpha: http://www.wolframalpha.com/input/?i=%284%2B%28ContinuedFractionK%5B5%2C+8%2C+%7Bi%2C+1%2C+infinity%7D%5D%29%29%5E2
You don't have to choose a perfect square less than or equal to $n$, however. You can choose a perfect square larger than $n$. $\sqrt{21} \ = \ \sqrt{25 - 4} \ = \ \sqrt{5^{2} - 4}$, which becomes:
$$
\sqrt{21} = 5 - \frac{4}{10-\frac{4}{10-\frac{4}{10-\frac{4}{10-\frac{4}{10-\ddots}}}}}
$$
You could even do it this way: $\sqrt{21} \ = \ \sqrt{\pi^{2} + (21 - \pi^{2})}$, which gives the result:
$$
\sqrt{21} = \pi + \frac{21 - \pi^{2}}{2\pi+\frac{21 - \pi^{2}}{2\pi+\frac{21 - \pi^{2}}{2\pi+\frac{21 - \pi^{2}}{2\pi+\frac{21 - \pi^{2}}{2\pi+\ddots}}}}}
$$
For more information on continued fractions and roots, read Manny Sardina's General Method for Extracting Roots using (Folded) Continued Fractions: http://myreckonings.com/Dead_Reckoning/Online/Materials/General%20Method%20for%20Extracting%20Roots.pdf
