In an past exam paper, one of the questions is to work out the continued fraction representaton of $-\sqrt{2}$.
I worked this out as follows:
$x_0=-\sqrt{2}$, $a_0=[-\sqrt{2}]=-1$ where $[x]$ is the integer part of $x$
$x_1=\tfrac{1}{-\sqrt{2}--1}$, $a_1=[x_1]=-2$
with $a_i=-2$ for $i\geq2$, giving the continued fraction representation as $[-1;\overline{-2}]$
This differs from the model solution as in it, he gives the integer part of $-\sqrt{2}$ as $-2$ and then the continued fraction representation as $[-2;1,1,\overline{2}]$
My question is, why did he set the interger part of $-1.41421....$ as $-2$ and not $-1$, and assuming I'm wrong, when my answer is plugged into a calculator why does it also get very close to $-\sqrt{2}$?
Thank you