Locate $\sqrt{2}$ on the number line.

It is easy to see how the Greeks might have discovered $\sqrt{2}$. Consider a square $OABC$,with each side $1$ unit in length. Then you can see by the Pythagoras theorem that $OB = \sqrt{1^{2} + 1^{2}} = \sqrt{2}$.
To represent $\sqrt{2}$ on the number line,place the square $OABC$ on the number line such that the vertex $O$ coincides with zero and the side $OA$ lies along the positive direction of the number line.
We have just seen that $OB = \sqrt{2}$. Using a compass with centre $O$ and radius $OB$,draw an arc intersecting the number line at the point $P$. Then the point $P$ corresponds to $\sqrt{2}$ on the number line.