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 (see Fig. $1$). Then you can see by the Pythagoras theorem that $OB =\sqrt{1^{2}+1^{2}}=\sqrt{2}$. How do we represent $\sqrt 2$ on the number line ? This is easy. Transfer Fig $1$. onto the number line making sure that the vertex $O$ coincides with zero (see Fig.$2$)

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 $P$ corresponds to $\sqrt 2$ on the number line.