A coin of diameter 1 unit has a dot where it touches the number line at 0. It rolls along the line without slipping for exactly one full turn, until the dot touches the line again at P. Which number does P represent?

Every real number has a unique location on the number line.

- Rationals: easy to plot. $\frac{3}{4}$ → 3 quarters past 0.
- Irrationals like $\sqrt{2}$ can be plotted geometrically: a right triangle with legs 1 and 1 has hypotenuse $\sqrt{2}$ (Pythagoras). Use a compass to mark off $\sqrt{2}$ on the number line from 0.

You can extend this — $\sqrt{3}$ from a triangle with legs $\sqrt{2}$ and 1, $\sqrt{4}=2,\sqrt{5}$ from legs 2 and 1, etc. (the spiral of Theodorus).