Successive magnification is used to locate a decimal on the number line: the orange part of each line is magnified into the line below it. Which number is marked by the orange point “?”

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).