AI Tutor
If the decimal expansion of 2\sqrt{2} never ends and never repeats, how can we be certain it marks an exact, unwavering spot on the number line?
Dynamic geometry software like GeoGebra lets us construct exact irrational lengths physically on a coordinate plane and swing them straight onto the number line.

Constructing Exact Roots

To construct an exact root, we rely on the Pythagorean theorem, which states that in any right triangle with legs aa and bb, the hypotenuse cc satisfies a2+b2=c2a^2 + b^2 = c^2, so c=a2+b2c = \sqrt{a^2 + b^2}.
When you draw a right triangle with base 1 and height 1 on a coordinate grid, the length of its hypotenuse is exactly 12+12=2\sqrt{1^2 + 1^2} = \sqrt{2}.
By drawing a circle in GeoGebra centered at the origin through (1,1)(1,1), the radius transfers that exact length of 2\sqrt{2} directly onto the horizontal axis.
What if you want to find higher roots like 3\sqrt{3}, 4\sqrt{4}, or 5\sqrt{5} using this same geometric technique?

The Spiral of Theodorus

In the 5th century BCE, the Greek mathematician Theodorus of Cyrene constructed square roots of consecutive integers up to 17\sqrt{17} by chaining right triangles into a continuous spiral.
If you attach a new leg of length 1 perpendicular to the 2\sqrt{2} hypotenuse, the next hypotenuse becomes (2)2+12=2+1=3\sqrt{(\sqrt{2})^2 + 1^2} = \sqrt{2 + 1} = \sqrt{3}.
Continuing this spiral allows you to construct any n\sqrt{n} geometrically and drop a compass arc down to locate it precisely on the real number line.