Heron's Square Roots — Fold a Rectangle Toward a Square

What is the square root of 10? It has no neat answer. Over 2,000 years ago Hero of Alexandria found a way to close in on it by averaging. The square root of a number is the side of a square with that area -- so you will chase a square.

You want the square root of 10, so draw a rectangle with area 10 square units -- say 10 by 1, or 5 by 2. It is the wrong shape (a long thin rectangle), but it has the right area.

Take the two side lengths and AVERAGE them to get a new width; the new height is the area divided by that width (so the area stays 10). Draw the new, less lopsided rectangle. Repeat two or three times: 5 and 2 average to 3.5 (height 2.857), then 3.18 (height 3.14), then 3.162... The rectangle squares up, and its side is the square root of 10, about 3.162.

What your squaring-up shows. (1) If a guess is too big, the area divided by it is too small, and the true root lies between -- so their average is a better guess. (2) The accuracy roughly DOUBLES each step ('quadratic convergence'), so about five rounds reach full calculator precision. (3) Start with a sensible guess, near the root, and it races in immediately. (4) Heron's averaging is exactly what you get by applying Newton's method to 'x squared minus S' -- but Hero found it about 1,600 years before Newton, and it is still how calculators and computer chips extract square roots today.