Heron's method and irrational numbers
Approximate a square root with the Heron method (a rectangle that becomes a square), then see why √2 is not a fraction.
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Guide
Heron's method finds a square root by a rectangle: to find √a, start with a rectangle of area a, for example 3 by 2 for a = 6. Replace the side x by the mean of x and a : x. The new rectangle is closer to a square.
In symbols: x₁ = ½ · (x₀ + a : x₀). Repeat the step: the digits settle down more and more. The figures are drawn to scale, 1 cell is 1 cm.
A number is rational if it can be written as a fraction p/q with whole numbers p and q. √2 is not rational. The proof is indirect: assume √2 = p/q in lowest terms, and show that p and q would both be even, which contradicts lowest terms.