Babylonian Square Root Method – Iterations

The Babylonian (Heron’s) method refines a square-root guess by averaging it with S divided by it. If the guess x is too high, S/x is too low, and their average is closer to √S than either. For S = 2 starting at x₀ = 1: x₁ = (1 + 2)/2 = 1.5; x₂ = (1.5 + 2/1.5)/2 = 1.41667; x₃ = 1.4142169 — already within 0.0000031 of √2 ≈ 1.4142136. The calculator lists every iterate, its square (to show the residual error), and the number of correct digits, which roughly doubles per step — quadratic convergence, the same engine inside Newton’s method applied to f(x) = x² − S. Any positive seed works; poor guesses just add a step or two (x₀ = 7 for √2 catches up by the fourth iterate). The method doubles as the classic divide-and-average algorithm taught before calculators.

1. 1

   Enter S and a starting guess

   Any positive guess; 1 or S/2 are common defaults

2. 2

   Average x with S/x

   Each iterate xₖ₊₁ = (xₖ + S/xₖ)/2 is printed

3. 3

   Watch errors halve their digits

   Correct digits double per iteration

4. 4

   Stop at tolerance

   The run halts when |xₖ² − S| falls below your threshold

• 1

  Averaging x and S/x always moves toward √S

• 2

  Digits double each iteration — three steps beat six decimal places

• 3

  The seed hardly matters; bad guesses cost one extra step

• Averaging x with x/2 instead of S/x

• Stopping after one iterate and expecting full precision

• Using a negative seed, which sends the iteration complex

• Confusing the method with digit-by-digit longhand square rooting