Babylonian Method for Roots

Mental Math · Axiom Academy

LESSON Babylonian Method for Square Roots An ancient algorithm for refining square root estimates through iterative averaging 1. The Core Idea: Averaging Opposites To find the square root of a number S , we start with any positive guess x . If our guess is too small, then S/x must be too large (and vice versa). The brilliant insight: the average of these two values gives us a better estimate! where x n is our current guess and x n+1 is our improved guess Watch how a poor initial guess quickly converges to the true value: 2. Step-by-Step Example: Finding √10 Let's compute √10 starting with an initial guess of x₀ = 3. Watch as each iteration brings us closer to the true value (≈ 3.162...): Iteration 1: x₁ = ½(3 + 10/3) = ½(3 + 3.333...) = 3.167 Iteration 2: x₂ = ½(3.167 + 10/3.167) = ½(3.167 + 3.158) = 3.162 Iteration 3: x₃ = ½(3.162 + 10/3.162) = 3.162278... 3. Visual Understanding: Geometric Mean The Babylonian method can be understood geometrically. When we have a rectangle with area S, we're trying to find the side length of a square with the same area. Starting with dimensions x by S/x, we iteratively make the rectangle "more square" by averaging the dimensions. Iteration 1: Average sides → (2+8)/2 = 5, giving 5×3.2 rectangle Iteration 2: Average sides → (5+3.2)/2 = 4.1, giving 4.1×3.9 rectangle Iteration 3: Converges to 4×4 square ✓

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