Babylonian Algorithm for Computing Cube Roots Without a Calculator

The Babylonian algorithm for computing square roots can be adapted for the calculation of cube roots as well. The modified Babylonian method is a divide-and-average process, except that you take the weighted average of two numbers. This iterative technique converges rapidly if you choose your initial value wisely. You can apply it as an alternative to the long division method for calculating cube roots.

Step 1: Call the number whose cube root you wish to calculate N and your initial guess C(0). You can pick any number for C(0), but the algorithm will converge faster when you choose a starting value close to the actual cube root.

For example, suppose you want to find the cube root of 400 (that is, N = 400). Since 7³ = 343 and 8³ = 512, we know the cube root of 400 is between 7 and 8, so let's choose C(0) = 7.

Step 2: Compute the first iteration, C(1), with the equation

C(1) = [2C(0) + N/C(0)²]/3

In this example, we have

C(1) = (2*7 + 400/49)/3 = 362/49 (≈7.388) after simplification.

Step 3: Calculate the second iteration, C(2) with the relation

C(2) = [2C(1) + N/C(1)²]/3

Continuing the example, we have

C(2) = [2*362/49 + 400/(362/49)²]/3 = 35483864/4815867 (≈7.368) after simplifying.

Step 4: Find each successive approximating fraction with the recursion formula

C(k) = [2C(k-1) + N/C(k-1)²]/3

Each new iteration is the weighted average of the previous term and N divided by the square of the previous term. The previous term is weighted twice as much.

Like the Babylonian algorithm for square roots, the modified cubic algorithm also converges quickly. The cube root of 400 is about 7.368062997.

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