Finding Square Roots

The "Babylonian method" for finding square roots is based on an iterative procedure that is sometimes called "divide and average." We will illustrate it for k = 2, but it works just as well for any k > 0. Suppose someone guesses that

.

We can check whether this guess is correct by multiplying 3/2 by itself. In so doing we find that 3/2 x 3/2 = 9/4 > 2, which shows that 3/2 is too large. To improve this initial guess, we note that there is a number for which 3/2 x [number] = 2. A little arithmetic (dividing 2 by 3/2) shows that this number is 4/3 - i.e., that

3/2 x 4/3 = 2.

This leads to the conclusion that the square root of 2 lies between 4/3 and 3/2. Now the fact that the "average" (i.e., the arithmetic mean) of 4/3 and 3/2 also lies between these two numbers leads us to try

as a (hopefully improved) second guess for the square root of 2. Again, we can check whether this guess is correct by multiplying 17/12 by itself. In so doing we find that

17/12 x 17/12 = 289/144 > 2,

which shows that 17/12 is also too large. This leads us to seek a number for which 17/12 x [number] = 2. This time the number is 24/17 (it can be obtained by dividing 2 by 17/12), leading to the fact that 17 /12 x 24/17 = 2. Since the square root of 2 lies between 24/17 and 17/12, we again calculate their arithmetic mean to obtain a (hopefully improved) third approximation. This leads to

which is correct up to the 6th decimal place! While we could continue this process indefinitely, let us instead pause and take stock.

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