The Babylonian Method, Part II

In order to compute , we begin by making a first "guess" (it doesn't need to be a "good" guess, even = 1 will do). Then, as we've seen, the number / is also an estimate for .

Now we claim that

the average of our first two estimates, is an even better estimate for . Assuming this for the moment then means that

should improve upon both of our previous estimates.

But why stop there? Continuing, we define

As increases, each successive estimate should be ever closer to . In fact, we have

provided that we begin with (and > 0).

Before we check this claim, let's look at a few examples.

Neal Carothers -