## The Babylonian Method for Computing Square Roots

Anyone who's ever taken a Chemistry or Physics or Statistics course has seen those tables of important numbers at the back of the textbook. Like me, you may even own a big, fat book entirely devoted to such tables. And you may have wondered, at one time or another, who made these tables and how (or why!). Especially when you consider the fact that all of the important methods that require these tables invariably seem to be attributed to "some old dead guy".

Just how did those old timers compute, say, square roots? Or logarithms? Or trigonometric functions? After all, on the mathematical time scale, computers were invented just yesterday -- and electronic calculators only the day before.

Is there an effective way to compute square roots, or any of the so-called transcendental functions, by hand?

The answer is: Yes! One technique for computing square roots, which dates back to ancient Babylon (roughly 4000 years ago), is easy to describe (and even easier to implement).

Let's suppose that we're interested in finding the square root of a positive number . That is, we're looking for a positive number satisfying .

To begin, let's suppose that is a "good guess" for ; in other words, suppose that is approximately equal to . Then, we can say that / is also a "good guess" for , since

Notice, too, that if is smaller than , then / is larger than , and conversely. For example,

In short, given one estimate for , we automatically have a second estimate, and the two estimates lie on opposite sides of the actual value of .

But now there's an easy way to improve on both estimates. Take the average!

In a moment, we'll check the details behind this claim. For now, let's turn this observation into an algorithm for computing .