## 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 .

**Continue**
with the mysteries of the ancient world, or
return to my home page.

Neal Carothers -
carother@bgnet.bgsu.edu