Already the Babylonians knew how to approximate square roots. Let's consider the example of how they found approximations to .

Let's start with a close approximation, say
*x*_{1}=3/2=1.5. If we
square *x*_{1}=3/2, we obtain 9/4, which is bigger than 2.
Consequently
.
If we now consider 2/*x*_{1}=4/3, its
square 16/9 is of course smaller than 2, so
.

We will do better if we take their average:

If we square *x*_{2}=17/12, we obtain 289/144, which is bigger
than 2. Consequently
.
If we now consider
2/*x*_{2}=24/17, its square 576/289 is of course smaller than 2,
so
.

Let's take their average again:

*x*_{3} is a pretty good rational approximation to the square root
of 2:

but if this is not good enough, we can just repeat the procedure again and again.

Newton and Raphson used ideas of the Calculus to generalize this
ancient method to find the zeros of an arbitrary equation

Their underlying idea is the approximation of the graph of the function

Let *r* be a root (also called a "zero") of *f*(*x*), that is *f*(*r*)
=0. Assume that
.
Let *x*_{1} be a number close to
*r* (which may be obtained by looking at the graph of *f*(*x*)).
The tangent line to the graph of *f*(*x*) at
(*x*_{1},*f*(*x*_{1})) has
*x*_{2} as its *x*-intercept.

From the above picture, we see that *x*_{2} is getting closer to
*r*. Easy calculations give

Since we assumed , we will not have problems with the denominator being equal to 0. We continue this process and find

This process will generate a sequence of numbers which approximates

This technique of successive approximations of real zeros is
called **Newton's method**, or the **Newton-Raphson Method**.

**Example.** Let us find an approximation to
to ten
decimal places.

Note that
is an irrational number. Therefore the
sequence of decimals which defines
will not stop.
Clearly
is the only zero of
*f*(*x*) = *x*^{2} - 5 on
the interval [1,3]. See the Picture.

Let
be the successive approximations obtained through
Newton's method. We have

Let us start this process by taking

It is quite remarkable that the results stabilize for more than ten decimal places after only 5 iterations!

**Example.** Let us approximate the only solution to the
equation

In fact, looking at the graphs we can see that this equation has one solution.

This solution is also the only zero of the function
.
So now we see how Newton's method may be used to
approximate *r*. Since *r* is between 0 and
,
we set *x*_{1} = 1. The rest of the sequence is
generated through the formula

We have

**Exercise 1.** Approximate the real root to two four decimal
places of

**Exercise 2.** Approximate to four decimal places

**Exercise 3.** Show that Newton's Method applied to
*f*(*x*)=*x*^{2}-2 and *x*_{1}=3/2 leads to exactly the same
approximating sequence for the square root of 2 as the Babylonian
Method.

**
**

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Helmut Knaust

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