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

*f*(

*x*) by the tangent lines, which we discussed in detail in the previous pages.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

*x*

_{3}through the equation

This process will generate a sequence of numbers which approximates

*r*.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

*x*

_{1}= 2.

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