This problem builds on the one in May on calculating Pi. This brilliant man Archimedes managed to establish that 3 1/10 < π < 3 1/7.

The problem is how did he calculate the lengths of the sides of the polygons, which needed him to be able to calculate square roots? He didn't have a calculator but needed to work to an appropriate degree of accuracy. To do this he used what we now call numerical roots.

How might he have calculated √3?

This must be somewhere between 1 and 2. How do I know this?

Now calculate the average of 3/2 and 2 (which is 1.75) - this is a second approximation to √3.
i.e. we are saying that a better approximation to √3 is (3/n + n)/2 where n is an approximation to √3 .

We then repeat the process to find the new (third) approximation to √3

${\displaystyle \sqrt{3}\approx \frac{(3/1.75+1.75)}{2}=1.73214...}$

to find a forth approximation repeat this process using 1.73214 and so on...

How many approximations do I have to make before I can find √3 correct to five decimal places.

Why do you think it works?

Will it always work no matter what I take as my first approximation and does the same apply to finding other roots?