Archimedes and numerical roots
Problem
This problem builds on Approximating Pi. This brilliant man Archimedes managed to establish that $3 1/10 < \pi < 3 1/7$.
The problem is how did he calculate the lengths of the sides of the polygons, which he needed 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 $\sqrt{3}$?
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 $\sqrt{3}$. i.e. we are saying that a better approximation to $\sqrt{3}$ is $(3/n + n)/2$ where $n$ is an approximation to $\sqrt{3}$.
We then repeat the process to find the new (third) approximation to $\sqrt{3}$. $$\sqrt{3} \approx {(3 / 1.75 + 1.75) \over {2}} = 1.73214\dots$$ to find a fourth approximation repeat this process using 1.73214 and so on...
How many approximations do I have to make before I can find $\sqrt{3}$ 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?
Getting Started
Student Solutions
There was a correct solution from Andrei Lazanu (School 205, Bucharest). The first part is very clear but I have tried to simplify his solution to the second part for inclusion here. Perhaps someone could improve on this for us. Thank you for your hard work Andrei. First, I approximated $\sqrt{3}$ using the method given in the problem. I know that $\sqrt{3}$ is between 1 and 2 because $1^2 < (\sqrt{3})^2 < 2^2$ or $1 < 3 < 4$.
I know that the approximation of $\sqrt{3}$ correct to five decimal places is: $$\sqrt{3} \approx {1.73205}$$
Now I show each of the approximation steps:
First approximation: $$\sqrt{3} \approx {2}$$Second approximation: $$\sqrt{3}\approx {{{3\over{2}} + 2} \over {2}} ={1.75}$$ Third approximation: $$\sqrt{3} \approx {{{3\over{1.75}} + 1.75} \over {2}} = {1.732142857}$$ Fourth approximation: $$\sqrt{3} \approx {{{3\over{1.732142857}} + 1.732142857} \over {2}} = {1.73205081}$$ So, four approximations are sufficient to approximate $\sqrt{3}$ correct to 5 decimal places.
You could think of the above as $$\sqrt{a^2}\approx {{{a^2\over{n}} + n} \over {2}} ={m}$$
Where n is the approximation to the root of a 2 (that is "a") and m the next approximation.
The first approximation (n) differs from a by k. I can therefore write n as a + k where k is numerically less than a (k could be negative).
So I have $$\mbox{The next approximation} = {{{a^2\over{a+k}} + a+k} \over {2}}$$ But $${{{a^2\over{a+k}} + a+k} \over {2}} = {{2a^2 + 2ak + k^2} \over{2(a+k)}}$$ and $${{2a^2 + 2ak + k^2} \over{2(a+k)}} = {{2a(a+k)+ k^2} \over{2(a+k)}}= {{2a(a+k)} \over{2(a+k)}} + {{k^2} \over{2(a+k)}} = a + {{k^2} \over{2(a+k)}}$$ While a is positive, $${{k^2} \over{2(a+k)}}$$must be positive as k is numerically less than a.
So $$a< {{{a^2\over{a+k}} + a+k} \over {2}}$$But the same equation could be written as: $$ {{2a^2 + 2ak + k^2} \over{2(a+k)}} = {{a^2+ (a+k)^2} \over{2(a+k)}}= {{(a+k)^2} \over{2(a+k)}} + {{a^2} \over{2(a+k)}} = {{a+k} \over{2}} + {{a^2}\over {2(a+k)}}$$The following number is equal to a+k: $${{a+k} \over{2}} + {{a^2}\over {2(a+k)}} + {{2ak+k^2}\over{2(a+k)}} = {{(a+k)^2 + a^2 + 2ak + k^2}\over{2(a+k)}} = {{a^2 + k^2 + 2ak + a^2 + 2ak + k^2}\over{2(a+k)}} = {{2(a^2 + 2ak + k^2)}\over{2(a+k)}}={{(a+k)^2}\over{(a+k)}} = (a+k)$$This means that $${{{a^2\over{a+k}} + a+k} \over {2}}< {a+k}.$$
From the two inequalities I obtain that: $$a< {{{a^2\over{a+k}} + a+k} \over {2}}< {a+k}.$$
This means that the solution obtained goes closer and closer at each step to the real value of whether k is positive or negative.