##### Question 10.8.3

In Section 5.1.2: Calculating Square Roots we studied one method for calculating square roots. In this question we’ll discover another, which is actually an example of the Newton-Raphson iterative method, and is equivalent to a method used by the ancient Babylonians to find square roots.

Suppose we want to find the square root of ten. Perhaps we want to know what side length a square field should have if we want its area to be ten square kilometres. Since $3^2 = 9$, and $4^2 = 16$, we know that the side length will be three and a bit, so we might start by imagining a square of side length $3$ inside our square of area $10$. The area not covered by the smaller square is $10 - 3^2 = 1$. What we need to find out is the extra length, marked with ‘$a$’ in the diagram.

The region with area $1$ can be considered as two rectangles and a little square.

1. Since the little square is much smaller than the rectangles, we might decide to ignore it, and get an approximation for $a$ by just considering the combined area of the two rectangles to be $1$. What would $a$ be in this case? Show hint

2. $3+a$ is now a decent approximation for $\sqrt{10}$. Is $(3+a)^2$ bigger or smaller than $10$ and by how much? Show answer

3. To get an even better approximation, we can repeat this process. This time we need to reduce the side length by a small amount, which I’ll call $b$. Find $b$ so that $3+a-b\approx \sqrt{10}$ by considering the area of the little square, $b^2$, to be negligible. Show hint