Mentally estimating square roots

What's $\sqrt{10}$? I'd reckon it's about $3 + \frac{10-9}{2 \times 3} = 3.166$. The actual answer is 3.162. What about $\sqrt{18}$? Should be about $4 + \frac{18-16}{2 \times 4} = 4.250$. The actual answer is 4.242. I found out about this calculation from this video but there was no explanation for why it works. Here's an explanation.

So the method here is as follows.

1. Let the number you want to find the square root of be $a^2$.
2. Let the largest square number which is less than $a^2$ be $b^2$ such that $b^2 <= a^2 < (b+1)^2$. For example if $a^2$ is 10 then $b^2$ is 9, if $a^2$ is 18 then $b^2$ is 16.
3. The square root of $a^2$ is approximately $b + \frac{a^2 - b^2}{2 b}$.

This method is easy to carry out mentally but why does it work? The trick here is that the graph of the square root function grows so slowly that we can approximate the curve between two adjacent square numbers as a line.



We can use the line to approximate the square root of any number between two square numbers. The first thing we need to know is the gradient of the line. The vertical distance between two adjacent square numbers on the square root curve is 1, since the two square numbers are the squares of two consecutive numbers. The horizontal distance changes and becomes larger as the adjacent square numbers become larger but we can calculate it as follows:

$$(b+1)^2 - b^2 = b^2 + 2b + 1 - b^2 = 2b + 1$$

So the horizontal distance is twice the square root of the smaller square number plus one. Therefore the gradient of the line is $\frac{1}{2b+1}$. Once we know by how much the line grows vertically for every horizontal unit, we can then determine how much higher than $b$ the point on the line will be at $a$ by multiplying the gradient by $a^2-b^2$, as shown below:



Since the difference in height is less than 1, it is going to be the part of the square root that comes after the decimal point, with the whole number part being $b$.

It might be hard to mentally divide by an odd number in $\frac{a^2-b^2}{2b+1}$ so we further approximate it as $\frac{a^2-b^2}{2b}$ instead. And that's why this method works.