So the method here is as follows.
- Let the number you want to find the square root of be a2.
- Let the largest square number which is less than a2 be b2 such that b2<=a2<(b+1)2. For example if a2 is 10 then b2 is 9, if a2 is 18 then b2 is 16.
- The square root of a2 is approximately b+a2−b22b.
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−b2=b2+2b+1−b2=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 12b+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 a2−b2, 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 a2−b22b+1 so we further approximate it as a2−b22b instead. And that's why this method works.
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