Most of us take our calculators for granted and press the square root button when we want to find a square root. To impress your friends as a mental gymnast you can follow a method that was used before the days of caculators
Here’s how to do it.
First, recall what is meant by a square root. A square root is a number that can be multiplied by itself gives a specified number. These numbers are very useful when calculating areas an often crop up in scientific calculations.
Now look at the multiplication tables that we learnt at school. These can give us the square roots are some whole numbers.
For the next part of the calculation it is useful to memorise the squares of all integers from one to twelve.
Numbers which have exact whole number square roots are unusual. Most square roots are irrational numbers which means that they can never be fully expressed as a fraction or decimal. In practice this does not matter, very few practical calculations require that square roots are accurate to more than four or five significant figures.
Now turn to the practical calculation itself.
Our calculation requires a first guess for the square root which we will progressively refine. Although it does not mater if we are widely wrong with the guess because the calculation will correct itself a good guess will speed up the calculation and reduce the number of refinements we need to make.
We can get the first guess by looking at the number we are about to square root.
Count the number of decimal places before we come to the most significant figure in the number. Count to the left if the number is greater than one. Count to the right if the number is less than one.
Lets call the count C.
If C is odd look at the two digits of the number If C is even look at the first three digits of the number. Then estimate the square root of the first two or three digits using your memory of the basic multiplication table. To complete the estimate align the guess C/2 columns to the left or right of the decimal place as appropriate.
For example calculate a first guess for the square root of 12345. C = 4 so look at the first three digits. The square root of 123 is just over 11. So a first guess for the root could be 110. The square root is actually 111.18
As another example find the square root of 0.001. C=4 so consider the square root of 10. This is just over 3. So the square root of 0.001 is about 0.03. Actually it is 0.031623.
Once we have the initial estimate the next step is to refine it using the formula:
New guess = Half * (Old Guess + Number to be rooted / Old Guess)
So for the root of 12345
New guess = 0.5 * ( 110 + 12345/110)
New guess = 111.11
And for the root of 0.001
New guess = 0.5 * (.03 + 0.001/.03)
New guess = 0.031667
The new guess can then be fed back into the formula and the cycle continued until the required accuracy is achieved.
This is a very ancient method known as the Babylonian, or sometimes Hero’s method. It is a very useful way in which to calculate square roots.