Estimating Square Root of Any Number

Estimating square root of any number | mathjokes.net
I use this trick quite often while computing square roots. It is extremely simple and utilizes the binomial expansion. It assumes that you know a couple of common squares and their roots. For example, 5 is √25, 12 is √144 etc.

Estimating Square Root of Any Number

Estimating Square Root of Any Number
Estimating Square Root of Any Number

Example 1:
Let's take a number 86.
Get the closest square. In this case 81 it's square root is 9.
Divide 86 by the square root, 9 we get 9.555
Take the average of 9 and 9.555 = 9.277.

Well the exact answer is 9.273, pretty close isn't it ;)

Example 2:
What is the square root of 420 ?
Closest square is 400, square root is 20.
420/20 = 21.
Average of 20 and 21 is 20.5.

Well, the exact answer is 20.494 ;)

Finding the square root of any number (Trick 2)

I'll explain it with an example.

Let's take a number 67.
(I am taking the number as 67 because it is a prime number.Choosing prime numbers ensured that I will have to answer in decimals, so the extra challenge was preferred to be thrown my way)

The two numbers closed to 67 that are perfect squares are 64 and 81. (8 is the square root of 64 and 9 is of 81)
So we know that the square root of 67 will be a number between 8 and 9

Do a square of 8.5 in your head, which would be (8*9).25 = 72.25. So your answer would be lesser than 8.5.
Since 72.25 shot way beyond our required number, your next guess would be 8.2
Square for 8.2 would be 67.24 (which overshot our number 67, but just by a slight margin. If I wanted to be more precise, I would say the answer would be close to 8.18 or 8.19)

Correct answer = 8.18535277187

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