How do I subtract a square root from a decimal number?

Answer

Evy, given your age, maybe read this answer with your math teacher.

Every positive number has a square root, but that’s usually a number with an infinite number of digits after the decimal point, so you’ll never know it exactly. You can calculate it to a certain number of digits.

The ancient Greek Heron knew a method to find the root of a number, possibly with digits after the decimal point, and the Babylonians would also have known this method. It’s an “iterative” method: you make a guess, and from that guess you calculate a better result, then a better one, and so on until you get enough numbers right.

Let me explain with an example: we want to know the square root of 151.37 to three digits after the decimal point. We now call that “your number”. Then do the following steps:

1) make an estimate. Take a number whose square is close to your number. For example, take 12 as an estimate here, because 12 x 12 = 144 and that is close to your number 151.37

2) then calculate as a better estimate : 0.5 times (this estimate + your number divided by the estimate)
so : 0.5 ( 12 + 151.37 / 12 ) = 12.30708344

now you always repeat this 2nd step, always with the improved estimate:

so : 0.5 ( 12.30708344 + 151.37 / 12.30708344) = 12.30325220
with that : 0.5 ( 12.30325220 + 151.37 / 12.30325220 ) = 12.30325160
with that : 0.5 ( 12.30325160 + 151.37 / 12.30325160 ) = 12.30325160

You can see that the estimate doesn’t change anymore, so as the square root of 151.37 we can take : 12.30325160.
This result is therefore correct to within 10 significant digits. This is more than accurate enough for all practical applications.

Such a method is called “iterative”, because you calculate the result in successive steps, which bring you closer and closer to the result. Iterative comes from the Latin word “iter” which means step.

Try it yourself on an example. Use the calculator to calculate the square root of, say, 641.75, and then find it using this method. If you do it right you should get the same result as your calculator.