Problem understand about mental math square root of number

I found a equation that can obtain approximately answer from square root of numbers without using calculator.

Please refer to Mental calculation - Wikipedia, the free encyclopedia

I didn't not understand the following equation how they formed.

Root(squared)=x

Root= a - b

Please explain briefly, thanks in advanced.

Re: Problem understand about mental math square root of number

Quote:

Originally Posted by

**Dark7568** I found a equation that can obtain approximately answer from square root of numbers without using calculator.

Please refer to

Mental calculation - Wikipedia, the free encyclopedia
I didn't not understand the following equation how they formed.

Root(squared)=x

Root= a - b

Please explain briefly, thanks in advanced.

x is the number you want the square root of. a is the root of a square close to x and b is the (unknown difference) between root of x and the known root of a.

So the root of x is approximately equal to a and b is the error in this crude the approximation and what follows will explain how to get an estimate of b.

CB

Re: Problem understand about mental math square root of number

Say you have a number that is not a perfect square, like 8, and you want to find the square root of it. For this formula you need the nearest perfect square to 9, which is 9 (3 * 3 = 9).

So, according to the formula in that article, you can get an approximation of the square root of 8 by subtracting it from 9, dividing the difference by two times the root of 9 (2 * 3 = 6), then subtracting all of that from the root of the known square, so the equation becomes:

$\displaystyle \sqrt {8} \approx 3 - \frac {9-8}{2(3)} = 3- \frac {1}{6} = 2 \frac {5}{6} = 2.83333333...$

So,

$\displaystyle \sqrt {8} \approx 2.83 (2d.p.)$

Using a calculator, I can find that:

$\displaystyle \sqrt {8} = +/- 2.828427... or +/- 2.83 (2d.p.)$

I hope that was clear enough.

P.S. I've never heard of this method before, it's actually quite clever (providing it always works)!

Re: Problem understand about mental math square root of number

looks like using a single iteration of Newton's method to find the roots of the equation $\displaystyle x^2 - 8 = 0$

$\displaystyle x_1 = x_0 - \frac{f(x_0)}{f'(x_0)}$

$\displaystyle x_0 = 3$

$\displaystyle x_1 = 3 - \frac{9 - 8}{2 \cdot 3}$

will work fine as long as the value of interest is relatively close to a perfect square number.