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Math Help - Problem understand about mental math square root of number

  1. #1
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    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.
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  2. #2
    Grand Panjandrum
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    Re: Problem understand about mental math square root of number

    Quote Originally Posted by Dark7568 View Post
    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
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  3. #3
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    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:

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

    So,

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

    Using a calculator, I can find that:

    \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)!
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  4. #4
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    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 x^2 - 8 = 0

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

    x_0 = 3

    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.
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