This is an old trick I discovered a few years ago for calculating the square root of any square natural number.



Let u be the number under scrutiny, and x is the positive square root of u.

The method is as follows:

u-2(1)-2(2)-2(3)-2(n)
 When zero is reached while doing the subtraction, stop. The number that was last used in subtracting from u is the square root of u.

 For example, u=25

25-2(1)-2(2)-2(3)-2(4)-2(5)=0

 n=5 and so x=5

Note that this method only returns the positive value for x. The negative value can be simply inferred from then on.

    25
      1
-_____
    24
      1
-____
     23
       2
-______
     21
       2
-______
      19
        3
-______      16
        3
-______
      13
        4
-______        9
        4
-______
        5
        5
-______
        0
Extension

      There is also a shortcut to this method which requires knowledge of another square number, less than u.

For example, we are given that 20^2=400.
and x^2=u=576

Let 20^2=z^2

The method is as follows:

u-z^2-z-2(z+1)-2(z+2)-2(z+n)

    576
    400
-______
    176
      20
-______
     156
       21
-______
     135
       21
 -______
     114
       22
 -_____

        92   
        22
 -_____

        70
        23
 -_____
       47
       23
 -_____
       24
       24
 -_______
       0


If the remainder is zero, than u is a square number. However, I would be curious to know if the remainder could be used to approximate the square root of any natural number.