2007-03-11 About Square Roots

Nerd stuff!

How to compute the square root without a calculator:

Basic idea:

sqrt( a2+2ab+b2 ) = sqrt( (a+b)2 ) = a+b
     -a2
     ----------
    (    2ab+b2 ) / ( 2a+b ) = b

Almost the same as division!

It's cryptic, I know. Here is an example:
   sqrt( 7563 )
From right to left, divide the number in groups of two digits: 75 63
Find the largest number a with a² ≤ 7500. Obviously, a=80, and hence

sqrt( 75 63 ) = 80+b
     -64 00
     ------
      11 63

To find out b, we need to compute
   1163 / ( 160+b ) = b.
Note that b is a natural number and 1 ≤ b ≤ 9. If there is no b that solves the equation (which is very likely), choose the largest value for which
   1163 / ( 160+b ) > b
holds. Here, b=6.
We need to continue, because 1163 / 166 ≠ 6:

 1163 / 166 = 6
  996
 ----
  167

For the next iteration, we set a=86, append two zeros out of thin air, put a comma behind the current result and continue.

Here is the complete calculation for the first two decimal places:

sqrt( 7563 ) = 86,96...
     -64
      ----
      1163 / 166 = 6
     - 996
      ------
       16700 / 1729 = 9
     - 15561
      ------
        113900 / 17386 = 6
     -  104316
      --------
          958400 / 17392_ = _
     - ...

As you can see, the divisor gets bigger and bigger as the result get more and more precise. That's the big disadvantage of this technique and the reason it's not used inside of computers. I still like it, because it takes the horror from the square root. I mean, c'mon, if they can be computed using only paper and a pencil, how difficult can they be?