Formula for Calculating Square Roots

Date: 28 Jan 1995 13:11:21 -0500
From: Roger Gillies
Subject: SQUARE ROOT

Thanks for the information on exponent zero.

Could you tell me:

What is the formula for calculating square roots?

R. Gillies

Date: 28 Jan 1995 23:48:52 -0500
From: Dr. Ken
Subject: Re: SQUARE ROOT

Hello there!

I found this description of a square-rooting algorithm in 
"Mathemagics" by Arthur Benjamin and Michael Brant Shermer.  
I'll copy it more or less verbatim, and we'll both learn, because I 
think I was about eight or nine years old the last time I saw this.

              4. 3 5 8
Example:    {19.00000000000
       4^2 = 16
             -----
              3 00
    83 x 3 =  2 49
              ------
                5100
   865 x 5 =    4325
                ------
                 77500
  8708 x 8 =     69664

Step 1:

If the number of digits to the left of the decimal point is odd, then 
the first digit of the answer will be the largest number whose square 
is less than the first digit of the original number.  If the number of 
digits to the left of the decimal point is even, then the first digit of 
the answer will be the largest number whose square is less than the
first _two_ digits of the dividend.  In this case, 19 is a 2-digit 
number, so the first digit of the quotient is the largest number 
whose square is less than 19, i.e. 4.  Write that number above 
either the first digit of the dividend (if odd) or the second digit of 
the dividend (if even)

Step 2:

Subtract the square of the number in Step 1, then bring down two
more digits.

Step 3:

Double the current quotient (on top, ignoring any decimal point), and 
put a blank space in front of it.  Here 4 x 2 = 8.  Put down 8_ x _ to 
the left of the current remainder, in this case 300.

Step 4:

The next digit of the quotient will be the largest number that can 
be put in both blanks so that the resulting multiplication problem 
is less than or equal to the current remainder.  In this case, the 
number is 3, because 83 x 3 = 249, whereas 84 x 4 = 336 is too 
high.  Write this number above the second digit of the next two 
numbers; in this case the 3 would go above the second 0.  We now 
have a quotient of 4.3.

Step 5:

As long as you want more digits, subtract the product from the 
remainder (i.e. 300 - 249 = 51) and bring down the next two digits, 
in this case 51 turns into 5100, which becomes the current 
remainder.  Now repeat steps 3 and 4.

If you ever get a remainder of zero, you've got a perfect square 
on your hands.  I hope this makes sense to you.

-Ken "Dr." Math