Square Roots: Estimate, Divide, Average

Date: 02/04/97 at 12:25:05
From: Dorothy L. Mason-Schweitzer & Dan Schweitzer
Subject: Square roots

Hello!

When I went to school about umpteen zillion years ago (pardon the
inexact expression, but it was back some time before the pyramids), I
recall a longhand-division-and-guess method used to determine square 
roots, at least until we ran out of papyrus. Nowadays, the only answer 
I can seem to find is, "Well, you hit this button on the
calculator..."; not quite what I'm looking for. Even math teachers 
only seem to know which buttons to push, including an engineering math
teacher in a technical school!

Do you know this method, or perhaps know some mathematically inclined
archaeologist I could ask?

Thank you much!

Dan Schweitzer

Date: 02/06/97 at 10:50:45
From: Doctor Mitteldorf
Subject: Re: Square roots

Dear Dan,

If you need to find a square root and have only a paper and pencil, 
the easiest way is probably to Estimate, Divide and Average.

Here's how it works.  Suppose you want to find sqrt(10).  Estimate 
that the answer is about 3.  Divide 3 into 10 to get 3.333333.  
Average 3 with 3.33333 to get 3.16666667.

To get more and more accurate, repeat the process, using your last 
answer as your next estimate: Divide 3.166667 into 10 to get 3.157894.
Average that number with 3.1666667 to get 3.162280.  The actual answer
is 3.162277, so you're already very close.  If you go through another
round of this procedure, you should have 5 more correct decimal 
places.

The reason this works is that if y equals sqrt(x), then x/y=y; but
if y is a little less than sqrt(x), then x/y will always be a little 
more than sqrt(x), and vice versa.

----------------------------------------------------------------------

There's another way that's a little harder to understand and to 
remember.  It's based on the formula (x+a)^2 = x^2 + 2ax + a^2.  Use 
the formula with x being the part of the number you've got so far and
a as the new digit.  We start by writing the square root out like a 
long division problem, but with no divisor.  

For an example, I'm going to write 
      ________
     /10.00000

because I can't e-mail the right symbol for square root.

Just as when you do long division, since you get each decimal place by 
making an estimate, and sometimes your estimate turns out wrong and 
you need to erase it and try a bigger or smaller one, you have the 
same problem in doing square roots in this method.  The problem is 
worse, however, because the estimate you're making is more complex 
than the one in long division.

Here's an example of how to do the square root of 10.  Try 3 as your 
first guess.

      _3______
     /10.00000

Then square the 3 and subtract.  Carry down TWO more digits from the 
top line:

      _3._____
     /10.00000
       9
      -----
       1.00 

Guess the next digit.  This is the hard part.  We guess "1" because 
we know it's the right answer; in practice, you often will guess high 
or low, go through the next step, and guess better the next time.  

For the number you've got so far, write the "3" as "30" by tacking on 
one zero.  Multiply the new digit by TWICE the number you've got so 
far (twice 30).  1*2*30=60.  Also, subtract the square of the digit 
(1^2=1).  

Your next step looks like this 

      _3.1____
     /10.00000
       9
      -----
       1.00 
        -60 
       -----
         40
         -1
       -----
         3900

When you "guessed" the 1, you had to have this whole process in mind.  
If you'd guessed too high, then the numbers you'd be subtracting 
would be too large, and your "remainder" would have been <0;  this 
part is just like long division.  If you guessed too low, the way you 
would know is that the number on the bottom line would have one digit 
too many (in this case, it would be 10000.

I've brought down two more digits (zeros) from the top line.  I'll 
show you one more step.  The next digit is 6 - found by guessing, or 
trial and error.  6*2*310=3720, and 6^2=36:

      _3.1____
     /10.00000
       9
      -----
       1.00 
        -60
       -----
         40
         -1
       -----
         3900
        -3720
        -----
          180
          -36
        -----
          144
  
Hope that's enough for you to get the idea. 

-Doctor Mitteldorf,  The Math Forum
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