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Q&A #3880

Teachers' Lounge Discussion: Square Root

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From: Loyd <loydlin@aol.com>
To: Teacher2Teacher Public Discussion
Date: 2002030516:16:00
Subject: Re: Square Root; Old Fashioned Way . Calculator Anyone?

Square root was taught to all students in the 30's and 40's. New York
State requires, or did in the 60's, to find the square root of numbers
between 1 and 100 to the nearest tenth.  Here is how you do it the
long old fashioned way:

Starting at the decimal, group digits in groups of two toward the left
for integers and toward the right for decimal digits. Notes for each
line will be printed below the example for the square root of 7878. 
1.)            __8___8_._7__5_
2.)            /78  78 .00 00
                64
             -----------
 3.)     160 /  14  78
 4.)     168/   13  44                  
            -----------------            
5.)    1760/     1  34  00
6.)   1767/      1  23  69
               ----------------
7.)    17740/       10  31  00
8.)   17745/         8  87  25
               -------------------
repeat.

Take note that the actual divisor is always the trial divisor plus the
digit that will go into the dividend.  The trial divisor is alway
found by multiplying the quotient by 2 and by 10 1.e., 20.  The ten
adjustS the place value and the 2 comes about because the 2 in the
square of a binomial (X^2 + 2XY + Y^2)=(X+Y)(X+Y)=(80 + 8.75)(80 +
8.75)

Many algebra books (or they used to)show the reasoning behind this
algorithm.  
1.)  Line one is the quotient to 2 decimal places.
2.)  The dividend is divided in groups of two starting at the decimal
and going left for the whole numbers and going right for the decimal
digits.  
3.) Find the biggest square in the first group, 78, which is 64.
Subtract and bring down next group
4.) Trial divisor is 8 x 20 or 160 which goes into 1478 about nine 
times.  Add 9 to 160 = 169 and find the product 9x169.  If product is
greater than 1478 which it is, the actual divisor will be 168.  168 is
the actual divisor and 8x168=1344.
5.) Subtract 1344 from 1478 and find the trial divisor which is
20x88=1760. 
6.) Divide 1760 into 13400 which gives 7 so try 1767 as actual divisor
by seeing if 7x1767 is less than 13400.  It is and equals 12369.
7.)  The next trial divisor is 887x20=17740. 
8.)  The actual divisor turns out to be 17745 so we multiply  17745x5
abd get 88725.  Subtract and continue with the process.  The next
trial divisor will be 8875x20.  

An easier method may be to divide the number you want to find the
square root of by any estimate of your choosing.  Average that number
with the quotient and use the average as the next estimate.  Keep
repeating the process and very soon the quotient will be the same as
your estimated divisor.       

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