Finding Square Roots
by
Erik Oosterwal
Sure, finding square roots isn't so tough, all you have to do is enter the
number and hit the square root button on the calculator. What's so
hard about that, right? The real question is how did people find square
roots before there were calculators and computers?
One way, I suppose, is to take an educated guess, then do a whole bunch of
math to see how close you are, then take another guess and do some more math
to refine your guess. For instance, we know that the square root of
40 should be somewhere between the square roots of 36 (6) and 49 (7), so
we could guess that the square root of 40 is about 6.5. If we multiply
6.5 by itself we get 42.25, so we know that's too big. Half-way between 6.5
and 6 we get 6.25, but 6.252 is only 39.0625, so that's too small.
Half-way between 6.25 and 6.5 is 6.375. 6.3752 is
40.64. We're getting closer, but we're still not there. It's
easy to see that this is going to take us a long time to find out that the
square root of 40 is 6.324555... There's got to be a better way,
right?
Right. So here it is.
Finding square roots is similar to doing long division, but there are some
subtle differences, so watch for them.
Step A. Take whatever number you're trying to find the
square root of and group it in lumps of two digits per group starting at
the decimal point and working towards your left. For instance:
891 becomes 
,
1684 becomes 
,
and 38431 becomes
In your result, one digit will go over each pair of digits, digits are 'brought
down' in pairs, and when adding 0's, they, too, will be added in pairs.
Step B. Find the greatest square that is smaller than
the left-most pair; write the root of this square as the first digit of the
result.
In this case, 4 is the largest square that is smaller than 8. We write the
number 2 in our result, and proceed to subtract 4 from the original number
to make a new 'dividend'.
Bring the 91 down and we get 491 as the new 'dividend'.
Step C. We now find a 'trial divisor' by multiplying
the existing result by 20, and write it to the left of our square root division.
In our case, we multiply the current value in the result area (2) by 20 to
get 40.
To find the next number to put in the result area, choose a digit that we
can add to the existing trial divisor and then multiply by that resulting
sum to get close to, but not over, the new 'dividend'. In our case we will
choose 9. 40+9 = 49, and 49x9 = 441.
For sake of reference, 10 would have been too big since 10+40 = 50 and 50x10
= 500, which is larger than 491.
Step D. Now add the resulting number from Step
C to the trial divisor (9 + 40) = 49. This now becomes the real divisor.
At this point we also put the number found in Step C as the
next digit in our result.
Note that in this case, the remainder (50) is larger than the divisor (49);
this happens sometimes and is not necessarily wrong.
Step E. Continue Step C and Step
D until all the pairs have been used, or until the desired accuracy
is attained.
 |
Multiply the existing result (29) by 20 to get 580.
Add 8 to 580 to get 588, then multiply 588 by 8 to get 4704.
Write down '8' in the result area.
Multiply the new result (298) by 20 to get 5960.
Add 4 to 5960 to get 5964, then multiply 5964 by 4 to get 23856.
Write down '4' in the result area. |
Feel free to try it out with your own favorite numbers...
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Copyright E. Oosterwal - 2004
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