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Square Roots by Hand

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					                        Square Roots by Hand
  Instructions:                                                            Example:
1 Count the number of digits your number has that are on the left          28.6, breaks up into
  side of the decimal point. If the number of digits is odd, then          "28" "60" "00".
  add a "0" to the front of your number. Obviously that doesn't            However, 9.1 breaks
  change your number's value, but it's important because now you           up as "09" "10"
  need to break the number into groups of two digits, and it's             "00".
  important that you break your number up on the proper
  boundries.
2                                                                          The first group is 28,
  Look at the first group of two digits as its own number. Find the
                                                                           which gives us 5
  largest number whose square is equal to or less than that. This is
                                                                           since 52 = 25, which
  the first digit of the exact square root.
                                                                           is less than 28.
3 Square the first digit of the square root and subtract the result        28 - 25 = 3
   from the first group of two digits.
4 Now append the next group of two digits to the result of step 3          "3" + "60" yields
   to obtain the partial dividend.                                         "360"
5 Now multiply what you have of the square root so far by 2, and           5 x 2 = 10
   discard any decimal points. This is the partial divisor.
6 Divide the partial divisor into the partial dividend and take the        360/10 = 36, which
   first digit of the result. This is the next digit of the square root.   yields "3"
   Since we worked with the first digit after the decimal in our
   original number (the 6 in 28.6) to figure out the new digit, we
   put a decimal in front of the new digit. So far, our square root
   is:                                                                     5.3
7 Take the new digit from step 6 and append it to the partial              "10" + "3" yields
   divisor from step five.                                                 103
8 Multiply the result by the new digit.                                    103 x 3 = 309
9 Subtract that from the old partial dividend.                             360 - 309 = 51
10 Append the next group of two digits (00) to the result of step 9        "51" + "00" yields
   to get a new partial dividend.                                          5100
   At this point the algorithm starts repeating steps:
11 Repeat step 5: Multiply what you have of the square root by
                                                                           53 x 2 = 106
   two and discard decimal points, to get a new partial divisor.
12 Repeat step 6: Divide the partial divisor into the partial divided      5100/106 = 48.11...,
   and take the first digit of the result.                                 which yields 4
   Tack the new digit onto the end of the square root, to get:             5.34
13 For more digits of accuracy, just repeat steps 7 through 10, and
   then loop back to step 5 as many times as you want. Add more
   groups of "00" as necessary.
As if all that weren't enough, there is one thing to watch out for. In fact, if you
tried to follow that set of instructions with some other number than 28.6, you may
well have found out what that is: The result of step 9 may be a negative number!
"So what?", you may ask. Well, since the results of step 9 are the basis for the
next partial dividend, getting a negative number means having a negative partial
dividend. And that, when you get to step 12 (or the repeat of step 6), means that
the next digit of the square root will be negative! "Goodness!" you exclaim, "what
on earth does it mean to have a negative digit in a number?"

Fortunately, the answer to that question is "nothing". If you think back to when
you were in the first or second grade when your math teacher explained the
decimal notation system to you, you'll remember that decimal notation is a (very
convenient) shorthand for a sum. For example, 127.6 is 1 in the 100's place, plus
2 in the 10's place, plus 7 in the one's place. Or, 1x100 + 2x10 + 7x1 + 6x.1. The
digits are simply shorthand for various products that you add together. So with
that in mind, we see that having a negative digit is no big deal (aside from being
awkward to write). It just means you have to do a little math in order to come up
with a "normal" number that has all positive digits. For example, what if instead of
127.6, the 2 was negative? That would be 1x100 + (-2)x10 + 7x1 + 6x.1, which is
100 - 20 + 7 + .6, or 87.6

So if the result of step 9 is a negative number, don't worry about it. Just proceed
as normal. Let's look at an example that has this phenomenon (I'll just show the
steps here, I won't repeat the descriptions of them):

The square root of 77.15 is:

   Step:                                   Result:
0 sqrt(77.15) =                            ?.???
1 77.15 yields "77" "15" "00"
2 82 = 64, and 64 < 77                     8.???
3 77 - 64 = 13
4 "13" + "15" = "1315"
5 8x2 = 16
6 1315/16 yields 8                         8.8?? (= 8 + 8x10-1)
7 "16" + "8" = "168"
8 168x8 = 1344
9 1315 - 1344 = -29
10 "-29" + "00" = "-2900"
11 88x2 = 176
12 -2900/176 yields -1                     8.79 (= 8 + 8x10-1 + (-1)x10-2)
13 "176" + "-1" = 1761
   The derivation of that 1761 is actually a little tricky. Remember that you have to
   append the new digit to the partial divisor, then multiply the result by the new digit.
   Appending a negative digit is a different operation than tacking a new digit on to the
   end of your result, because now you're really performing the calculation (-(176x10 +
   1)), which you then multiply by -1 to get 1761. The minus signs nicely cancel out,
   which is good.
   So what you really do is append the absolute value of the new digit to the partial
   divisor, and make the result the same sign as the new digit. Then multiply by the new
   digit as normal. You will always get a positive result from this.
14 -2900 - 1761 = -4661
15 "-4661" + "00" = "-466100"
16 879x2 = 1758
17 -466100/1758 yields -2                   8.788 (= 8 + 8x10-1 + (-1)x10-2 + (-2)x10-3)

And indeed, that's not a bad answer since 8.7882 = 77.228944, which is only off
by .078944. Not bad at all, for only three cycles through the algorithm.

You may wonder if this business with negative digits ever stops. The answer is
no. The reason is because of the nature of the algorithm. This is a successive
approximation algorithm, which means that each step gets you a little closer to
the correct answer. So if your initial approximation is too small, the algorithm will
have to add a small positive amount at each step in order to make progress
towards the correct answer. But if the initial approximation is too large, the
algorithm has no choice but to take small amounts away from the approximation
at each step. Otherwise, the approximation would just get worse.

But how does it get like this? Notice that steps 1 through 4, are different, and
they're not repeated. They are engineered to always give you a positive first digit
that is an under-approximation of the true answer. This is not difficult to see; step
2 tells you straight away to find a number whose square is less than the first
group of digits. Therefore, that number's square root is less than the square root
of the first group of digits. It is guaranteed to be the correct first digit because it is
the square root of the largest perfect square that's less than the first group of
digits, but nonetheless as a value in itself it is less than the true answer.

Since it is easy to figure out that the first digit of the answer is an under-
approximation of the true answer, it is also pretty easy to see that the second
digit of the answer will be positive. The second digit has to be positive in order to
bring the approximation closer to the true answer. However, it is just part of the
nature of the algorithm that the second digit can "overshoot" the true answer.
When that happens, the approximation changes from an under-approximation to
an over-approximation, and so all your remaining digits will be negative. I leave it
to cleverer minds than my own to figure out why it is possible for this second digit
to overshoot the true answer (and hopefully to fix the algorithm).
Finally, one last note. In the event that you choose to find the square root of a
number that has a rational square root (such as 17.64, whose square root is
exactly 4.2), you will find that at some instance of step 9 the subtraction comes
out to exactly zero and there are no more groups of digits left to deal with that
aren't "00" themselves. If that happens, you know you're done, and you have
found the exact square root you're looking for (not merely an approximation of it).
This is easy to prove; if you assume that the result of step 9 is exactly 0, and you
follow that zero through the calculation, you'll see that all other digits produced by
the algorithm will also be 0.

				
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posted:10/16/2012
language:English
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Prateek Bhuwania Prateek Bhuwania
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