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Square Root by Hand

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Brian M. Scott

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Jun 24, 1997, 3:00:00 AM6/24/97
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On Tue, 24 Jun 97 08:18:11 GMT, rghu...@lava.net (Robert G.. Hughes)
wrote:

>Does any one have a reference or know (Remember is probably more appropriate)
>how to take the squaree root of a number using just a pencil and paper. I
>remember something that looked a division problem but had something about
>doubling some of the terms.

This question came up a few months ago; here's what I wrote then.

It's not so much a formula as a method. In fact, the method is
somewhat similar to long division. It's easiest to explain with an
example; I'll calculate sqrt(1123.4) to two decimal places.

1. Starting at the decimal point and working both left and right,
divide the number into pairs of digits: 11 23 . 40 00 Each pair will
produce one digit of the square root, so I've added the extra 00 to
get two places to the right of the decimal point.

2. Find the largest integer whose square will 'fit' into the first
block. In this case the block is 11, and the integer is 3. Write the
3 above the 11 (as if you were writing a quotient in a long division),
write 3^2, or 9, under the 11, and subtract. Then bring down the next
2-digit block. If you use a fixed-pitch font, the result will look
something like this:

3
sqrt(11 23.40 00)
9
-----
2 23

3. Take the partial square root, which is 3, double it, and append a
0 to get 60. Estimate how many times 60 will go into 223: 3. Add
that estimate to the 60 (to get 63), and multiply by the estimate:
63 x 3 = 189. Since this is no more than 223, we're in business. If
it were bigger, you'd have to decrease the estimate and try again.
(Make sure that you use the largest digit that will work, however.)
This estimate, 3, is the next digit of the quotient, and the 189 is
the amount to be subtracted from 223 in preparation for the next step.

3 3
sqrt(11 23.40 00)
9
-----
2 23
1 89
---------
34 40

4. Again bring down the next block. Double the partial square root
and append a 0 to get 660. This certainly won't fit more than 5 times
into 3440; and 665 x 5 = 3325, which is still small enough, so the
next digit of the square root is 5.

3 3. 5
sqrt(11 23.40 00)
9
-----
2 23
1 89
--------
34 40
33 35
-----
1 05 00

5. Do it all again: 2 x 335 = 670, and I attach a 0 to get 6700.
This goes at most once into 10500, and 6701 x 1 = 6701, so at the end
of the next stage my calculation looks like this:

3 3. 5 1
sqrt(11 23.40 00)
9
-----
2 23
1 89
--------
34 40
33 35
-----
1 05 00
67 01
-------
37 99

6. If I were going to continue, I'd bring down the next 00 block, and
I'd take 67020 as the next trial 'divisor'. The next digit of the
square root would therefore be 5 (67025 x 5 = 342125, and 67026 x 6 =
402156 is too large), so for 2-place precision I round to 33.52.

7. Ultimately the method is based on the formula
(a + b)^2 = a^2 + 2ab + b^2. In the first step we took a to be 30 and
b to be whatever was left over: 1123.4 = 30^2 + 2(30)b + b^2. After
subtracting 30^2, or 900, we had 223.4 left, and it has to be
60b + b^2, or (60 + b)b. At the next stage we estimated b to be 3, so
that (60 + b)b = 189. That's an underestimate: we're still short by
34.4, so we continue. We now think of 33 as our a, so that a^2 =
1089, and the missing 34.4 has to be 2(33)b + b^2, or (66 + b)b. If
we try b=.2, we find that it's too big, so we take b=.1. And so on.
It's just successive approximations.

Brian M. Scott

Jerry Beeler

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Jun 29, 1997, 3:00:00 AM6/29/97
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Robert G.. Hughes wrote:
>
> Does any one have a reference or know (Remember is probably more appropriate)

By example, let's take SQRT(75):

STEP 1: 75 is between 64 and 81; therefore SQRT(75) is between SQRT(64)
and SQRT(81) ..

64 < 75 < 81
8 < SQRT(75) < 9

STEP 2: 75 is closer to 81 than to 64, and so 9 is a closer
approximation of SQRT(75). Therefore, divide 75 by 9. divide until the
quotient has one more decimal place than the divisor ...

(75.0)/9 = 8.3

STEP 3: Find the average of 9 and 8.3. the average should have the
same number of decimal places as the quotient ..

(9+8.3)/2 = 8.7

STEP 4: Repeat steps 2 and 3 using the average 8.7 as the divisor ..

(77.000)/8.7=8.62 (8.7+8.62)/2 = 8.66

Hence, SQRT(75) is approximately equal to 8.66. Each successive
repetition yields a number closer to the irrational number.

This is the only "by hand" method that I am aware of and teach - I'm
sure there are others and would be interested in any other (hopefully
easier) solutions.

Jerry

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