hand calculating square roots
suppose one wants to
calculate the square root of a
number (let's say 7684324) with nothing else but paper and pencil (a little 4
operations calculator is
recommended, but not necessary), much the same way we used to do
long division in third grade.
to start our
quest, divide the
digits in
pairs (always in pairs) starting from the right:
7'68'43'24 | *result*
------------
|
| *scratch area*
|
the first digit we can do
mentally, we need a number that multiplied by itself gives the closest to 7 without actually passing it. the number is 2 since 2*2=4 (if we chose 3 we would have that 3*3=9 which is greater than 7). Put the result on the top right box and subtract it's
square from 7:
7'68'43'24 | 2
-4 ------------
-- | 2
3 | *2
| --
| 4
"lower" the next pair of digits to proceed:
7'68'43'24 | 2
-4 ------------
-- | 2
3 68 | *2
| --
| 4
now pay
attention for this is the crucial part.
Multiply the result we got in the previous step (2) by 2 (we always multiply by 2, we are
not multiplying the
result by
itself here). We need the a
digit (x) in the form:
2*2=4
4x
* x
----
(read as fourty-x times x equals - note that 4x is actually 4*10+x)
again, we need the result to be the
closest to 368 without actually passing it. we find that x=7 is what we're looking for:
47*7=329
7 is then the second
digit of our
final answer:
7'68'43'24 | 27
-4 ------------
-- | 2 47
3 68 |* 2 * 7
| -- ---
| 4 329
once again subtract the result from 368 and lower the next pair of
digits:
7'68'43'24 | 27
-4 ------------
-- | 2 47
3 68 |* 2 * 7
-3 29 | -- ---
----- | 4 329
39 43 |
repeating once again, multiply 27 by 2 (54) and find x so that five-hundred-forty-x times x is closest but not passing 3943. 7 is once again our number:
547
* 7
----
3829
7 is our third digit, subtract 3829 from 8943 and lower the next (last) pair of digits:
7'68'43'24 | 277
-4 --------------
-- | 2 47 547
3 68 |* 2 * 7 * 7
-3 29 | -- --- ----
----- | 4 329 3829
39 43 |
-38 29 |
------ |
1 14 24 |
multiply 277 by 2 (554), find x so that 554x * x is closest but smaller than 11424 (as you begin to see, the
greater the
precision you need, the "bigger" the calculations get). x=2 is the number we are searching for:
5542
* 2
-----
11084
we then have:
7'68'43'24 | 2772
-4 --------------
-- | 2 47 547
3 68 |* 2 * 7 * 7
-3 29 | -- --- ----
----- | 4 329 3829
39 43 |
-38 29 | 5542
------ | * 2
1 14 24 | -----
-1 10 84 | 11084
-------- |
3 40 |
this is our
answer cut at the decimal point. if we wanted decimal places we could continue to our
heart's
desire by adding
zeros (in pairs):
7'68'43'24.00'00... | 2772.
-4 --------------
-- | 2 47 547
3 68 |* 2 * 7 * 7
-3 29 | -- --- ----
----- | 4 329 3829
39 43 |
-38 29 | 5542
------ | * 2
1 14 24 | -----
-1 10 84 | 11084
-------- |
3 40 |
the actual result we were searching for is:
sqrt(7784324)=2772.06132688293893019385871363441
as can see, our
result is
correct (cut at the decimal point).
check
this search for other pages on the
subject.