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When using the phrase "square root", one normally refers to the positive, real root.

But, any complex number has n nth roots, thus two square roots. Take a real number

x = x + 0*i = x * e0*i = x * e2*pi*i

The square roots are

x1/2 * e1/2*0*i = x1/2 * 1 = x1/2
and
x1/2 * e1/2*2*pi*i = x1/2 * -1 = -x1/2

Update 2002-11-12: The definition of a square root has disappeared! I suggest the following, simple definition:

The sqare root of a number x is a number whose square is x.

I am sure someone can node a much more mathematical definition than this.

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.

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