about complex number

In fact there's no definition on if a complex number is positive or negative but separately, its real and imaginary parts can be said to be positive or negative.
To find out the square root of a complex number, we have to express the complex number in polar form followed by using the De Moivre's Theorem:
For an ordinary complex number a + bi, where a and b are real values, we can change it to:

圖片參考：http://i117.photobucket.com/albums/o61/billy_hywung/Maths/Polar1.jpg
where K = √(a2 + b2) and θ = tan-1 (b/a)
Then:

圖片參考：http://i117.photobucket.com/albums/o61/billy_hywung/Maths/Polar4.jpg

For this problem, there's no need to consider more on if the outcome value of K and θ should be positive or negative.
Taking the example below:

圖片參考：http://i117.photobucket.com/albums/o61/billy_hywung/Maths/Polar3.jpg

In addition, the negative of the answer, i.e. √5(0.447-0.894i) can also give a square value of -3-4i, just like when √9 = 3, -3 can also make a square value of 9 too.

First, the square root of any positive real number is not always positive
For example 4=2^2=(-2)^2
So, the root may be positive or negative
We always choose the positive one is just the convention only
The second one is that we cannot compare two complex number directly, we can only compare their absolute value (or modulus or magnitude). Specifically , for any complex number,we cannot say it is greater than 0 or less than 0. So, we cannot classify a complex number as negative or positive.
Why don't use modulus to say which one is bigger?
Consider 1+3i and 1-3i
then their modulus are both √10
then we say that they are equal? But actually they are two numbers
(Compare with real number system, if there is one x its magnitude is √10, then x should be √10)
To find the square root of a complex number in which both the real and imaginary parts are not equal to zero
Let see an example.
Consider we want to find √(1+i)
It is equivalent to say that, we want to find
a+bi such that (where a and b are real)
(a+bi)^2=1+i
a^2+2abi-b^2=1+i
(a^2-b^2)+2abi=1+i
compare real and imaginary parts
a^2-b^2=1...(1)
2ab=1...(2)
from (1)
a^2=1+b^2
from (2)
4(a^2)(b^2)=1
4(1+b^2)(b^2)=1
4b^4+4b^2-1=0
b^2=(-1+√2)/2
b=+-√[(-1+√2)/2]
a^2=(1+√2)/2
a=+-√[(1+√2)/2]
so
√(1+i)
=+-√[(1+√2)/2]+-√[(-1+√2)/2]i
The method can be generalised to other situations.