What do you not fully understand?

[YADD]

Originally Posted by ngtridung

2) Is there an alternative for using complex number?

My answers would be:
2) Yes, there is. If someone would have defined operator g(k, f) as the kth root of a function f and if this operator can be used in all the places where complex number ‘i’ is used, then complex number are not inevitable. The math might be more complicated though. (another operator is h(a, b) = a^b. here i = (-1)^(1/2)) is just one special case.

I beg to differ.

Complex numbers are more than simply notations for 'counting' or arithmetic manipulations. I believe that you overlooked a significant aspect of analyticity. Complex analysis uncovers a rich structure manifest in, say, the Cauchy integral theorem or the residue theorem. It is the 'internal coherence' and 'harmony' of the complex number set that bring out wonders.