What is the confusion surrounding complex numbers and their representation?

[Muhammad Ali]

I am familiar with complex nos. I know about their algebric manipulations, for example. But I could not understand the notion of ´iota´. I know that complex numbers are extremely essential for solving the equation involving cube roots or higher or negative square root of a number. I am also aware of their use in Quantum Physics and I, therefore, know about the importance of ´iota´. But I am unable to understand their working.
It appears to me that we have assume a greek letter for say finding the negative squareroot of number and thus everything is ok with it. If, for example, this is the way of finding the solutions then why don´t we take any real no divided by zero as equal to another Greek letter, say, omega.
Secondly, I am confused with the graphical representation of the complex numbers. We represent them using Argand Diagram and whenever I compare the Argand Diagram to the two dimensional real plane I don't find any difference. What is this nonsense?
Finally, Do we have Argand diagram in more than two dimension?
I am very irritated with these questions please help me out.

 

[matt grime]

If, for example, this is the way of finding the solutions then why don´t we take any real no divided by zero as equal to another Greek letter, say, omega.

who says that we don't? There are many places where do precisely this (as long as we are not dividing zero by zero).

Secondly, I am confused with the graphical representation of the complex numbers. We represent them using Argand Diagram and whenever I compare the Argand Diagram to the two dimensional real plane I don't find any difference. What is this nonsense?

this is a good observation, apart from the last sentence. The argand plane and the real plane are obviously different in one respect - one has axes labelled Re, and Im, and the other doesn't. Of course the complex numbers are in bijection with the real plane, which is what you're noticing. But pairs of real numbers (a,b) do not have any nice arithmetic defined on them a priori. For instance definin (a,b)*(c,d)=(ac,bd) is bad because you now have non-zero numbers that multiply to zero. The operation (a,b)*(c,d)=(ac-bd,ad+bc) is a good operation, and is precisely how one can define the complex numbers without mentioning i (and it is i, not iota. i has a dot above it, iota doesn't), but you'd only come to this after defining C=R.Note also that C is isomoprhic to R[x]/(x^2 +1), as well. this is the polynomials over R modulo the poly x^2 +1.

Finally, Do we have Argand diagram in more than two dimension?
I am very irritated with these questions please help me out.

there are several intepretations of this. I will go with: we have the quartenions (R^4 with a multiplication) but they are non-commutative. We have octonions, and sedonions (R^8 and R^16) but they are even more badly behaved. Apart from that any other multiplication operation will introduce zero divisors.

http://www.maths.bris.ac.uk/~maxmg/maths/introductory/complex.html

is something that explains all this in more detail