Sorry for a long story, which only address the title of your question (and not the inner questions).
I remember the first time I was introduced to complex numbers at school. The teacher (of mathematics, not physics) was explaining us how to solve quadratic equations (
a.x^2+b.x+c=0). After giving us the method, he ended up with the well known solution for the roots:
Of course it didn't take long for a bright student to tell the teacher: "Hey, but then what happen if the expression in the square root is negative?"
For example solve
x^2+1=0, your roots will be:
All (or most of) the class understood the conundrum and started to scratch their head as they knew for sure that no number could be squared and retain a negative sign ...
The teacher continued, completely undisturbed, "It's not a problem, we can make tools for that. Let's just use a quantity
i defined such as
i^2=-1". And he went on to introduce the complex numbers and the rules in the complex plane.
Again it wasn't long before a voice from the baffled audience shouted "So this is a actually a convoluted way to bypass rules you taught us previsouly (like a squared number will always be positive). What use is that? why go to such complexity ? (no pun intended, although I now wonder how the complex numbers got their name from initially).
So the teacher put it this way:
There are many physics equations which follows a quadratic law, or
even more complicated laws where the solutions involved square roots
of potentially negative numbers, and (before the Complex numbers) the
physicians couldn't solve their system fully so they asked
mathematicians to define a new domain (larger than the
domain) where these systems would be solvable. The Complex numbers are
the tool mathematicians came up with.
By now my understanding of the complex numbers is a bit deeper, but this simple description still holds true. The Complex numbers are just a mathematical tool. A complex number do not have other physical equivalent than the one you give to them.
Same can be said about the
Real numbers. I work with a multi sensor tool which measure 10 different parameters in parallel. The output for anyone is just a list of numbers, only myself knows that:
- the first number represents a Weight, in [N],
- the second is a Moment, in [N.m]
- the third is an acceleration, in [G]
- and so on ...
All different physical dimensions, yet on my screen they're all just numbers, only in my head do I know this one represent this, this one represent that...
For complex numbers, you have 2 components. Each may represent a different physical dimension (electric field and magnetic field for EM). The
i part is only the mathematical tool allowing you to handle these numbers in a more graceful form (because you could also describe each components separately with real numbers only, but the equations become real ugly). The
i in itself means nothing physically.