Complex theory in physics I'm a physics graduate and usually encounter with complex numbers in physics.
For example, in electrical engineering,
Why do we express capacitive reactance as an imaginary number..?
Can't it be expressed as real, as we generally do?
Forgive me, if my question is logically incorrect, But I do want to know why "i" is involved..?
 A: For your engineering problems, the complex numbers are usually introduced to simplify the problem solving. Just take the complex reactance as an example. If you trace back to the place where you introduce the complex numbers, you will find that it is when you solve the "forced oscillation" equations$$\frac{d^{2}q}{dt^2}+2\beta\frac{dq}{dt}+\omega^2q=F_0cos\omega t$$Indeed, this equation can be solved purely using real numbers. However, it saves time by converting it into a complex equation, solving it in the complex field and then converting the solution back to the real number field (check any classical mechanics book for a detailed discussion. Don't read engineering physics textbooks if you want to understand the logic behind it.)
There are fields of physics in which complex numbers are widely "believed to be intrinsic" rather than just a problem-solving short-cut. Indeed, complex number field and real number field have different algebraic structures and sometimes phenomena are better described using the complex algebra. However, even this does not mean that complex numbers are the only choice. For example, one can work with a 2-dimensional real space equiped with a structure matrix $$\left( \begin{matrix} 0 & -1\\ 1 & 0 \end{matrix} \right)$$This space is, in some sense, equivalent to the complex field and both work well for describing the nature.
Physics can go even beyond complex numbers. For example, Grassmann numbers are used in describing spinor fields, accounting for the anti-commutativity. On the other hand, one is free to work with a set of matrices possessing the same anti-commutative property.
In my opinion, it is not the complex numbers themselves that are important; rather, it is the algebraic properties they possesses which makes them the right language to describe the nature. If you learn some serious algebra, you will have many substitutes to the complex numbers, but complex field is definitely easy to compute.
