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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..?

sherlockz
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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.

Zheng Liu
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  • Would be there any physical significance for that..? – sherlockz Oct 05 '14 at 05:09
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    +1 "Don't read engineering physics textbooks if you want to understand the logic behind it" good advice indeed. Also, the field formed with your matrix is isomorphic to $(\mathbb{C},+,\times)$, so your statement "in some sense, equivalent" could be misleading: you need to say it stronger. You might be interested in my own answer to a very like here as well as this one here – Selene Routley Oct 05 '14 at 05:44
  • @WetSavannaAnimal aka Rod Vance Thanks for showing me that answer. I should have read your answer first. Your answer is given in a more precise and explicit way. By saying "in some sense equivalent", things in my mind was that the field I mentioned was isomorphic to the complex field in the sense of Lie group isomorphism, but I didn't learn enough physics and not sure whether this Lie group isomorphism is enough to say that they work well for physics. – Zheng Liu Oct 05 '14 at 07:34
  • @ZhengLiu Your answer is a good one: it's just that in most cases we would say that the matrix field and $\mathbb{C}$ are exactly the same. – Selene Routley Oct 05 '14 at 07:43
  • @Abhishek K Physics chooses the suitable math objects to describe the world. There is no physical significance for complex number field itself. We use it because its math structure is nice describing the nature. After all, real numbers do not have more physical significance than complex numbers. – Zheng Liu Oct 05 '14 at 07:47
  • @AbhishekK A similar example occurring to my mind( though it may not be a best example) is the definition of temperature. Why is temperature a real number? Well, one can say that 0th and 2nd laws indicate temperature has properties parallel to reals. However, one can possibly build the entire theory on rational number set only. moreover, when one extends it to statistical mechanics, one faces negative temperature objects, which is "hotter" than any positive temperature objects. If one has a critical math mind, one can say that real number may not be the best language to describe temperature. – Zheng Liu Oct 05 '14 at 08:01