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Recommendations for Introductions to Complex Numbers

On this page you can find some resources that introduce you to complex numbers. The main focus is on the application of complex numbers in electronics (for the analysis of AC circuits), but I hope that some resources will also be useful for students of other engineering disciplines or physics.

One minor detail, but potential source of confusion, is the symbol that is used. Mathematicians (and physicists) use, for obvious reasons, the symbol "i" for the imaginary part. But in electronics this symbol has always been used for the current. So in electronics the symbol "j" is used instead.

The first resource is only useful for electronics: In the chapter on complex numbers in allaboutcircuits there is a good explanation why something more complicated than a real number is needed to represent voltages and currents in AC circuits. The concept of the phase of a voltage or current is very well explained. However, the explanation of complex numbers is not so good: The crucial property that j * j = -1 is not mentioned at all, in fact complex numbers are treated as if the real and the imaginery part were just 2 components of a vector. Nevertheless, if you are new to AC circuits, you will find the discussion of "phase" useful, just make sure that afterwards you have a look at one of the following resources that include better explanations of complex numbers.

One such resource is Pauls's Online Math Notes - Complex Numbers by Paul Dawkins from Lamar University, a very nice introduction, no matter whether you need complex numbers for electronics, physics or any other subject.

Very short and exactly what you need to know about complex numbers if you want to analyse AC circuits is the Crash Course in Complex Numbers by Tom Mobley from Collin College.

If you need to learn about complex numbers because you are interested in physics, you should have a look at some books that sum up all areas of mathematics that are most important for physics. One such book that you can get for free and that is very good is Mathematical Tools for Physics by James Nearing, University of Miami. Chapter 3 is on Complex Algebra. It does not mention AC circuit analysis (electrical circuits are covered in the chapter on differential equations, section 4.8), but if you are studying EE, you may find the book very interesting in general: It includes good explanations of some very useful mathematical techniques, together with many problems.

Slightly easier than Nearing is the very popular Mathematical Methods in the Physical Sciences by Mary L. Boas. Complex numbers are covered in chapter 2, p.46-81.

To practice the handling of complex numbers, you should play with them using your calculator or favourite math program. A place where you can do so for free is Wolfram Alpha.

If you have a question about complex numbers, a very good place to ask is Physics Forums. If you have difficulties solving a problem, you should ask in the Homework - Calculus and Beyond subforum. If there is something you do not understand about complex numbers in general, ask in the General Math subforum. But before you register and ask your question, reading some info about Physics Forums may be useful.

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