Applications of Complex Numbers

Question

For my Complex Analysis course, we are to look up applications of Complex Numbers in the real world. The semester has just started and I am still new to the complex field. I want to get a head start on my research for the course. Anything I have seen on the complex field has only been in passing from my other course like ODE, Linear Algebra, and Abstract Algebra. I was wondering if someone can lead me into the right direction about what applications of complex numbers I can look into for my research topic. Recommended books I can refer to would also help.

Thank you for your time and thanks in advanced for your feedback.

Answer 1

You'd be hard pressed to find examples of real-life quantities that are better explained by Complex numbers than the Reals, but there are plenty of real-life phenomena which, although they are observed on the real number line, can be much better explained and, despite their name, simplified through the math of complex numbers. You see this a lot in Physics.

Answer 2

Complex numbers are used quite extensively in electrical engineering. Capacitors and inductors behave as resistors with a complex magnitude. One has units like volt-amperes (reactive power), vs watts (real power).

On the other hand, when you see in relativity, $x^2+y^2+z^2+(ict)^2$ as a distance, the $i$ does not serve the same role as the complex number, since you never find anything akin to eg $2+i$. The relativity, and the electromagnetism it is found from, is actually based on quarterions, or ''four vectors'' rather than complex numbers.

Answer 3

First and foremost, complex numbers are a mathematical tool to solve tricky equations in a nice way.

In particular, they simplifly a lot if you'd like to calculate the behavior of resistors, capacitors and impedances if you want to use them with alternating current (which usually comes out of your power supply). These kind of calculations can be done without complex numbers, but would involve unwieldly sine and cosine functions. With complex numbers, one can employ $$e^{i\omega t}=\cos(\omega t)+i \sin(\omega t),$$ and things can be dealt with using complex numbers.

This link is taken from a previous question and explains how complex numbers are used in electrical engineering.

Answer 4

$i$ is a rotation operator, so $i$ (and $j$ and $k$) can be used to model rotations of a rigid body in space. Look up quaternions.

Answer 5

A good place to look for a list of applications of complex numbers would be Wikipedia's article on complex numbers (They even have a section on applications).

One of those that is worth pointing out is the use of complex numbers in Quantum Mechanics, in particular in the Schrödinger equation.

Answer 6

The Schrodinger Equation from quantum physics is most neatly written using complex numbers and the (complex-valued) field $\psi$

$$i \hbar \frac{\partial}{\partial t} \psi = \hat{H} \psi$$

It is possible to split this equation up into a few real-valued equations, but you lose some elegance in the process.