# Applications of Complex Numbers

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.

• By applications do you mean to other math fields, or in "real life?" Don't want to assume too much since this is undergraduate research
– MT_
Commented Feb 11, 2014 at 3:15
• Real life applications. Commented Feb 11, 2014 at 3:15
• math.stackexchange.com/questions/285520/… Commented Feb 11, 2014 at 3:16
• Please avoid the tags undergraduate-research and research. We are trying to remove them. Commented Jul 24, 2016 at 19:57

## 7 Answers

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.

• Every complex equation can be formulated into real numbers only if you repeat the definition of complex numbers I suppose. Commented Jan 16, 2019 at 19:09

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.

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.

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.

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.

$i$ 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.

Complex numbers make 2D analytic geometry significantly simpler. The discovery of analytic geometry dates back to the 17th century, when René Descartes (https://en.wikipedia.org/wiki/Ren%C3%A9_Descartes) came up with the genial idea of assigning coordinates to points in the plane. Now it seems almost trivial, but this was a huge leap for mathematics: it connected two previously separate areas. Suddenly, you could do geometry by doing calculations with numbers! Getting a new point of view like this one is *huge *and it usually leads to lots of new interesting results: because now you can use a new, better language that allows you to *think *about new concepts in an easier way. Example: There were many open problems in ancient Greek geometry. Among them, Angle trisection (https://en.wikipedia.org/wiki/Trisection), Squaring the circle (https://en.wikipedia.org/wiki/Squaring_the_circle), and Doubling the cube (https://en.wikipedia.org/wiki/Doubling_the_cube). All of these are impossible when using just a compass and a straightedge. These problems were open for centuries because there is basically no way you can prove that they cannot be solved, just by thinking in terms of geometry. We needed algebra. Once we started studying the algebraic properties of geometric constructions, we discovered, for example, that all lengths constructible using a compass and a straigthedge are algebraic numbers such that the degree of their minimal polynomial is a power of 2. This, among other things, rules out the constructibility of $\sqrt[3]{2}$. Now, analytic geometry gave us a nice new tool that was easy to work with -- as long as you dealt with points and linear objects only. Checking whether two lines are parallel? Easy. Finding the intersection of two line segments? Easy. Rotating an object around a point? Possible, but painful. Once you start dealing with angles and rotations, the notation starts to be really clumsy. Part of the reason is that you have to work with each coordinate separately, and you don't really see the connections between the coordinates and the angles. This all changed once we realized that the Complex plane (https://en.wikipedia.org/wiki/Complex_plane) is isomorphic to the standard Cartesian plane. Thus, when doing analytic geometry in 2D, instead of representing a point by a pair of reals, we can represent it by a single complex number. Instant profit! Why? Because for complex numbers we have the polar form (see Complex number (https://en.wikipedia.org/wiki/Complex_numbers#Polar_form)) and we have a very good idea how they relate to angles: namely, when you multiply two complex numbers, you multiply their sizes (absolute values) and add their polar angles (arguments). When doing 2D analytic geometry using complex numbers, operations that involve angles and rotations become as simple as translations and resizing. Here's one nice example. Go ahead and try solving it without complex numbers, before reading the solution. The question is simple: what is the sum of the three angles shown in the picture?

Here's the answer: The three angles correspond to the complex numbers $1+i$, $2+i$, and $3+i$. To add those three angles together, we simply multiply those three numbers. We get: $(1+i)(2+i)(3+i) = 10i$. Hence, the sum of those three angles is precisely the right angle. Neat, right?