# Imaginary numbers outside of math

In Algebra we are learning and using imaginary numbers. Someone asked if imaginary numbers are ever used outside of math, and our teacher said he talked to an electrical engineer who used imaginary numbers all the time. Our teacher didn't know how or why they were used in electrical engineering, just that they were. I was wondering how the imaginary numbers are used in electrical engineering and/or anywhere else other than math.

Dale
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2020 Award
Imaginary numbers come up in electrical engineering, signal processing, and control systems analysis all the time. I primarily use them for magnetic resonance imaging in for quadrature detection and the Fourier transform. Basically a signal that is a cosine is real and a signal that is a sine is imaginary.

I suppose you've been taught DeMoivre's Theorem/Formula?
ie.
$$e^{ix} = cos(x)+isin(x)$$
In engineering we deal A LOT with sinusoidal signals with phases. Since 'i' is just a phase shift of pi/2, we can represent phases as a "complex quantity". This allows us to convert all types of sinusoidal signals into exponentials rather than trig, where you need big trig identities to simplify or manipulate.

In physics and engineering, a lot of processes and theories deal with differential equations. There solutions are in the form of (real) trig equations ie. asin(x) + bcos(x). But again, these are too long and tedious to deal with so we put it complex exponential form and 'remember' that the solution is always the real part of exp(ix).

That is the very very basics of it, it off course is much more complicated and hopefully I havn't miss-explained parts of it. Someone will correct me though :)

Use it all the time... it's no joke. Seriously.

in control systems you are always working with 'i'.
wherever we have to deal with frequency, 'i' comes into picture.

also for your information, in electrical engineering most of the time, we write 'j' instead of 'i'. (i thought this would add to your curiousity) I don't know the exact reason, but may be because we use 'i' for representing current, thats why.

Well, to be precise its for the mathematical analysis that you use complex numbers. There are obviously ways to analyze without using complex numbers, but they're tedious. As n0_3sc has pointed out, Euler's Theorem expresses a direct mapping between complex exponentials and sinusoids. From that property follow some other properties of complex exponentials which form the basis for phasor analysis, frequency domain computation and (therefore) the mathematics of control system analysis. Fundamentally, Euler's Theorem is an important connection between one of the most important pair of signals in electrical engineering and complex numbers.

Another reason you will like to use the complex exponential in expressing solutions to linear differential equations is its relation to eigenvalue problems. Specifically,

$$D_{x}(e^{j\omega x}) = je^{jx}$$
$$D_{x}^{2}(e^{j\omega x}) = -\omega^{2}e^{jx}$$

Sure enough you could do this with sinusoids too (in fact thats what you're doing right now), but doing it with complex exponentials is a whole lot neater. Quite a few problems of interest in the time domain involve the solution of eigenvalue equations, where these properties are often used.

Finally, even though all this is sufficient reason to use complex exponentials and more generally complex numbers, both the Laplace and Fourier Transforms have kernels that are complex exponentials and are members of a more general class of transforms whose variants are heavily used in signal processing, control systems and lot of other allied areas of electrical engineering and mathematics.

And yes, you encounter j in EE rather than i, because historically $i(t)$ has been used to denote "instantaneous" current, as Varun points out. (http://www.perl.com/doc/manual/html/lib/Math/Complex.html differs...apparently, i has also been used to denote intensity :tongue:)

stewartcs
in control systems you are always working with 'i'.
wherever we have to deal with frequency, 'i' comes into picture.

also for your information, in electrical engineering most of the time, we write 'j' instead of 'i'. (i thought this would add to your curiousity) I don't know the exact reason, but may be because we use 'i' for representing current, thats why.

That's correct. A lower case "i" is reserved for current in EE.

CS

Lets not forget that complex numbers are just about the best way to represent resistance in AC signals. We call it impedance....where the system has both a real and imaginary resistance to a given frequency....as pointed out, its particular to the phase info and is important when looking to match up several modules for proper power matching...

Phasors, Impedance, Complex Power... list goes on.