# Why are waves represented as complex numbers?

Superposed_Cat
Why do we represent waves as complex numbers? Why wont real suffice? Thanks for any help.

## Answers and Replies

Homework Helper
Gold Member
Dearly Missed
We COULD represent them only with reals.
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However, by using the fact that the complex exponential is, simply, an expression involving the trigs, using the complex exponential is perfectly valid, but ALSO allows us the particularly simple properties of the exponential when dealing with trig. functions.

Astrum
The equations representing waves generally come from solving second order PDEs/ODEs, and hence can be represented using complex numbers because of Euler's formula.

$$e^{i\theta} = \cos(\theta) + i\sin(\theta)$$

I'm sure somebody else can give a more enlightening answer, but that's the way that I understand it.

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Gold Member
That's it: mathematical ease. However, in wave mechanics, complex numbers are an essential part of the Physics.

Superposed_Cat
Thanks all,

Mentor
2021 Award
Why do we represent waves as complex numbers? Why wont real suffice? Thanks for any help.
There is not really any difference between a pair of real numbers (with a given relationship) and a single complex number. In the frequency domain you need amplitude and phase, so that is two numbers and so complex numbers is a reasonable mathematical representation.

We can for example represent any solution to the source-free Maxwell equations (which decouple into two wave equations, one for the magnetic field and one for the electric field) as the real part of a Fourier transform which is, roughly speaking, a continuous linear superposition of plane waves of the form ##\vec{E} = \vec{E}_0 e^{i(\vec{k}\cdot \vec{r} - \omega t)}## which is extremely useful because we can restrict ourselves to analyzing plane waves and then build any other vacuum solution via a Fourier transform. All we have to do is carefully take the real part at the end of a calculation in order to get physical quantities.

Mentor
However, in wave mechanics, complex numbers are an essential part of the Physics.

Just to clarify: here "wave mechanics" means specifically "quantum mechanics."

Mentor
2021 Award
All we have to do is carefully take the real part at the end of a calculation in order to get physical quantities.
But there are cases where the physical quantity inherently requires two real numbers to represent it and those two real numbers are related in such a way that representing them as a single complex number is reasonably, both mathematically and physically.

For example, in MRI you detect the amount of magnetization in the plane transverse to the main magnetic field. There is a strength of the magnetization and a direction, requiring two real numbers to describe. In such cases, the physical quantity of interest is actually a complex number.

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