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## Main Question or Discussion Point

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

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

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arildno

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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.

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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.

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

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

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Philip Wood

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Thanks all,

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Dale

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

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WannabeNewton

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jtbell

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Just to clarify: here "wave mechanics" means specifically "quantum mechanics."However, in wave mechanics, complex numbers are an essential part of the Physics.

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Dale

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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.All we have to do iscarefullytake the real part at the end of a calculation in order to get physical quantities.

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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WannabeNewton

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Thanks for the cool example! MRIs are pretty sweet :) but I didn't mean to say that we would need to take the real part to get the physical quantity in all physics problems making use of complex numbers and such; I just meant that we would do that in order to get the physical electric and magnetic fields from the respective plane wave solutions in the above example.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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meBigGuy

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