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Would it be fair to state that most if not all of the practical applications of complex numbers stem from the relationship between $\pi$, $e$, and $i$ described in Euler's identity ($e^{i \pi} +1 = 0$)?

I've found that some of the most common practical applications of complex numbers include the study of brain waves, electrical engineering, and wireless technologies such as cell phones and radar. It appears as though all of the practical applications of complex numbers are in some way related to waves and oscillation, and therefore to $\pi$.

Clarification: I'm looking for a way I can boil down the practical/real-world applications (thus leaving out their utility in providing completeness to the fundamental theorem of algebra) of complex numbers to someone who just now for the first time was introduced to the idea that $\sqrt{-1}$ exists, and who has only a very basic understanding of trig, and no knowledge of polar coordinates or quantum theory. If the statement "most of the practical applications of complex numbers stem from the relationship between $\pi$, $e$, and $i$ described in Euler's identity ($e^{i \pi} +1 = 0$)" is a true statement, and no simpler way exists to concisely describe to practical usefulness of complex numbers, that's what I'm going to go with.

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  • $\begingroup$ I think the first sentence would be considered reductive by most people familiar with applications of complex numbers. As for the title question, that would be a duplicate of many questions, for example math.stackexchange.com/q/285520/29335 $\endgroup$
    – rschwieb
    Mar 25, 2020 at 14:27
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    $\begingroup$ Try quantum mechanics. $\endgroup$
    – justadzr
    Mar 25, 2020 at 14:29
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    $\begingroup$ One major feature of the complex numbers is its relation to rigid motions in Euclidean geometry. Another feature is its algebraic-closedness. I wouldn't emphasize the symbols $e,i,\pi$ at the cost of these other things. $\endgroup$
    – rschwieb
    Mar 25, 2020 at 14:29
  • $\begingroup$ @rschwieb I agree that there are similarities, but what i'm looking for is more specific. I'm trying to find a way to boil down complex numbers (which would seem to someone with little knowledge of advanced mathematics to be entirely un-useful) to something which, although they may not fully understand, they can still relate to. $\endgroup$
    – Elem-Teach-w-Bach-n-Math-Ed
    Mar 25, 2020 at 14:35
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    $\begingroup$ @Elem-Teach-w-Bach-n-Math-Ed IMO 2-d geometry is a LOT more relatable than $\pi$, $e$ or Euler's formula. I have heard about a "buried treasure" problem that can be solved using $i$, which I think is related here. That's fairly convincing of the practicality of solving a problem that can't be solved with the reals alone. $\endgroup$
    – rschwieb
    Mar 25, 2020 at 14:51

2 Answers 2

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This is like saying all practical uses of mathematics come from numbers. Obviously they are at the core, but there is so much more you can do with math. $\pi$ shows up in complex numbers because of their polar form. Every complex number $z$ can be written as \begin{equation*} z = r(\cos{\theta} + i\sin{\theta}) \end{equation*} which in turn can be written as \begin{equation*} z = re^{i\theta}. \end{equation*} The formula you wrote is a special case of this, when $\theta=\pi$ and $r=1$.

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Rotations aren't the only reason complex numbers are useful. They also naturally arise when two quantities are coupled. They are helpful, for example, with Hamilton's equations and 2D fluid flow.

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