Yesterday, my son asked me of what use was the fact that $e^{i \pi}=-1$. I told him that he could use it to impress a girl, which is true but probably an incomplete response. So, I ask you, is there any practical application of the fact that $e^{i \pi}=-1$?
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What's really useful is Euler's formula connecting the complex exponential to the trigonometric functions. That allows for uniform treatments of linear differential equations. It's also surprisingly crucial in quantum mechanics, where you need spaces of complex valued functions. The identity at $z=-1$ is a consequence.
If your son is old/advanced enough to understand where the identity comes from those examples might convince him that the complex exponential is useful.
You might want to check out William Dunham's Euler: The Master of Us All
Ethan Bolker
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@MichaelStern While we're advertising Durnham / Euler, you and your son might enjoy the following entertaining 55 minute video: https://www.youtube.com/watch?v=fEWj93XjON0 – peter a g Nov 20 '17 at 13:52
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I have one suggestions about it. The fist, for example, is for give a answer the following question: the number $e^{\pi}$ is algebraic, i.e., there exists a polynomial with integers coefficients such that $e^{\pi}$ is a root?
The answer is not, because $e^{\pi.i}=-1$ implies that
$$(e^{\pi.i})^{-i}=(-1)^{-i} \Rightarrow e^{\pi.(-i^2)}=(-1)^{-i} \Rightarrow e^{\pi}=(-1)^{-i}$$
But, by Gelfand-Schneider Theorem, (see https://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_theorem) this number is not algebraic. By other hand, the problem if $\pi^e$ is algebraic or transcendent is still a open problem in Mathematics.
Cgomes
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It's used primarily in the identity $e^{i\theta} = \cos\theta + i \sin\theta,$ which means the theory of Fourier series and Fourier transforms can exist.