Easy example why complex numbers are cool

I am looking for an example explainable to someone only knowing high school mathematics why complex numbers are necessary.
The best example would be possible to explain rigourously and also be clearly important in a daily day sense.

I.e. complex Laplace transform has applications in pricing of options in mathematical finance which is somewhat easy to sell as important, but impossible to explain the details of.
It is easy to say: Then we can generalise the square root! – but it is harder to argue why that makes any difference in the real world.

The question has edited the wording cool out of it replaced with a description to stop it from being opinion based. I hope it helps 🙂

Answer

Using eiθ=cosθ+isinθ it is very easy to find (and remember) many trigonometric identities.

For example, ei(α+β)=eiαeiβ gives the sine-of-sums and cosine-of-sums formulas.


ei(α+β)=eiαeiβcos(α+β)+isin(α+β)=(cosα+isinα)(cosβ+isinβ)=(cosαcosβsinαsinβ)+i(sinαcosβ+cosαsinβ)

Equating the real parts,

cos(α+β)=cosαcosβsinαsinβ

Equating the imaginary parts,

sin(α+β)=sinαcosβ+cosαsinβ

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