What's your favorite illustration of the usefulness of complex numbers?
The fundamental laws of the Universe are written in complex numbers, e.g. Schroedinger's equation: i dψ/dt = Hψ For everything in the world - every thought anyone's ever had, every object, every action - complex numbers are at the bottom in some really deep way.
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I enjoyed this article by Scott Aaronson on what the implications would be if probability amplitudes were reals or quaternions instead: https://www.scottaaronson.com/blog/?p=4021
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E.g., faster than light signalling with quaternionic QM
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That's a statement on how basic they are. "Complex" and "imaginary" are huge misnomers (unlike quaternions from "four"). They're good captivating learners but are bad at describing what is going on conceptually. They don't have to do with complexity. Maybe Duogonal numbers?
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This makes more sense
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Absolutely agreed, though I have to add that Feynman made a point to show that complex numbers were in no way necessary to qm, they just made it more convenient.
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"Complex" numbers are a basic tool. A normed division algebra over the field of real numbers. Blame Gauss for coming up with the horrible name.
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Cheeky hbar = 1 there
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Can we say something about calculations on these numbers?
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Isn't that going little bit too far though? Quantum mechanics does explain lot of stuff, but claiming that every thought that anyone's had involves complex number, too far. It's like claiming that universe evolves unitarily. You can predict, but cannot claim!
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