Q: What the heck are imaginary numbers, how are they useful, and do they really exist?

Q: What the heck are imaginary numbers, how are they useful, and do they really exist?

Fine, so imaginary and complex numbers arise naturally and are extremely useful, but do they really exist? Sure, numbers (complex or otherwise) exist as concepts in our brains, but no numbers ever appear in the physical world. Imaginary numbers, like real numbers, are simply ideas without any physical existence. But it is hard to see how one would (convincingly) argue that real numbers actually exist while imaginary numbers do not. Some commentators have taken my argument above to imply that I think numbers (including imaginary numbers) actually exist (perhaps in some platonic sense).

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