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> But “imaginary” numbers are no more or less legit than, say, transcendental numbers.

I disagree? The ratio of a circumference to a diameter is clearly a positive number between 3 and 4, and it sure feels more legit than something whose square is a negative number.

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Well it's not that the square is a negative number but rather that you were only taught to do arithmetic with 1x1 matrices so you don't have a nice abstraction in your head to convince yourself. It's similar to people who insist that 0 is not a natural number because they haven't studied abstract algebra and don't care about the underlying algebraic structure (nevermind that peano arithmetic starts by postulating zero and succ...)

Why would abstract algebra suggest 0 is a natural number? The natural numbers aren't so much as a group, with or without 0. Peano starts with 0 because it's easier that way, but starting with 1 is easier for other things.

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