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I think anybody who thinks real numbers exist will also say that imaginary numbers exist. "real" and "imaginary" in the context of numbers are just arbitrary labels assigned for historical reasons, they're not meant to convey any philosophical judgements about existence.

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I don't think so. You can have imaginary rationals, or even imaginary natural numbers (that is, a+bi, where both a and b are rational numbers or integers).

Although I will admit that a big part of the point of the reals was solutions to polynomials, and it takes complex numbers to be able to solve them all.

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