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[naddy]

(12-10-2024 10:53 AM)carey Wrote:  What did it for me in the debate whether complex numbers are a necessity or a mathematical convenience occurred in high school when my chemistry teacher introduced the Schrödinger equation (thank you Mr. Cardulla!) and there it was, i, in the equation itself, suggesting a fundamental role of complex numbers in nature. That didn't stop diehard deniers of the necessity of complex numbers from pointing out that "complex numbers are just ordered pairs of real numbers." However the argument that complex numbers can be expressed using real numbers behaving like complex numbers doesn't appear to me to undercut the necessary role of the complex number structure.

The designations "real" and "imaginary" are not to be taken literally. Like natural, integer, rational, or real numbers, complex numbers are a mathematical construct that happens to have proven useful for modeling some aspects of physical reality. Whether that implies that any numbers "exist" in a meaningful way is one for the philosophers and oddly irrelevant for mathematicians and physicists/engineers.

It's also worth pointing out that unless you are into pure mathematics, all the math you will ever encounter is from a subset selected because we have found applications for it, with the parts underlying cryptography the most esoteric because they are purely structural and don't model physical systems. That doesn't mean that this math is more "real" than the one we haven't found applications for.