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[carey]

(12-10-2024 12:44 PM)naddy Wrote:  

(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.

I appreciate your comments but I think they refer to a different debate (i.e., the philosophical debate whether "real" or "imaginary" are to be taken literally or "exist") vs the pragmatic debate whether complex numbers are necessary to describe nature. For example, many types of numbers can be defined (e.g., Abraham Robinson's hyperreals) and can be useful, but strictly speaking, may not be necessary. However, the square root of -1 occurs in the Schrödinger equation, the fundamental equation describing atoms and other systems, hence complex numbers (irrespective of the meaning of the name "complex" :) are essential in physics.