For what it’s worth, I dislike the term “real number” precisely because it suggests that there’s something particularly real about them. Real numbers have a consistent and unambiguous mathematical definition; so do complex numbers. Real numbers show up in the real world; so do complex numbers. If I were to tell someone about real numbers, I would immediately mention that there’s nothing that makes them any more real or fake than any other kind of number.
Unrelatedly, my favorite mathematical definition (the one that I enjoy the most, not the one I think is actually best in any sense) is essentially the opposite of Up-Goer Five: it tries to explain a concept as thoroughly as possible using as few words as possible, even if that requires using very obscure words. That definition is:
The complex numbers are the algebraic closure of the completion of the field of fractions of the initial ring.
I thought I might get some pushback on taking the word “real” in “real number” literally, because, as you say, real numbers are just as legitimate a mathematical object as anything else.
We probably differ, though, in how much we think of real & complex numbers as showing up in the real world. In practice, when I measure something quantitatively, the result’s almost always a real number. If I count things I get natural numbers. If I can also count things backwards I get the integers. If I take a reading from a digital meter I get a rational number, and (classically) if I could look arbitrarily closely at the needle on an analogue meter I could read off real numbers.
But where do complex numbers pop up? To me they really only seem to inhere in quantum mechanics (where they are, admittedly, absolutely fundamental to the whole theory), but even there you have to work rather hard to directly measure something like the wavefunction’s real & imaginary parts.
In the macroscopic world it’s not easy to physically get at whatever complex numbers comprise a system’s state. I can certainly theorize about the complex numbers embodied in a system after the fact; I learned how to use phasors in electronics, contour integration in complex analysis class, complex arguments to exponential functions to represent oscillations, and so on. But these often feel like mere computational gimmicks I deploy to simplify the mathematics, and even when using complex numbers feels completely natural in the exam room, the only numbers I see in the lab are real numbers.
As such I’m OK with informally differentiating between real numbers & complex numbers on the basis that I can point to any human-scale quantitative phenomenon, and say “real numbers are just right there”, while the same isn’t true of complex numbers. This isn’t especially rigorous, but I thought that was a worthwhile way to avoid spending several introductory paragraphs trying to pin down real numbers more formally. (And I expect the kind of person who needs or wants an up-goer five description of real numbers would still get more out of my hand-waving than they’d get out of, say, “real numbers form the unique Archimedean complete totally ordered field (R,+,·,<) up to isomorphism”.)
As far as I know, the most visible way that complex numbers show up “in the real world” is as sine waves. Sine waves of a given frequency can be thought of as complex numbers. Adding together two sine waves corresponds to adding the corresponding complex numbers. Convoluting two sine waves corresponds to multiplying the corresponding complex numbers.
Since every analog signal can be thought of as a sum or integral of sine waves of different frequencies, an analog signal can be represented as a collection of complex numbers, one corresponding to the sinusoid at each frequency. This is what the Fourier transform is. Since convolution of analog signals corresponds to multiplication of their Fourier transforms, now a lot of the stuff we know about multiplication is applicable to convolution as well.
For what it’s worth, I dislike the term “real number” precisely because it suggests that there’s something particularly real about them. Real numbers have a consistent and unambiguous mathematical definition; so do complex numbers. Real numbers show up in the real world; so do complex numbers. If I were to tell someone about real numbers, I would immediately mention that there’s nothing that makes them any more real or fake than any other kind of number.
Unrelatedly, my favorite mathematical definition (the one that I enjoy the most, not the one I think is actually best in any sense) is essentially the opposite of Up-Goer Five: it tries to explain a concept as thoroughly as possible using as few words as possible, even if that requires using very obscure words. That definition is:
I thought I might get some pushback on taking the word “real” in “real number” literally, because, as you say, real numbers are just as legitimate a mathematical object as anything else.
We probably differ, though, in how much we think of real & complex numbers as showing up in the real world. In practice, when I measure something quantitatively, the result’s almost always a real number. If I count things I get natural numbers. If I can also count things backwards I get the integers. If I take a reading from a digital meter I get a rational number, and (classically) if I could look arbitrarily closely at the needle on an analogue meter I could read off real numbers.
But where do complex numbers pop up? To me they really only seem to inhere in quantum mechanics (where they are, admittedly, absolutely fundamental to the whole theory), but even there you have to work rather hard to directly measure something like the wavefunction’s real & imaginary parts.
In the macroscopic world it’s not easy to physically get at whatever complex numbers comprise a system’s state. I can certainly theorize about the complex numbers embodied in a system after the fact; I learned how to use phasors in electronics, contour integration in complex analysis class, complex arguments to exponential functions to represent oscillations, and so on. But these often feel like mere computational gimmicks I deploy to simplify the mathematics, and even when using complex numbers feels completely natural in the exam room, the only numbers I see in the lab are real numbers.
As such I’m OK with informally differentiating between real numbers & complex numbers on the basis that I can point to any human-scale quantitative phenomenon, and say “real numbers are just right there”, while the same isn’t true of complex numbers. This isn’t especially rigorous, but I thought that was a worthwhile way to avoid spending several introductory paragraphs trying to pin down real numbers more formally. (And I expect the kind of person who needs or wants an up-goer five description of real numbers would still get more out of my hand-waving than they’d get out of, say, “real numbers form the unique Archimedean complete totally ordered field (R,+,·,<) up to isomorphism”.)
As far as I know, the most visible way that complex numbers show up “in the real world” is as sine waves. Sine waves of a given frequency can be thought of as complex numbers. Adding together two sine waves corresponds to adding the corresponding complex numbers. Convoluting two sine waves corresponds to multiplying the corresponding complex numbers.
Since every analog signal can be thought of as a sum or integral of sine waves of different frequencies, an analog signal can be represented as a collection of complex numbers, one corresponding to the sinusoid at each frequency. This is what the Fourier transform is. Since convolution of analog signals corresponds to multiplication of their Fourier transforms, now a lot of the stuff we know about multiplication is applicable to convolution as well.