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 th...
xkcd's Up-Goer Five comic gave technical specifications for the Saturn V rocket using only the 1,000 most common words in the English language.
This seemed to me and Briénne to be a really fun exercise, both for tabooing one's words and for communicating difficult concepts to laypeople. So why not make a game out of it? Pick any tough, important, or interesting argument or idea, and use this text editor to try to describe what you have in mind with extremely common words only.
This is challenging, so if you almost succeed and want to share your results, you can mark words where you had to cheat in *italics*. Bonus points if your explanation is actually useful for gaining a deeper understanding of the idea, or for teaching it, in the spirit of Gödel's Second Incompleteness Theorem Explained in Words of One Syllable.
As an example, here's my attempt to capture the five theses using only top-thousand words:
If you make a really strong computer and it is not very nice, you will not go to space today.
Other ideas to start with: agent, akrasia, Bayes' theorem, Bayesianism, CFAR, cognitive bias, consequentialism, deontology, effective altruism, Everett-style ('Many Worlds') interpretations of quantum mechanics, entropy, evolution, the Great Reductionist Thesis, halting problem, humanism, law of nature, LessWrong, logic, mathematics, the measurement problem, MIRI, Newcomb's problem, Newton's laws of motion, optimization, Pascal's wager, philosophy, preference, proof, rationality, religion, science, Shannon information, signaling, the simulation argument, singularity, sociopathy, the supernatural, superposition, time, timeless decision theory, transfinite numbers, Turing machine, utilitarianism, validity and soundness, virtue ethics, VNM-utility