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 r...
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