I agree. I think complex numbers are best explained in the context of motivations for number systems (and I made a comment explaining complex numbers this way about a month ago[1]). This way takes longer than just answering the question, “what’s a complex number?”, but it also builds a better intuition of why complex numbers aren’t silly (or at least, why they’re only as silly as anything that isn’t a natural number). If you can get a student to be okay with the idea that we define new number systems to resolve problems in prior ones, you can get them to be okay with a construction of complex numbers from the reals, just as they’ll accept a construction of the irrationals from the rationals, and integers from the naturals, etc.
I find that a really rigorous (albeit...spartan) first pass for understanding complex numbers can be developed by reading through Chapter 1 of Rudin’s Principles of Mathematical Analysis. He doesn’t go quite as far as developing Peano arithmetic from first principles, but he does build up the number systems successively, beginning with the natural numbers if I recall correctly. That was the first book I read where I felt like I really understood what complex numbers were, because up until that point I was only dealing with them algebraically using an explicit i in the form a + bi. Rudin’s development of the complex plane and representation of real and complex numbers as points (a, b) is much better for intuition, in my experience, and it lights the way to an even deeper intuition of what complex numbers are in the context of Euclidean spaces later on.
That said, Rudin is terse to the point that many would call him pretentious, so if there is another book that does the same thing with better exposition, that might be a better choice...
> But there is one further thing to note, which is the inherent ambiguity between i and -i. We can't tell these apart. John Conway says this well, when asked about the square root of -1 his reply is "which square root of -1 do you mean?" (paraphrasing.)
It’s been a while since I did this exercise, but if I recall correctly this ambiguity is part of the proof that we cannot order the complex field, right? We end up with an absurdity because i should be greater than -i, but i^3 = -i, which evidently resists coherent ordering. And similarly i x i = -1, but -i x i = 1.
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