Kinda funny, I was thinking about the idea of explaining imaginary numbers to high school students by saying "actually, there's no magic.
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Like is "there's this thing called i and it's 'imaginary' but we can use it like any other number" more or less intuitive than "you're sitting here doing algebra and suddenly a number explodes into 4 numbers"
@ZachWeinersmith Explain ideals and construct ℂ as ℝ[X]/(X²+1).
I actually do think this is (appropriately simplified) a far better explanation than the "add an imaginary i". Especially if you go all the way ℕ→ℤ→ℚ→ℝ→ℂ. because once you get to ℂ, the quotient construction is already familiar.
Really, the actually weirdest and hard to understand step is ℚ→ℝ, which is also the most badly explained in school.
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@ZachWeinersmith Explain ideals and construct ℂ as ℝ[X]/(X²+1).
I actually do think this is (appropriately simplified) a far better explanation than the "add an imaginary i". Especially if you go all the way ℕ→ℤ→ℚ→ℝ→ℂ. because once you get to ℂ, the quotient construction is already familiar.
Really, the actually weirdest and hard to understand step is ℚ→ℝ, which is also the most badly explained in school.
@ZachWeinersmith The thing that irks me about the "i is the solution to X²+1=0" explanation is, that it is unclear *which* of the two it is (and whether it matters).
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@gregeganSF @ZachWeinersmith I remember the maths teacher in school telling us that square roots can not be calculated, but you can only guess, then square, guess again, and get closer.
I always thought that to be BS, but never dug deeper because I already had a calculator back then.Sometimes I wonder where I would have gotten in life if I had real teachers, instead of these bottom of the barrel failures.
@stefanie @gregeganSF @ZachWeinersmith I got lucky in HS I did have one maths teacher that looked up a "long-hand square root" and taught it to me, tho I never got good at it.
But, I think a lot of the generation that would be teaching me maths hadn't learned square-root long-hand, and the students that were interested in such stuff where taught roots on the slide rule.
