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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@ZachWeinersmith > "POOF, a rotation matrix appears"
Next step: quaternions.
@ZachWeinersmith (I am sure you'll manage before recess)
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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"
Some people will find the algebraic idea simpler, some people the geometric.
Some people will like the argument “You know how there is no rational number you can square to get 2, but we can introduce a number with that property and see where it takes us? Let’s do the same thing with a number you can square to get minus one, as well!”
And some people will like the argument “You know how multiplying by minus one takes you 180 degrees around zero on the number line? Let’s introduce the idea of multiplying by something that takes you 90 degrees around zero, which sort of makes us have to have a number plane, and see where *that* takes us!”
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Kinda funny, I was thinking about the idea of explaining imaginary numbers to high school students by saying "actually, there's no magic. You can do the whole thing using only reals." But then basically you have to be like "x^2+1=0" doesn't appear to have a solution until POOF, a rotation matrix appears. Which is maybe more confusing?
@ZachWeinersmith I’d say matrices are more confusing, and you’re gonna need the i notation later anyway.
You can’t write reals either in a conventional way, so technically they’re weird too! (I realized later.) The whole “real” and “imaginary” names are a little misleading, but that’s a discussion that’ll quickly derail into philosophy.
[edit: it helps to realize that a significant part of the reals are literally called “irrational”
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@ZachWeinersmith I’d say matrices are more confusing, and you’re gonna need the i notation later anyway.
You can’t write reals either in a conventional way, so technically they’re weird too! (I realized later.) The whole “real” and “imaginary” names are a little misleading, but that’s a discussion that’ll quickly derail into philosophy.
[edit: it helps to realize that a significant part of the reals are literally called “irrational”]@Wlm @ZachWeinersmith you can do it without matrices by going with rotations. And by going the Geometric Algebra route, one can even develop frameworks where everything (complex numbers, split-complex numbers, quaternions etc) can be derived the same way.
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Some people will find the algebraic idea simpler, some people the geometric.
Some people will like the argument “You know how there is no rational number you can square to get 2, but we can introduce a number with that property and see where it takes us? Let’s do the same thing with a number you can square to get minus one, as well!”
And some people will like the argument “You know how multiplying by minus one takes you 180 degrees around zero on the number line? Let’s introduce the idea of multiplying by something that takes you 90 degrees around zero, which sort of makes us have to have a number plane, and see where *that* takes us!”
@gregeganSF @ZachWeinersmith Wait. That's what sqrt(2) is? Literally?
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@gregeganSF @ZachWeinersmith Wait. That's what sqrt(2) is? Literally?
There are many different ways you can do all of these things!
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@ZachWeinersmith I’d say matrices are more confusing, and you’re gonna need the i notation later anyway.
You can’t write reals either in a conventional way, so technically they’re weird too! (I realized later.) The whole “real” and “imaginary” names are a little misleading, but that’s a discussion that’ll quickly derail into philosophy.
[edit: it helps to realize that a significant part of the reals are literally called “irrational”]@ZachWeinersmith I always liked the escalating order of discovery*:
1 + x = 1? Zero
1 + x = 0? Negative numbers
3 * x = 1? Rational numbers
x * x = 2? Irrational numbers
x * x = -1? Imaginary numbers
x * y - y * x = 1? Quaternions
(xy)z - x(yz) = 1? Octonions
where every step is a 🤯.(*or invention if you will)
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There are many different ways you can do all of these things!
@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.
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@Wlm @ZachWeinersmith you can do it without matrices by going with rotations. And by going the Geometric Algebra route, one can even develop frameworks where everything (complex numbers, split-complex numbers, quaternions etc) can be derived the same way.
@oblomov @ZachWeinersmith Yeah it’s important to realize that they aren’t matrices per se, but matrices are one way to represent them (same with tensors). But maybe not yet in high school.
(NB not a mathematician, so apologies for any sloppy terminology.) -
Kinda funny, I was thinking about the idea of explaining imaginary numbers to high school students by saying "actually, there's no magic. You can do the whole thing using only reals." But then basically you have to be like "x^2+1=0" doesn't appear to have a solution until POOF, a rotation matrix appears. Which is maybe more confusing?
@ZachWeinersmith If you're introducing it, I would start with a number line. Explain that all real numbers are there. Now show them that sqrt(-1) is not there.
Now, assuming that's a number at all, you can multiply that number by any real number and you get a second number line. 1i, 2i, 3i, etc ...
EXCEPT 0 times anything is 0. So those two points have to be the same. That's when you erase the second number line and make it vertical. THAT'S where your complex plane comes from.
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@ZachWeinersmith If you're introducing it, I would start with a number line. Explain that all real numbers are there. Now show them that sqrt(-1) is not there.
Now, assuming that's a number at all, you can multiply that number by any real number and you get a second number line. 1i, 2i, 3i, etc ...
EXCEPT 0 times anything is 0. So those two points have to be the same. That's when you erase the second number line and make it vertical. THAT'S where your complex plane comes from.
@ZachWeinersmith Also, for the love of god, explain to them that they're called "imaginary" because they're NOT REAL NUMBERS.
It's a pun. It doesn't actually mean anything.
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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.
