But it drives home the reason why I parted ways with advanced math. There just came a point when the math no longer resonated with the natural world that I was comfortable living in.
Imaginary (and complex) numbers were the first real schism where my mind repelled the very idea.
Nonetheless, decades later, I would be talking with a friend who is much smarter than me. We were talking about the lossy nature of converting audio from the time domain to the frequency domain and back. I understood that phase information was lost in the translation (I also thought it was curious that the human brain seemed to be phase-agnostic and so could not perceive the difference when audio was round-tripped through the frequency domain).
"Oh," he said, "when you do a Fast Fourier transform to move audio into the frequency domain the phase information is represented by the imaginary part of a complex number."
Wait, what? So complex numbers have a real-world analog in constructing an FFT — a "real" algorithm that does real work?
I see now that the really smart people just understand the natural world much better than I do — or perhaps see it differently than I do.
I think it's less about some people being especially good at math than mathematics education being especially terrible and uninspiring. 3blue1brown's approach to math pedagogy had really changed how I'm able to see math and to understand it's profundity (as someone who didn't get any further than you, it sounds like, in formal math education [I never took calculus in school.]).
The second playlist is great, thank you! As someone with near zero math knowledge, it was very exciting learning that imaginary numbers are a beautiful and intuitive concept. Really makes me want to start playing with frequency domain stuff. Also, the reveal of the mapped video visualization... the whole series is worth watching just for that moment.
(In the following, mathematical uses of terms are given in "quotes".)
So many of the names of mathematical objects are arbitrary and I think that can be harmful. "Real" numbers are just as imaginary as "imaginary" ones! They both have use in physics as models for natural phenomena, but neither are real like an atom. You may still argue that "natural" numbers are real because we really can have n of something, but general real numbers can't be written down or computed or used to divide physical space and time as we understand them now. The names are mostly historical artefacts that serve as mnemonics, nothing more.
Don't be intimidated! The person who first invented complex numbers called them "useless" and "mental torture."[1] Subsequent mathematicians have simply gotten used to them.
You'd invent them, too, if you spent a few weeks building a computer program to output the values of `t` that make `t^3+pt+q=0` (when `4p^3 + 27q^2 < 0`, there are 3 real solutions, but they can't be expressed in general without involving complex numbers).[2]
> Imaginary (and complex) numbers were the first real schism where my mind repelled the very idea.
If extending the number line bothers you, you should also have a problem with negative numbers. And, in fact, many ancient mathematicians did, so you're in good company :)
Complex numbers behave quite differently than vectors/matrixes, and their discovery wasn't motivated by needing to represent coordinates. They were discovered (or invented, for the nonplatonists) in the process of resolving the paradox that 3rd degree polynomials must have at least 1 (real) root (because they always stretch out in opposite directions, and so cross 0), but you end up needing to square root a negative number to solve for it (the paradox being, this was thought to be impossible, since the square of any real number is positive).
Their existence is a natural consequence of the rules of the algebraic operations (addition, subtraction, multiplication, division, powers and logarithms); they aren't incidental, they are integral. Without imaginary numbers, the algebraic operations aren't closed.
(This being a brief summary of the thesis of the videos I linked in a sibling comment.)
That being said, they are useful as coordinates, as well.
Imaginary numbers are important in a lot of real world applications. One of the most important ones at the moment is in quantum mechanics and quantum algorithms. You can't model quantum algorithms without imaginary numbers.
The problem isn't you! "Imaginary" numbers are terribly named, and I can see why that would put anyone off engaging with them. They are just as "real" as 2d co-ordinates are.
Not exactly "physical" in the same sense as a normal Rubik's cube, since there isn't any physical restriction that makes illegal moves of taking it apart and putting it back together anymore difficult than performing legal moves.
(person who made that video and just found this post through youtube analytics here)
There are 8 magnets on each side of each cube, for a total of 48 per cube.
The puzzle can be assembled to invalid states, so yes it does come to the user to ensure that only allowed or "canonical" moves are used.
Unbelievable. I used to be a collector of twisty puzzles and there was quite a few levels of challenging puzzles beyond my ability, but seeing this is just astounding! I only have barely a clue what each of the moves are doing but it makes me smile knowing someone cared enough to make this. Science has gone too far
Does the limitations of 3D physics just mean that the thing can't really be connected, and you are responsible for making a subset of the moves by free hand, and if you screw up you either cheated, or made the equivalent of a corner twist on a Rubik's cube, leaving the resulting configuration unsolvable?
(person who made that video and just found this post through youtube analytics here)
Yes, it is a lot more complicated this is because of the nature of what the gyro is doing. The gyros are effectively a 4 dimension rotation of the puzzle that actually doesn't change the state of the puzzle, but just the orientation. This doesn't work out to be very pretty in 3d.
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But it drives home the reason why I parted ways with advanced math. There just came a point when the math no longer resonated with the natural world that I was comfortable living in.
Imaginary (and complex) numbers were the first real schism where my mind repelled the very idea.
Nonetheless, decades later, I would be talking with a friend who is much smarter than me. We were talking about the lossy nature of converting audio from the time domain to the frequency domain and back. I understood that phase information was lost in the translation (I also thought it was curious that the human brain seemed to be phase-agnostic and so could not perceive the difference when audio was round-tripped through the frequency domain).
"Oh," he said, "when you do a Fast Fourier transform to move audio into the frequency domain the phase information is represented by the imaginary part of a complex number."
Wait, what? So complex numbers have a real-world analog in constructing an FFT — a "real" algorithm that does real work?
I see now that the really smart people just understand the natural world much better than I do — or perhaps see it differently than I do.
https://youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFit...
This is also a great series of videos about how, like it says on the tin, the imaginary numbers are real:
https://youtube.com/playlist?list=PLiaHhY2iBX9g6KIvZ_703G3KJ...
So many of the names of mathematical objects are arbitrary and I think that can be harmful. "Real" numbers are just as imaginary as "imaginary" ones! They both have use in physics as models for natural phenomena, but neither are real like an atom. You may still argue that "natural" numbers are real because we really can have n of something, but general real numbers can't be written down or computed or used to divide physical space and time as we understand them now. The names are mostly historical artefacts that serve as mnemonics, nothing more.
You'd invent them, too, if you spent a few weeks building a computer program to output the values of `t` that make `t^3+pt+q=0` (when `4p^3 + 27q^2 < 0`, there are 3 real solutions, but they can't be expressed in general without involving complex numbers).[2]
[1] https://en.wikipedia.org/wiki/Complex_number#History [2] https://en.wikipedia.org/wiki/Cubic_equation#Cardano's_formu...
If extending the number line bothers you, you should also have a problem with negative numbers. And, in fact, many ancient mathematicians did, so you're in good company :)
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No. Complex numbers are "simply" neat construct to represent 2D coordinates algebraically - i.e. without matrices.
Their existence is a natural consequence of the rules of the algebraic operations (addition, subtraction, multiplication, division, powers and logarithms); they aren't incidental, they are integral. Without imaginary numbers, the algebraic operations aren't closed.
(This being a brief summary of the thesis of the videos I linked in a sibling comment.)
That being said, they are useful as coordinates, as well.
Deleted Comment
The puzzle can be assembled to invalid states, so yes it does come to the user to ensure that only allowed or "canonical" moves are used.
Thank you! It means a lot! Especially because I am just a high schooler...
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Yes, it is a lot more complicated this is because of the nature of what the gyro is doing. The gyros are effectively a 4 dimension rotation of the puzzle that actually doesn't change the state of the puzzle, but just the orientation. This doesn't work out to be very pretty in 3d.