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I guess whoever taught you complex numbers didn't do it with a focus on diagrams? I agree math should be taught as visually as possible, with algebra-only expl

I guess whoever taught you complex numbers didn't do it with a focus on diagrams? I agree math should be taught as visually as possible, with algebra-only explanations being minimised. Even things like the expansion of (a+b)^2 = a^2 + 2ab + b^2 or a^2 - b^2 = (a + b)(a - b) should be motivated by diagrams. Anyway it turns that multiplication of complex numbers is adding the polar angles. This explains why multiplying two negative real numbers gives you a positive despite multiply two positive also giving a positive. If you then go from that and say "what number when multiplied by itself would give me minus 1" you get i. Also I don't think you've hit an IQ limit, it's not like you're lifting some weight that's too big for you. You just haven't seen an explanation that links to your existing corpus of knowledge.

at_compile_time 4y ago
It doesn't help that i is given names like "imaginary" and "complex". We may as well call them "too-hard-for-you" numbers, and laugh condescendingly when students ask about them. Same with quaternions and dual quaternions. They perform rotation and scaling in 3D space. Calling them "hypercomplex numbers" makes it sound like an advanced concept only to be understood after years of dedicated study. I get that naming things is hard, but they could have gone with something that didn't sound like it was intended to stroke the egos of the learned few who understand. I prefer the term rotor, because they perform rotation. The actual math involved isn't that difficult if you've learned the basics of geometric/Clifford algebra.
- lordnacho 4y ago
  Someone on HN proposed "Lateral Number" which I thought was a good one. It's a number, just to the side of the numbers you know.
  - cyberbanjo 4y ago
    "If we call +1, -1, and √-1 had been called direct, inverse and lateral units, instead of positive, negative, and imaginary (or impossible) units, such an obscurity would have been out of the question." --Gauss
    - cestith 4y ago
      One of the most helpful things for me when handling imaginary and complex numbers came not from a math class, but a physics class. My HS physics instructor said rather than "imaginary" we could think of -1 as a "hidden" number, because after you do the math fairly often you see where the number would be, but you just can't see it among the "real" numbers. This was literally an aside to something else he was talking about at the time. Calling the imaginary numbers "lateral" and saying they were "to the side of" other numbers would've been helpful, too. I'm not sure it would have been quite as helpful to me, personally, but either is better than "imaginary" just to juxtapose with the "real" numbers.
      - lordnacho 4y ago
        > My HS physics instructor said rather than "imaginary" we could think of -1 as a "hidden" number, because after you do the math fairly often you see where the number would be, but you just can't see it among the "real" numbers. IIRC you often encounter a system where the energy is constant, but only if you consider the imaginary part of some equation holding the potential energy, which then exchanges with "actual" kinetic energy.
      - cestith 4y ago
        Somehow I missed the √ symbol in √-1 there but I think it was understood I meant i, not -1.
- fjeifisjf 4y ago
JW_00000 4y ago
To add to this, a website I've found useful is betterexplained.com. Specifically for complex numbers, take a look at [1] and [2]. But they also have good articles with visualizations to help you understand the Fourier transform, the number e, integrals and derivatives, linear algebra, etc. A starting point is [3]. One useful idea is that they color the different parts of math equations to explain each part separately, see [4]. Another great source of math visualizations is the YouTube channel 3Blue1Brown, in particular the series on linear algebra [5]. (I didn't intuitively understand what a determinant represented or what the determinant of a transformation being zero really meant, until watching [6].) [1] https://betterexplained.com/articles/a-visual-intuitive-guide-to-imaginary-numbers/ https://betterexplained.com/articles/a-visual-intuitive-guid... [2] https://betterexplained.com/articles/intuitive-arithmetic-with-complex-numbers/ https://betterexplained.com/articles/intuitive-arithmetic-wi... [3] https://betterexplained.com/articles/developing-your-intuition-for-math/ https://betterexplained.com/articles/developing-your-intuiti... [4] https://betterexplained.com/articles/colorized-math-equations/ https://betterexplained.com/articles/colorized-math-equation... [5] https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2x... [6] https://www.youtube.com/watch?v=Ip3X9LOh2dk&list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab&index=6 https://www.youtube.com/watch?v=Ip3X9LOh2dk&list=PLZHQObOWTQ...
dylan604 4y ago
While it has been multiple decades now since I've been in school, this by far, was the best explanation I've read on how/when/where/why of imaginary numbers. I don't recall ever hearing polar anything related to discussions of "i". Thanks. You've now filled my "learn something new every day" quota.

at_compile_time 4y ago

It doesn't help that i is given names like "imaginary" and "complex". We may as well call them "too-hard-for-you" numbers, and laugh condescendingly when students ask about them. Same with quaternions and dual quaternions. They perform rotation and scaling in 3D space. Calling them "hypercomplex numbers" makes it sound like an advanced concept only to be understood after years of dedicated study. I get that naming things is hard, but they could have gone with something that didn't sound like it was intended to stroke the egos of the learned few who understand. I prefer the term rotor, because they perform rotation. The actual math involved isn't that difficult if you've learned the basics of geometric/Clifford algebra.

lordnacho 4y ago
Someone on HN proposed "Lateral Number" which I thought was a good one. It's a number, just to the side of the numbers you know.
- cyberbanjo 4y ago
  "If we call +1, -1, and √-1 had been called direct, inverse and lateral units, instead of positive, negative, and imaginary (or impossible) units, such an obscurity would have been out of the question." --Gauss
  - cestith 4y ago
    One of the most helpful things for me when handling imaginary and complex numbers came not from a math class, but a physics class. My HS physics instructor said rather than "imaginary" we could think of -1 as a "hidden" number, because after you do the math fairly often you see where the number would be, but you just can't see it among the "real" numbers. This was literally an aside to something else he was talking about at the time. Calling the imaginary numbers "lateral" and saying they were "to the side of" other numbers would've been helpful, too. I'm not sure it would have been quite as helpful to me, personally, but either is better than "imaginary" just to juxtapose with the "real" numbers.
    - lordnacho 4y ago
      > My HS physics instructor said rather than "imaginary" we could think of -1 as a "hidden" number, because after you do the math fairly often you see where the number would be, but you just can't see it among the "real" numbers. IIRC you often encounter a system where the energy is constant, but only if you consider the imaginary part of some equation holding the potential energy, which then exchanges with "actual" kinetic energy.
    - cestith 4y ago
      Somehow I missed the √ symbol in √-1 there but I think it was understood I meant i, not -1.
fjeifisjf 4y ago

JW_00000 4y ago

To add to this, a website I've found useful is betterexplained.com. Specifically for complex numbers, take a look at [1] and [2]. But they also have good articles with visualizations to help you understand the Fourier transform, the number e, integrals and derivatives, linear algebra, etc. A starting point is [3]. One useful idea is that they color the different parts of math equations to explain each part separately, see [4]. Another great source of math visualizations is the YouTube channel 3Blue1Brown, in particular the series on linear algebra [5]. (I didn't intuitively understand what a determinant represented or what the determinant of a transformation being zero really meant, until watching [6].) [1] https://betterexplained.com/articles/a-visual-intuitive-guide-to-imaginary-numbers/ https://betterexplained.com/articles/a-visual-intuitive-guid... [2] https://betterexplained.com/articles/intuitive-arithmetic-with-complex-numbers/ https://betterexplained.com/articles/intuitive-arithmetic-wi... [3] https://betterexplained.com/articles/developing-your-intuition-for-math/ https://betterexplained.com/articles/developing-your-intuiti... [4] https://betterexplained.com/articles/colorized-math-equations/ https://betterexplained.com/articles/colorized-math-equation... [5] https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2x... [6] https://www.youtube.com/watch?v=Ip3X9LOh2dk&list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab&index=6 https://www.youtube.com/watch?v=Ip3X9LOh2dk&list=PLZHQObOWTQ...

dylan604 4y ago

While it has been multiple decades now since I've been in school, this by far, was the best explanation I've read on how/when/where/why of imaginary numbers. I don't recall ever hearing polar anything related to discussions of "i". Thanks. You've now filled my "learn something new every day" quota.