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Talking about rotation as if that is obviously relevant completely skips the surprising and interesting part of this identity. edit: Said another way, what you

by creato 7y ago

Talking about rotation as if that is obviously relevant completely skips the surprising and interesting part of this identity. edit: Said another way, what you said is an explanation for what (cos(pi), sin(pi)) is. That exp(i*pi) has anything to do with sine or cosine is what makes the identity interesting, and your explanation says nothing about that.

dnautics 7y ago
Yeah and also that an exponential is a turning function, or that you can even take an number to an imaginary power in the first place, or that turning the right amount just happens to take the power of e (and not, say, 2)
- username90 7y ago
  It's a lot less surprising if you look at the functions as Taylor series, then you'd see that exponentiation is basically the same thing as cos + sin. This also explains "why e and not 2?", because the Taylor series for e, cos and sin are very nice by definition (e is defined to have trivial derivative, and radians are defined to make sin and cos have simple derivatives) while the Taylor series for exponentiation by 2 is not.
  - dnautics 7y ago
    Agreed, but that is a lot of overhead to get to the insight.
- thaumasiotes 7y ago
  > that turning the right amount just happens to take the power of e (and not, say, 2) username90 already explained this one, but in my own rephrasing: The premise is incorrect. It isn't that "turning just happens to take the power of e". e is defined to be the number for which this is true.
  - bsaul 7y ago
    Those kind of « it’s defined like this » explanation has always confused me in math in general. there are two really different things : - the mathematical concept was initialy created as a way to solve this particular problem, and so this should come at no surprise And - after centuries trying to define this concept, we found that the best way to define this concept is like that. In your case, was « e » created to perform 2D rotations in the first place ?
    - Excel_Wizard 7y ago
      If you explored the problem domain, starting with an understanding of trigonometry and calculus, it will naturally emerge as a result. It was not arbitrarily defined. Imaginary numbers were not arbitrarily defined either, they precisely describe an actual phenomenon. At a certain level, it is difficult to distinguish between "the reality we decided is true" for math vs. "the reality that must be true underneath". If you accept empiricism, such that you trust observations we make about the physical world, this problem becomes irrelevant to this discussion.
    - username90 7y ago
      We can look at history: 1. e was discovered in 1618 2. i was introduced around 1637 4. calculus was formalized by Newton in 1687 3. derivatives of sine and cosine were discovered in 1722 4. de Moivre's formula (cos x + i sin x) ^ n = cos nx + i sin nx was discovered in 1730 5. Eulers fornula e ^ ix = cos x + i sin x was discovered in 1748 by using Taylor series (requires their derivatives). So we knew about e and i more than a century before Eulers fornula. However we discovered it pretty soon after we started applying calculus to trigonometrical functions. Also we can note that it is a strictly more powerful than de Moivre's formula since it shows that we can easily add angles just by multiplying their complex representations, so it is not just a simplification of old knowledge. So it seems like the discovery of Euler's fornula actually did help with working with rotations. However it was done during a time when we were still exploring the applications of Calculus so there were a lot of low hanging fruit like this to pick. So to directly answer your question, e was not created to perform 2D rotations, instead we discovered that putting i inside of e makes 2D rotations simple. But the actual important identity you care about is that you can add angles by multiplying which is easy to prove using eulers fornula: (cos x + i sin x) * (cos y + i sin y) = cos (x + y) + i sin (x + y)
garmaine 7y ago
Complex numbers are about rotation. That’s the only odd insight in Euler’s equation. Once you get that, it all falls into place (and in fact starts to explain all the use of complex numbers in engineering).
- xaedes 7y ago
  Hm does that mean you could say the complex quantum probability is just a probability over rotations?
  - tim333 7y ago
    Well you add up all the rotation like stuff and square the result to get the probability. In the book QED Feynman describes it as adding little arrows to explain it to a lay audience. In physical terms it's probably most like adding waves and then the square of the amplitude or wave energy is the probability but not quite - no one really knows what the actual reality is.
  - baq 7y ago
    Kinda sorta, imagine that probability has a direction. Interference is then about alignment of these directions.
- posterboy 7y ago
  what does electric engineering have to do with rotation? Generators perhaps run in circles, an inductor is a coil often enough, but a capacitor?
  - mikorym 7y ago
    Electromagnetism concerns movement of what appear to be moving circles. If you roll a circle with a pen marker it draws a sine or cosine curve. If you try to draw the spiral of a light wave you get perpendicular sine and cosine curves. Light is electromagnetic radiation and in turn both electricity and magnetism are fundamentally connected forces.
jacobolus 7y ago
Personally I think the cool part is that the complex logarithm takes the stereographic projection and turns it into the Mercator projection. That is, it conformally maps the plane onto an infinite two-ended cylinder. The expression log(–1) = iπ more or less just tells you how to normalize the coordinate system on the cylinder. If you wanted you could pick a somewhat different coordinate system. This one is conventional and often convenient though.

by creato 7y ago

dnautics 7y ago

Yeah and also that an exponential is a turning function, or that you can even take an number to an imaginary power in the first place, or that turning the right amount just happens to take the power of e (and not, say, 2)

username90 7y ago
It's a lot less surprising if you look at the functions as Taylor series, then you'd see that exponentiation is basically the same thing as cos + sin. This also explains "why e and not 2?", because the Taylor series for e, cos and sin are very nice by definition (e is defined to have trivial derivative, and radians are defined to make sin and cos have simple derivatives) while the Taylor series for exponentiation by 2 is not.
- dnautics 7y ago
  Agreed, but that is a lot of overhead to get to the insight.
thaumasiotes 7y ago
> that turning the right amount just happens to take the power of e (and not, say, 2) username90 already explained this one, but in my own rephrasing: The premise is incorrect. It isn't that "turning just happens to take the power of e". e is defined to be the number for which this is true.
- bsaul 7y ago
  Those kind of « it’s defined like this » explanation has always confused me in math in general. there are two really different things : - the mathematical concept was initialy created as a way to solve this particular problem, and so this should come at no surprise And - after centuries trying to define this concept, we found that the best way to define this concept is like that. In your case, was « e » created to perform 2D rotations in the first place ?
  - Excel_Wizard 7y ago
    If you explored the problem domain, starting with an understanding of trigonometry and calculus, it will naturally emerge as a result. It was not arbitrarily defined. Imaginary numbers were not arbitrarily defined either, they precisely describe an actual phenomenon. At a certain level, it is difficult to distinguish between "the reality we decided is true" for math vs. "the reality that must be true underneath". If you accept empiricism, such that you trust observations we make about the physical world, this problem becomes irrelevant to this discussion.
  - username90 7y ago
    We can look at history: 1. e was discovered in 1618 2. i was introduced around 1637 4. calculus was formalized by Newton in 1687 3. derivatives of sine and cosine were discovered in 1722 4. de Moivre's formula (cos x + i sin x) ^ n = cos nx + i sin nx was discovered in 1730 5. Eulers fornula e ^ ix = cos x + i sin x was discovered in 1748 by using Taylor series (requires their derivatives). So we knew about e and i more than a century before Eulers fornula. However we discovered it pretty soon after we started applying calculus to trigonometrical functions. Also we can note that it is a strictly more powerful than de Moivre's formula since it shows that we can easily add angles just by multiplying their complex representations, so it is not just a simplification of old knowledge. So it seems like the discovery of Euler's fornula actually did help with working with rotations. However it was done during a time when we were still exploring the applications of Calculus so there were a lot of low hanging fruit like this to pick. So to directly answer your question, e was not created to perform 2D rotations, instead we discovered that putting i inside of e makes 2D rotations simple. But the actual important identity you care about is that you can add angles by multiplying which is easy to prove using eulers fornula: (cos x + i sin x) * (cos y + i sin y) = cos (x + y) + i sin (x + y)

garmaine 7y ago

Complex numbers are about rotation. That’s the only odd insight in Euler’s equation. Once you get that, it all falls into place (and in fact starts to explain all the use of complex numbers in engineering).

xaedes 7y ago
Hm does that mean you could say the complex quantum probability is just a probability over rotations?
- tim333 7y ago
  Well you add up all the rotation like stuff and square the result to get the probability. In the book QED Feynman describes it as adding little arrows to explain it to a lay audience. In physical terms it's probably most like adding waves and then the square of the amplitude or wave energy is the probability but not quite - no one really knows what the actual reality is.
- baq 7y ago
  Kinda sorta, imagine that probability has a direction. Interference is then about alignment of these directions.
posterboy 7y ago
what does electric engineering have to do with rotation? Generators perhaps run in circles, an inductor is a coil often enough, but a capacitor?
- mikorym 7y ago
  Electromagnetism concerns movement of what appear to be moving circles. If you roll a circle with a pen marker it draws a sine or cosine curve. If you try to draw the spiral of a light wave you get perpendicular sine and cosine curves. Light is electromagnetic radiation and in turn both electricity and magnetism are fundamentally connected forces.

jacobolus 7y ago

Personally I think the cool part is that the complex logarithm takes the stereographic projection and turns it into the Mercator projection. That is, it conformally maps the plane onto an infinite two-ended cylinder. The expression log(–1) = iπ more or less just tells you how to normalize the coordinate system on the cylinder. If you wanted you could pick a somewhat different coordinate system. This one is conventional and often convenient though.