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> exp(iπ) = –1 ... How can this be? The “plain English” expression of this formula is: a half turn rotation in a plane is equivalent to a reflection across the
by jacobolus 7y ago
> exp(iπ) = –1 ... How can this be?
The “plain English” expression of this formula is: a half turn rotation in a plane is equivalent to a reflection across the axis of rotation.
YMMV, but I have found that small children can understand this statement.
- deleted 7y ago[deleted]
- ColanR 7y agoThat is the best explanation I've ever seen.
- creato 7y agoTalking 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 agoYeah 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 agoIt'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 agoAgreed, 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 agoThose 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 agoIf 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 agoWe 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 agoComplex 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 agoHm does that mean you could say the complex quantum probability is just a probability over rotations?
- tim333 7y agoWell 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 agoKinda sorta, imagine that probability has a direction. Interference is then about alignment of these directions.
- posterboy 7y agowhat 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 agoElectromagnetism 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 agoPersonally 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.
- ilyagr 7y agoFrom this point of view, it's not easy to see what exp(1) should be. The beauty is in the connection.
- jacobolus 7y agoWhat the precise value of exp(1) turns out to be isn’t really the point though. That’s one reason that e^x is a pedagogically problematic shorthand for exp(x).
- ilyagr 7y agoI disagree, I think it's very much essential to the point. For one, 10^(i*pi) is some horrible complex number, which is what you'd expect when exponentiating random complex numbers. Exercise: figure out what it is from the rotation point of view. Secondly, usual definitions of e don't involve complex numbers at all. Nor do questions such as whether e is rational, algebraic, etc. Thirdly, if you go with this argument too far, you'll conclude that the value of pi is also not important. After all, circles must have some length, who cares what it is precisely?
- jacobolus 7y ago> 10^(iπ) is some horrible complex number Sure, you want to use a convenient coordinate system. But you already have 2 horrible numbers in the conventional coordinate system: the π and e. In the conventional (“natural”) coordinate system for logarithms, the coordinates represent powers of e and rotations by 1/2π turns, respectively. You have just decided that you’ll use those specific horrible numbers in the logarithmic coordinates, because e and π are what you get out if you choose a coordinate system for the logarithm of quotients-of-vectors where the derivative of log(x) at the point x = 1 is 1. This turns out to be convenient for simplifying differential equations and related formulas. Personally I like using a coordinate system where the “scale” axis of the logarithm uses units of doublings, and the “rotation” axis of the logarithm uses units of full turns. If I use that one when making a picture, I can easily tell you exactly what the value is (as a rational number) at every grid intersection. e.g. here is a plot of a Möbius transformation, https://raw.githubusercontent.com/jrus/images-for-observable/master/v.-*(z%2B1)%7C(z-1).b.jpg https://raw.githubusercontent.com/jrus/images-for-observable... Every contour at the most prominent level is either a doubling or 1/12 of a turn. (You’ll notice that when we zoom on any part of the picture, we see little rectangles rather than little squares; the “natural” coordinate system is a square one, which is also often convenient.)
- gigatexal 7y agoI was never very good at geometry and this explanation has me feeling like a really stupid child.
- The_rationalist 7y agoIt's kinda sad that formalism here is far less clearer than natural language. I often wonder if instead of formalisms, mathematicians should just have used attempto controlled English.
- lonelappde 7y agoThey tried that before the nineteenth century and it stymied progress. Imagine multiplying just two-digit numbers without any notation.
- The_rationalist 7y agoThey didn't tried attempto controlled English in the nineteenth century as it only exists since end of twenty century. Imagine multiplying just two-digit numbers without any notation I agree, notation can be useful for concision. Far less for clarity. Instead of a black or white thinking, I'm wishing that mathematicians use more well defined English in their proof, but still use notation where it is clear and more concise.
- kittiepryde 7y agoYou explained that 'a half turn rotation in a plane is equivalent to a reflection across the axis of rotation' to a child and they were able to figure out how 'exp(i*pi)=-1' was derived? That's remarkable! I think these children were on a much higher level than most adults that we'd still consider smart, I wouldn't put down anyone for not immediately seeing these two things as equivalent.
- jacobolus 7y agoNo, the children were able to understand how if you rotate a piece of paper by half a turn, all of the points end up the same distance from the axis of rotation, but the opposite direction from where they started. All of the stuff about exponential functions and angle measures and “imaginary” numbers just obfuscates the core idea.
- SilasX 7y agoBut the core idea involves why imaginary exponentiation means a rotation in the complex plane! If you don't have that part, you're not talking about i at all, you're just teaching the concept of rotation, with the same symbols as are used in complex math. You would have the exact same explanation if you were teaching about rotation unconnected to imaginary numbers or exponentiation at all!
- matt-attack 7y agoI think it makes even more sense when written in terms of tau. e ^ (i * tau) = 1 In plain English: “rotating by a full turn is the same as doing nothing”.
- throwaway2048 7y agothis, and the unit circle are by far the most compelling arguments for tau. Euler's identity isn't some mysterious thing where all these mathematical building blocks meet, its a completely straight forward and fundamental consequence of circles existing!.
- lonelappde 7y ago2pi is a full turn. Using a different name doesn't change much.
- F-0X 7y agoBut that's just false. Rotations preserve orientation, reflections do not. They can't be equivalent. Two reflections can be equivalent to a rotation however, because that restores the original orientation.
- CogitoCogito 7y ago> But that's just false. Rotations preserve orientation, reflections do not. They can't be equivalent. Two reflections can be equivalent to a rotation however, because that restores the original orientation. I believe the parent meant the statement as it applies to a single point. E.g. meant to say "a half turn rotation in a plane [of a point] is equivalent to a reflection across the axis of rotation."
- jacobolus 7y agoA reflection across the axis of rotation, which is a point, not a reflection across a line. –1 is the quotient of any two vectors which have the same magnitude and point in opposite directions.
- tim333 7y agoThe surprising thing is that a bit of arithmetic results in something like geometry.
- SilasX 7y agoI've never considered that (in itself) a satisfying or complete explanation, because it just handwaves away the problem of how it is that imaginary exponentiation gets you a rotation in the complex plane, and feels like you're just defining it as such, which isn't imparting any insight. Yes, small children can understand the bit about "two half turns put you in the reflected position", but not "why is i rotating a complex number at all".
- heavenlyblue 7y agoWell, the first time it made a tick for me was when I realised that rotational exponentiation is what provides for perfectly smooth transitions between even and odd powers of negative numbers.
- jacobolus 7y ago> but not "why is i rotating a complex number at all". Yes, this part requires getting a good amount of practice with vectors, and is best saved for kids aged maybe 10+ (depends a lot on the kid and their level of preparation). The most straight-forward approach is to define I to mean a quarter turn rotation anticlockwise in the plane. Then try to figure out what a quantity like Z = a + bI (where a and b are scalars) would mean: if we multiply it by a vector v we get Zv = av + bIv, that is: a part of the vector av pointed in the same direction as v plus a part of the vector bIv pointed in a perpendicular direction. Now we can investigate what happens when we multiply I(Iv): we rotate v a quarter turn, then another quarter turn. Or in other words, I(Iv) = –v for any v. Because our multiplication is associative and vectors have inverses, we can write II = –1. The tricky part of the abstraction here is that we can treat scale + rotation transformations of the plane like numbers. Multiplying them corresponds to composition of transformations. Adding them and then applying the sum to a vector is the same as applying each separately and adding the parts afterward. Getting comfortable with this algebraic system is certainly not trivial. Then there is a neat insight that when we take a pure rotation’s logarithm (a.k.a. “angle measure”), that turns out to have magnitude proportional to the arclength between two rotated points on a circle.
- SilasX 7y ago
- deleted 7y ago[deleted]