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> 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

> 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
>The most straight-forward approach is to define I to mean a quarter turn rotation anticlockwise in the plane Like I said originally, if you're just defining i to work this way, you're not conveying any of the insight behind why this is a natural extension of the existing rules/definitions of i, sine, exponentiation, etc. All your latest comment is doing is meticulously spelling out the concept of rotation in a plane. Again, that's not the hard(est) part of proving this result or of conveying the intuition. You're handwaving away 90% of it. It's not reasonable to characterize someone as "understanding" it because they get rotation, the last 10%.
- jacobolus 7y ago
  The “existing rules/definitions of i, sine, etc.” are poor places to start. There are many ways to define these concepts. The traditional versions are needlessly obscurantist and get the appropriate pedagogical/conceptual order backwards. The proper pedagogical definition for I is “the ratio of two vectors of the same magnitude which point perpendicularly in the plane”, or “the transformation which rotates vectors in the plane by a quarter turn”. Defining I as √(–1) and then working with purely formal quantities of the type a + bI is much harder to follow. It just seems completely arbitrary and invented (which is why people had trouble with it historically, and many students still do today). If you define I to mean a quarter turn, then having it square to a half turn makes perfect sense. Once you know that a half turn is equivalent to a reflection across the axis of rotation, then it’s pretty clear to see why a half turn of planar vectors should be written as the scalar –1. So the property that I^2 = –1 emerges naturally. Then when you sometime later start talking about logarithms, there’s a nice opening to talk about what the logarithm should be of a rotation. This ties in nicely with a discussion about position, velocity, and acceleration in uniform circular motion, etc. * * * It seems like there is some “deep insight” when you start with very obscure concepts/notation defined purely formally/abstractly and then fiddle with them a bit and suddenly out pops something simple and concrete. It makes for a good magic trick or punchline at the end of the tedious slog that is a traditional math course. But it’s better to start with simple concrete ideas and notation which are meaningful a priori.
  - SilasX 7y ago
    >Then when you sometime later start talking about logarithms, there’s a nice opening to talk about what the logarithm should be of a rotation Right, so you agree your explanation isn't covering the connection to the existing rules of i, and is just explaining how rotation works and giving it a symbol. That's great, but it has nothing to do with the insight that people are actually impressed by, which is that the concepts and rules created for a different domain (e.g. exponentiation and i as square root of -1) naturally extend in such a way that gives the Euler equation and so on. You're not explaining that insight at all, and shouldn't consider anyone to actually understand the insight if they get it. All you're explaining is the concept of point rotation. You shouldn't represent that as "oh, a child understands why exp(pi i) = -1, see, I explain it just fine!" If you prefer teaching things in that order, great! If you're dismissive of formal and abstraction notation, great! But you're not actually explaining the amazing insight everyone here is celebrating.
    - jacobolus 7y ago
      You seem to be misunderstanding what I am trying to say. Sorry if it was unclear. If I am explaining to small children that a half-turn rotation is the same as a reflection through the axis of rotation, then there is no symbol at all. Just words and physical manipulation. It’s a simple idea. If I am explaining to a high school student (or a well prepared 11 year old) what complex numbers mean, then we are going to start by doing a bunch of discussion of vectors in the context of geometry and mechanics. If I am explaining how logarithms work, we are going to attack them from many directions: iterated multiplication and compound interest, exponential growth/decay, velocity proportional to current position, uniform circular motion, ... There are many subtle and interesting concepts involved here. It’s worth taking them slowly and spending a few years building up fluency. There is indeed an insight that the logarithm of a rotation is proportional to arclength. That’s really the key insight we are talking about here. Why does that happen? That part is pretty interesting and worth exploring (but not nearly so difficult or mind blowing as it is made out to be). Iterated multiplication of rotations behaves similarly to iterated multiplication of scalars, and we can take logarithms of rotations and then add them just like we could do with logarithms of scalars. Uniform circular motion turns out to be a type of exponential growth. Now if you take a rotation R and look at the arclength from some vector x to Rx on the circle (an abstract circle in the space of displacement vectors) of squared radius x^2, and then you iterate R, you get proportional arclengths. That is, arclength(arc of origin-centered circle from x to RRx) = 2 × arclength(arc from x to Rx). And the extends to any other number of iterations of R. More generally if we compose two rotations then we add their arclengths. So arclength of the circular arc is proportional in general to the logarithm of rotation, and we can therefore use arclength as a model of the logarithm. (This is called “angle measure”, and we can physically measure it with a protractor). > concepts and rules created for a different domain The history of the understanding of astronomy, complex numbers, etc. is not really relevant at the introductory level. Slide rules and protractors are however useful tools to teach about. Leading off with power series, classical trigonometry, complex numbers defined as purely abstract formal objects invented for finding roots of polynomials, etc. is not “insightful”, it is just obfuscatory. It is a product of our current anachronistic approach to teaching which is based on training human computers even though we no longer need them and neglecting problem solving, and focusing almost entirely on algebra and a symbol-heavy framing of “trigonometry” and calculus at the expense of geometric understanding. At some later point it’s all fine and dandy to talk about the history of astronomy and chord/sine tables, the development of differential equations and the understanding of 2D and 3D rotation: Hipparchus, Archimedes, Ptolemy, Madhava, Al Kashi, Bürgi, Napier, Mercator, Newton, the Bernoullis, Euler, Argand, Rodrigues, Gauss, Cauchy, Riemann, Hamilton, Grassmann, Cayley, Maxwell, Gibbs, Clifford, Möbius, Klein, Lie, and all the rest. But not as an introduction.

SilasX 7y ago

>The most straight-forward approach is to define I to mean a quarter turn rotation anticlockwise in the plane Like I said originally, if you're just defining i to work this way, you're not conveying any of the insight behind why this is a natural extension of the existing rules/definitions of i, sine, exponentiation, etc. All your latest comment is doing is meticulously spelling out the concept of rotation in a plane. Again, that's not the hard(est) part of proving this result or of conveying the intuition. You're handwaving away 90% of it. It's not reasonable to characterize someone as "understanding" it because they get rotation, the last 10%.

jacobolus 7y ago
The “existing rules/definitions of i, sine, etc.” are poor places to start. There are many ways to define these concepts. The traditional versions are needlessly obscurantist and get the appropriate pedagogical/conceptual order backwards. The proper pedagogical definition for I is “the ratio of two vectors of the same magnitude which point perpendicularly in the plane”, or “the transformation which rotates vectors in the plane by a quarter turn”. Defining I as √(–1) and then working with purely formal quantities of the type a + bI is much harder to follow. It just seems completely arbitrary and invented (which is why people had trouble with it historically, and many students still do today). If you define I to mean a quarter turn, then having it square to a half turn makes perfect sense. Once you know that a half turn is equivalent to a reflection across the axis of rotation, then it’s pretty clear to see why a half turn of planar vectors should be written as the scalar –1. So the property that I^2 = –1 emerges naturally. Then when you sometime later start talking about logarithms, there’s a nice opening to talk about what the logarithm should be of a rotation. This ties in nicely with a discussion about position, velocity, and acceleration in uniform circular motion, etc. * * * It seems like there is some “deep insight” when you start with very obscure concepts/notation defined purely formally/abstractly and then fiddle with them a bit and suddenly out pops something simple and concrete. It makes for a good magic trick or punchline at the end of the tedious slog that is a traditional math course. But it’s better to start with simple concrete ideas and notation which are meaningful a priori.
- SilasX 7y ago
  >Then when you sometime later start talking about logarithms, there’s a nice opening to talk about what the logarithm should be of a rotation Right, so you agree your explanation isn't covering the connection to the existing rules of i, and is just explaining how rotation works and giving it a symbol. That's great, but it has nothing to do with the insight that people are actually impressed by, which is that the concepts and rules created for a different domain (e.g. exponentiation and i as square root of -1) naturally extend in such a way that gives the Euler equation and so on. You're not explaining that insight at all, and shouldn't consider anyone to actually understand the insight if they get it. All you're explaining is the concept of point rotation. You shouldn't represent that as "oh, a child understands why exp(pi i) = -1, see, I explain it just fine!" If you prefer teaching things in that order, great! If you're dismissive of formal and abstraction notation, great! But you're not actually explaining the amazing insight everyone here is celebrating.
  - jacobolus 7y ago
    You seem to be misunderstanding what I am trying to say. Sorry if it was unclear. If I am explaining to small children that a half-turn rotation is the same as a reflection through the axis of rotation, then there is no symbol at all. Just words and physical manipulation. It’s a simple idea. If I am explaining to a high school student (or a well prepared 11 year old) what complex numbers mean, then we are going to start by doing a bunch of discussion of vectors in the context of geometry and mechanics. If I am explaining how logarithms work, we are going to attack them from many directions: iterated multiplication and compound interest, exponential growth/decay, velocity proportional to current position, uniform circular motion, ... There are many subtle and interesting concepts involved here. It’s worth taking them slowly and spending a few years building up fluency. There is indeed an insight that the logarithm of a rotation is proportional to arclength. That’s really the key insight we are talking about here. Why does that happen? That part is pretty interesting and worth exploring (but not nearly so difficult or mind blowing as it is made out to be). Iterated multiplication of rotations behaves similarly to iterated multiplication of scalars, and we can take logarithms of rotations and then add them just like we could do with logarithms of scalars. Uniform circular motion turns out to be a type of exponential growth. Now if you take a rotation R and look at the arclength from some vector x to Rx on the circle (an abstract circle in the space of displacement vectors) of squared radius x^2, and then you iterate R, you get proportional arclengths. That is, arclength(arc of origin-centered circle from x to RRx) = 2 × arclength(arc from x to Rx). And the extends to any other number of iterations of R. More generally if we compose two rotations then we add their arclengths. So arclength of the circular arc is proportional in general to the logarithm of rotation, and we can therefore use arclength as a model of the logarithm. (This is called “angle measure”, and we can physically measure it with a protractor). > concepts and rules created for a different domain The history of the understanding of astronomy, complex numbers, etc. is not really relevant at the introductory level. Slide rules and protractors are however useful tools to teach about. Leading off with power series, classical trigonometry, complex numbers defined as purely abstract formal objects invented for finding roots of polynomials, etc. is not “insightful”, it is just obfuscatory. It is a product of our current anachronistic approach to teaching which is based on training human computers even though we no longer need them and neglecting problem solving, and focusing almost entirely on algebra and a symbol-heavy framing of “trigonometry” and calculus at the expense of geometric understanding. At some later point it’s all fine and dandy to talk about the history of astronomy and chord/sine tables, the development of differential equations and the understanding of 2D and 3D rotation: Hipparchus, Archimedes, Ptolemy, Madhava, Al Kashi, Bürgi, Napier, Mercator, Newton, the Bernoullis, Euler, Argand, Rodrigues, Gauss, Cauchy, Riemann, Hamilton, Grassmann, Cayley, Maxwell, Gibbs, Clifford, Möbius, Klein, Lie, and all the rest. But not as an introduction.