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I've never considered that (in itself) a satisfying or complete explanation, because it just handwaves away the problem of how it is that imaginary exponentiati

by SilasX 7y ago

I'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 ago
Well, 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
  >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.

by SilasX 7y ago

heavenlyblue 7y ago

Well, 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
>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.