4 ms·

I think there's definitely something to your suggestion here. For instance, there is no result to sqrt(-1) on the real line, but it becomes possible in the comp

I think there's definitely something to your suggestion here. For instance, there is no result to sqrt(-1) on the real line, but it becomes possible in the complex plane because you have an additional degree of freedom. One qualifier I'd add though is that they're 'asymmetric' 2D numbers. The real vs imaginary axes have different behaviors. If I multiply by the positive unitary value on the imaginary axis (i.e. 'i'), I get CCW rotation by 90 degrees; with the negative imaginary unit I get CW rotation; positive real unit, no change occurs; negative real unit, rotation by 180 degrees. Actually that's something I wonder about—could there be a 'symmetric' version of that? Maybe something where the real axis behaves more like the imaginary axis, but maybe some signs are flipped or something. I guess an important aspect of how using the complex plane is beneficial is the fact that it combines these disparate elements though—we can start with something real, do certain transformations to it in our more broadly capable complex plane, then bring it back into the real line.

Jtsummers 8y ago
Reduce these to the integral powers of i: i^0, i^1, i^2, i^3. The symmetries occur two apart (0, 2; 1, 3). Higher integral powers are equivalent to the 4 listed, i^n = i^(n mod 4). Really, each power corresponds to an additional 90 degree rotation. 0th = 0 degree rotation (just scaling), 1st = 90 degree rotation, 2nd = 180 degree rotation, 3rd = 270 degree rotation (equivalently -90 degree). So that seeming CW rotation is also 3 CCW rotations.
- westoncb 8y ago
  Sure, I guess that's a consequence of general rotational symmetry. The main thing I was pointing out were the different geometric behaviors of multiplication by real vs imaginary units—not so much the difference between negative and positive units on either axis.
  - Jtsummers 8y ago
    Separate out the components, rotation and reflection and scaling. -1 is reflection. i is rotation by 90 degrees CCW. -i is both rotation by 90 degrees and reflection. Positive reals are scaling. So -ai is really rotation by 90 degrees, reflection, and then scaling by a.
    - westoncb 8y ago
      But none of that is related to my key point: multiplication by unitary values is significantly different depending on which axis the unit comes from (real vs imaginary), hence my usage of 'asymmetric'. In the last part of my first comment I point out the benefit (that I perceive) arising from the asymmetry. I also still wonder if a symmetric version (like I describe in the first comment) would be possible. Splitting the geometric behavior into rotation, reflection, and scaling is interesting—so thanks for the description—I just don't see how it relates to my comment.
marzell 8y ago
It would be great to get this level of explanation extrapolated all the way out to describe the behavior of octonions. Kidding, but not really ;)
yiyus 8y ago
You want geometric algebra. In 2D GA, you would have two unitary vectors, i and j, such that i * i = 1 and j * j = 1. The (non-commutative) product of between them (or their division) would be a bivector: i * j = ij = - j * i. You rotate 90 degrees the i and j vectors using the bivector. The good thing about GA is that the same concept can be easily extended to 3D (quaternions), and in fact to 4D and nD.
- westoncb 8y ago
  That does sound pretty close to what I had in mind. Are i and j from your example associated with orthogonal axes in a plane?
  - yiyus 8y ago
    Yes, they are. i would be the unitary vector in the horizontal direction, x, and j in the vertical one, y. In fact, geometric algebras are very general, and the one I briefly sketched is not the only possible interpretation. If you are interested, you can find many good introductions online, directed at different audiences. You can also search for the term "Clifford algebras" if you are interested in a more formal approach.

Jtsummers 8y ago

Reduce these to the integral powers of i: i^0, i^1, i^2, i^3. The symmetries occur two apart (0, 2; 1, 3). Higher integral powers are equivalent to the 4 listed, i^n = i^(n mod 4). Really, each power corresponds to an additional 90 degree rotation. 0th = 0 degree rotation (just scaling), 1st = 90 degree rotation, 2nd = 180 degree rotation, 3rd = 270 degree rotation (equivalently -90 degree). So that seeming CW rotation is also 3 CCW rotations.

westoncb 8y ago
Sure, I guess that's a consequence of general rotational symmetry. The main thing I was pointing out were the different geometric behaviors of multiplication by real vs imaginary units—not so much the difference between negative and positive units on either axis.
- Jtsummers 8y ago
  Separate out the components, rotation and reflection and scaling. -1 is reflection. i is rotation by 90 degrees CCW. -i is both rotation by 90 degrees and reflection. Positive reals are scaling. So -ai is really rotation by 90 degrees, reflection, and then scaling by a.
  - westoncb 8y ago
    But none of that is related to my key point: multiplication by unitary values is significantly different depending on which axis the unit comes from (real vs imaginary), hence my usage of 'asymmetric'. In the last part of my first comment I point out the benefit (that I perceive) arising from the asymmetry. I also still wonder if a symmetric version (like I describe in the first comment) would be possible. Splitting the geometric behavior into rotation, reflection, and scaling is interesting—so thanks for the description—I just don't see how it relates to my comment.

marzell 8y ago

It would be great to get this level of explanation extrapolated all the way out to describe the behavior of octonions. Kidding, but not really ;)

yiyus 8y ago

You want geometric algebra. In 2D GA, you would have two unitary vectors, i and j, such that i * i = 1 and j * j = 1. The (non-commutative) product of between them (or their division) would be a bivector: i * j = ij = - j * i. You rotate 90 degrees the i and j vectors using the bivector. The good thing about GA is that the same concept can be easily extended to 3D (quaternions), and in fact to 4D and nD.

westoncb 8y ago
That does sound pretty close to what I had in mind. Are i and j from your example associated with orthogonal axes in a plane?
- yiyus 8y ago
  Yes, they are. i would be the unitary vector in the horizontal direction, x, and j in the vertical one, y. In fact, geometric algebras are very general, and the one I briefly sketched is not the only possible interpretation. If you are interested, you can find many good introductions online, directed at different audiences. You can also search for the term "Clifford algebras" if you are interested in a more formal approach.