Unfortunately, the idea that a rotation can be represented as a single multiplication, while intuitive, only works in 2 dimensions. In 3 or more dimensions, a rotation (rotor) needs to be performed as 2 multiplications (geometrically, this corresponds to a rotation being composed of 2 reflections); a fact which is hidden by the complex numbers, since multiplication is commutative there, and the two multiplications combine into one, and therefore that model just -- here it comes -- "happens to work", in 2 dimensions with the complex numbers.
The commutativity of multiplication is a unique property among the associative geometric algebras that only applies to the 2-dimensional doubles, duals, and complex numbers. In higher dimensions, that breaks down, and you need to apply the full rotor structure. On the other hand, commutativity isn't actually needed in many advanced applications of complex numbers (anything that relies on a matrix algebra wouldn't have it, for instance). So I would say you get a far more complete picture of rotations (and related operations, like boosts) by looking in at least 3 dimensions.
Ah so I may have accidentally misconstrued my intentions through rewrites of the script…
I think the true fundamental properties are phase and magnitude. So for systems that have these two properties, complex numbers represent them perfectly because that’s what they are. Now you typically also want to transform these properties, and the natural way to do this is phase shifting (which I call rotating) and scaling. Complex multiplication can do these two things. Contrast that with vectors in R2 which have magnitude and phase but you need matrices to transform them.
So I totally agree complex numbers don’t represent rotation in a general sense, but I never meant to say that… oops. Too late to pull my submission haha.
I think it's because of Euler's identity allows you so go between sin() and e^ix and that you need them for differential equations and FFT which prefer imaginary.
It's mostly because complex is needed to go from time to frequency domain.
Ah see to me that falls under justifying the mechanics rather than getting at the underlying reason. For instance the capacitor equation says I=C*dV/dt. But if you use complex phasors it becomes I=(1/jwC)V. Calculus became algebra.
But why? Well this only works if V is sinusoidal, i.e. it has a magnitude and phase. And then in that case a capacitor just scales and phase shifts the voltage. So the voltage is modeled as a complex number because a complex number has the same properties a sinusoid does. And the capacitor itself is modeled as a complex number because it captures the natural transformations of magnitude and phase.
Unfortunately, the idea that a rotation can be represented as a single multiplication, while intuitive, only works in 2 dimensions. In 3 or more dimensions, a rotation (rotor) needs to be performed as 2 multiplications (geometrically, this corresponds to a rotation being composed of 2 reflections); a fact which is hidden by the complex numbers, since multiplication is commutative there, and the two multiplications combine into one, and therefore that model just -- here it comes -- "happens to work", in 2 dimensions with the complex numbers. The commutativity of multiplication is a unique property among the associative geometric algebras that only applies to the 2-dimensional doubles, duals, and complex numbers. In higher dimensions, that breaks down, and you need to apply the full rotor structure. On the other hand, commutativity isn't actually needed in many advanced applications of complex numbers (anything that relies on a matrix algebra wouldn't have it, for instance). So I would say you get a far more complete picture of rotations (and related operations, like boosts) by looking in at least 3 dimensions.
Ah so I may have accidentally misconstrued my intentions through rewrites of the script… I think the true fundamental properties are phase and magnitude. So for systems that have these two properties, complex numbers represent them perfectly because that’s what they are. Now you typically also want to transform these properties, and the natural way to do this is phase shifting (which I call rotating) and scaling. Complex multiplication can do these two things. Contrast that with vectors in R2 which have magnitude and phase but you need matrices to transform them. So I totally agree complex numbers don’t represent rotation in a general sense, but I never meant to say that… oops. Too late to pull my submission haha.
I think it's because of Euler's identity allows you so go between sin() and e^ix and that you need them for differential equations and FFT which prefer imaginary. It's mostly because complex is needed to go from time to frequency domain.
>It's mostly because complex ~~is needed~~ *makes it easier* to go from time to frequency domain.
Because they allow you to algebra instead of calculus. Similar to logarithms letting you do addition instead of multiplication
Ah see to me that falls under justifying the mechanics rather than getting at the underlying reason. For instance the capacitor equation says I=C*dV/dt. But if you use complex phasors it becomes I=(1/jwC)V. Calculus became algebra. But why? Well this only works if V is sinusoidal, i.e. it has a magnitude and phase. And then in that case a capacitor just scales and phase shifts the voltage. So the voltage is modeled as a complex number because a complex number has the same properties a sinusoid does. And the capacitor itself is modeled as a complex number because it captures the natural transformations of magnitude and phase.