if this definition obscures the geometric meaning, then why use it in the first place? Is there a good reason for using this definition? I've always interpreted complex numbers as being 2 dimensional vectors and never really understood why sqrt(-1) is needed
Max
You can define a complex number as a+bi, where i is a new type of unit such that i*i = -1. This tells you how to multiply complex numbers. Hence you could also say i = sqrt(-1), but saying so is not particularly meaningful.
I think interpreting complex numbers as 2D vectors is the better approach, but then again they're not exactly vectors, as they have this multiplication rule and form a field.
keenan
@idontknow In spite of what I say in lecture, there are good reasons for thinking of complex numbers in terms of roots and so forth---that's just not the most helpful point of view for doing computer graphics/geometry. (And is also not helpful for making the conceptual leap to quaternions.). If your interest is instead developing abstract algebra, then complex numbers are really nice because all polynomials have complex solutions. This fact in turn simplifies subjects that build on complex numbers, such as algebraic geometry. Occasionally, these topics do come back around to provide useful perspectives on computer graphics---for instance, the study of algebraic varieties shows up when trying to represent volumetric frame fields for 3D meshing. But for just performing spatial transformations, it really is better to start out by thinking of $i$ as a 90-degree rotation, $e^{i\theta}$ as a rotation by $\theta$ radians, $ae^{i\theta}$ as a scaling by $a$ and rotation by $\theta$, etc. The book Visual Complex Analysis takes this perspective much further in a very nice way.
if this definition obscures the geometric meaning, then why use it in the first place? Is there a good reason for using this definition? I've always interpreted complex numbers as being 2 dimensional vectors and never really understood why sqrt(-1) is needed
You can define a complex number as a+bi, where i is a new type of unit such that i*i = -1. This tells you how to multiply complex numbers. Hence you could also say i = sqrt(-1), but saying so is not particularly meaningful.
I think interpreting complex numbers as 2D vectors is the better approach, but then again they're not exactly vectors, as they have this multiplication rule and form a field.
@idontknow In spite of what I say in lecture, there are good reasons for thinking of complex numbers in terms of roots and so forth---that's just not the most helpful point of view for doing computer graphics/geometry. (And is also not helpful for making the conceptual leap to quaternions.). If your interest is instead developing abstract algebra, then complex numbers are really nice because all polynomials have complex solutions. This fact in turn simplifies subjects that build on complex numbers, such as algebraic geometry. Occasionally, these topics do come back around to provide useful perspectives on computer graphics---for instance, the study of algebraic varieties shows up when trying to represent volumetric frame fields for 3D meshing. But for just performing spatial transformations, it really is better to start out by thinking of $i$ as a 90-degree rotation, $e^{i\theta}$ as a rotation by $\theta$ radians, $ae^{i\theta}$ as a scaling by $a$ and rotation by $\theta$, etc. The book Visual Complex Analysis takes this perspective much further in a very nice way.
This slide reminds me of Mohamed Ababou's fantastic paper mind+logic https://vixra.org/abs/1910.0134