The complex numbers do not necessarily come from "conservation of energy"
Indeed, there are other ways to explain conservation of energy, basically by invoking kinetic energy; and there are other ways to justify complex numbers in wave phenomena.
the complex numbers are the eigenvalues of your basic oscillator x'' + w^2*x = 0
Let me take this apart:
those complex numbers are eigenvalues because eigenvalues are obtained by factoring the characteristic polynomial of the matrix.
And in this case, we are talking about the functions in a Hilbert space, with eigenvalues of the differential equation
x'' + w^2*x: The eigenfunctions are just e^-iwx and e^iwx, w is the eigenvalue. Our "matrix" in this case is the Hilbert space L2. It's not like you can take the determinate of a set of functions, the linear operator can be factored:
x'' + w^2*x=(D^2 + w^2)x=(D + w*i)(D - w*i)x, where D is the differential operator, Your eigenfunctions are the solutions to : (D + w*i) and (D - w*i) We can interchange the orders of (D + w*i) and (D - w*i) so, the set of eigenfunctions is the solutions to either (D + w*i)=0 or (D - w*i)=0
It's the same result you have for a matrix, except you have different linear operators, like matrix multiplication instead of the D's. Then, you find the eigenvalues of your matrix operations like, Ax-bI =0, where I is the identiry operator
The Fourier transform is an operation which can be performed using NxN matrix multiplication on a sampled signal of length N, although in the Hilbert space, we use the inner product on L2. It's no surprise that the matrix F, for the Fourier transform, X(f)=Fx(t), can be diagonalized, because it is unitary, and all of its eigenvalues are 1.
Factorization of elements of R[L] (polynomials with Real coefficients with a single variable called L) can yield irreductible elements of degree 2, that is, L*L + positive constant.
The reason for introducing complex numbers is that they make factorization smoother by allowing all polynomials to be factored down to terms of degree 1. And then "L*L + positive constant = 0" means "L*L = negative constant", so the only way to find L here is to invent a number whose square is a negative constant.
Inventing extra numbers is allowed as long as they stay consistent with the number system they are based on. So the Complex numbers are called an Extension of the Real numbers because + - * / on Complexes are intuitive extensions of those same operations on Reals.
Indeed, playing with Complexes feels like playing with a very limited version of polynomials on Reals, so you can do it with 8th grade algebra.
Mathieu Bouchard - tél:+1.514.383.3801 - http://artengine.ca/matju Freelance Digital Arts Engineer, Montréal QC Canada
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