Nevertheless, there are still things that I find fascinating. Take e.g. the imaginary number
i that we were talking about. With Euler's formula
e^ix = cos(x) + i*sin(x) we relate complex exponentials to oscillations. This gives rise to the
Fourier transform where signals in the time domain can be expressed by signals in the frequency domain. It can be extended with the
Laplace transform that is very useful to express differential equations as algebraic equations. In
Control Theory, working in the s-domain (Laplace Transform) is very common and it impresses me how we can mathematically construct complex dynamical systems by interconnecting simple block diagrams in the s-domain.
Of course, there is a caveat, and that is that it is only applicable to linear systems. I suspect that it is the the "superposition" property of linear systems that allows these mathematical tools to be so effective. In reality, systems tend to be nonlinear so we have to simplify by necessity.
At the university, I had a physics teacher who joked that phycisists would look at nonlinear systems and kept looking until it became a bit more linear and continued looking until it became fully linear so that they could start working
.
)
