> So you need at least that one, since it is by convention, and not really derivable[1]:
exp(ix) = cos(x) + i*sin(x)
I think if you've read the cognitive science POV on this (I think I read Lakoff's books first), it helps rederive that. Keep in mind that Euler's notation is just notion (I also recommend visual complex analysis). It's better from a cognitive neuroscience to pretend to be Euler himself and invent it yourself (literally embody his cognitive process). It's just concept blending between stuff you already know and stuff you didn't know you know but hadn't put a link or thought to denote a name, but are quite natural things if you want to develop the intuition for understand complexity.
From a computer engineering point of view, often a Fourier series expansion of an arbitrary periodic time signal. Like a sinc function. Usually in analog signal processing, it depends on approximation and either quadratic forms or hyperbolic transcendral forms are useful, and knowing in what dimension a "perfect" computational equivalent exists as function of a complex conjugate. Going back in time, this seems to be why the Shannon-Nyquist rate was incessantly in people's heads for about 30 years. It let us to the digital and discrete switchover in the '90s. It just caused a shift from the telecom companies, what "content" was worth, then a boom then bust, then boom cycle for the new internet companies. I think these days, with the advent of huge corpus of text and higher order content (whether it be source code, neural nets, etc) mined from the internet available for free, I think with all of the global issues taking place right now that probably require transnational mutually assured prevention of tragedy of the commons situations, but it seems like it needs much more interdisciplinary blurring of people's fields.
Now, if you think about where you need to actively design controllers (traditionally this was the realm of control theory) - think about non-linear dynamics and modern robotics, actually it doesn't even needed to be robotics, it can be any cyber-physical system, which is /all systems/ in general, this is why data flow architectures are being shifted towards rather than classical control flow architectures in a computer science/computer engineering POV. You will have to develop the intuition of stopping time, etc. There are "deeper" intuitions involved with combinatorics and natural numbers, this is probably the nexus of computer, neuroscience, medicine, physics, philosophy, linguistics, math, logic, science, engineering, etc. Actually, you can disregard numbers (but still appreciate use number theory, I think it's best to realize you're basically sharing some cross cultural understanding), while still doing arithmetic. Sometimes this is very useful for physical (real) systems where you have to make computation tradeoffs between mathematical physics, stability, natural language, etc.
If you have a deeper understanding, from my limited understanding of the biosciences, all of these are relatively translatable between mtRNA/RNA scales (where we understand the chicken/egg problem as simply codons, but proteins in effect are "simulating" what we call heat transference problems/radiative transfer problems. But a lot of genomic data isn't decoded yet.