From time to time, I think about how material from introductory courses like real analysis or linear algebra can be structured in a way I would have liked to see in my freshman days.
So recently, I was wondering about $\mathrm π$ and polar coordinates. I want them to be introduced as soon as possible and by the use of complex numbers.
What I am looking for is an elegant and elementary way of establishing the following:
The map $ℝ → S^1,~ϑ ↦ \mathrm e^{\mathrm i ϑ}$ is a surjective group homomorphism with kernel $\mathrm τℤ$ for some unique $\mathrm τ ∈ ℝ$ with $τ > 0$.
Then I would go on to define $\mathrm π = \frac {\mathrm τ} 2$ and so on. So my question is:
Given as few notions as possible from topology and differentiability, with the minimum of a solid foundation in real and complex sequences and series, how can this result be proven in a preferably conceptual way?
The main problem is that I don’t know of any way to prove the above mentioned statement without the use of much topology. What really bugs me is surjectivity and the kernel of the map.
For example, to establish that the kernel of the map given above is a cyclic subgroup, I would argue that, by continuity of $\mathrm{exp}$, it’s a closed subgroup which cannot be all of $ℝ$ and then show that its generated by the smallest positive number in it. Then I could prove the surjectivity by the connectedness-argument suggested by Robert Israel in this related question. Using a differential-equation-argument, the functional equation of $\mathrm{exp}$ follows easily.
But can you help me doing this with less prerequisites?
The German textbook "Analysis" by O. Forster does basically this (but does not use algebraic terms): Define the complex exponential by the power series, prove its continuity and then define $\cos(x) $ as real part of $\exp(ix) $ for real $x$. Then show that the cosine is monotonically decreasing on some appropriate interval (e.g [0,2]) from 1 to a negative value. Conclude that there is a smallest root in this interval and call it $\pi/2$. Everything else can than be deduced easily.
This is pretty quick but leaves open when all this has to do with the circle (after all, you could have defined the cosine as the real part of $a^x$ for some $a$...). To show that $\pi/2$ is the area of the half unit circle you can integrate the function $\sqrt{1-x^2}$ but this need a notion of integral and the fundamental theorem of calculus.