@Oscar Lanzi: the midpoint of the arc certainly doesn’t lie on any diagonal of the outer pentagon. Two respective isosceles triangles, mirror-symmetric, have angles 54° (marked with P on the drawing) and 63°, whereas all angles between sides and diagonals of a regular pentagon are multiples of 36°. Anyway some positions on the arc are not eligible due to collinearity constrain.

For c = ±1 al least two given points coincide and, moreover, there is a Re = const line (namely, Re = −1 for c = 1 and Re = 0 for c = −1) having three distinct points on it. The same for powers-of-ξ multiples. Why didn’t the author exclude bad values of c explicitly?

Just a quibble: $u\bar v + v\bar u$ is an inner product on ℍ, namely the standard inner product by the factor of 2.

Clifford algebra is a way more arcane than complex numbers. The fact that ℂ is a division algebra (while extensions of ℝ using square roots of positive numbers are not) explains why the minus sign in Clifford multiplication is preferable, but a Clifford algebra hasn’t necessary to be a division algebra, at the end. Ī̲ wouldn’t deem explaining simpler things via complicated things practical.

The same as for mathoverflow.net/a/30185/56921 — vector calculus is not complex calculus! Complex numbers certainly are a good example of an inner product space, and to teach vectors you can use the example of ℂ, but complex multiplication and (especially) division go beyond the vector-based intuition.

This is a conceptual mess. Basic probability theory absolutely doesn’t need complex numbers but—applied to the real world—it is a simplification just like any other theory, QM included. Surely we can (and do) reduce complex amplitudes to dumb probability measures using |·|², but it has nothing to do with “correct description of probability theory”. It is indeed about real-world randomness which extends far beyond Kolmogorov-style probability. Can anybody rewrite, please?

There is some hype about alleged physical relevance of p-adic numbers (in fact, Ī̲ know in person several researchers who built their career upon it), but—as for $\mathbb{Q}_p$—these are either overreaching generalizations or barren conjectures, whereas complex numbers are justified by quantum mechanics alone.

How does the vector calculus identity explain complex numbers? It is about an inner product space, true in every dimension (and not necessary for a positive-definite scalar product). There is dot product in every $\mathbb{R}^n, n\ge 1$, but only for $n = 1, 2, 4$ multiplication exists (and only for $n = 1, 2$ is it commutative). Downvote.

How is Dirac theory based on (parabolic) Schrödinger equation? We know that a solution to Dirac equation is a solution to Klein–Gordon equation which is hyperbolic.

Ī̲ wouldn’t say it demystifies anything.

To be pedantic: the empty subset is open everywhere. If you mean “on all fibers of $π$ over $U$”, then wouldn’t it be vacuously true for empty $U$?

Respect for the adjective “non-constant” (although in the full generality it should be stated more cautiously: not locally constant anywhere).

Wouldn’t be helpful to observe that both $R_2$ and $R_4$ are not faithful in the same manner? Ī̲ mean that −1 lies in their kernel, hence both, in fact, represent PSL(2)—for k = ℂ known in physics as the special orthochronous Lorentz group—and all spaces mentioned here are endowed with a PSL(2) representation, not merely SL(2).

Oops, Ī̲ missed that $X$ is not (necessarily) one-dimensional.