You seem to be misunderstanding what I am trying to say. Sorry if it was unclear.

If I am explaining to small children that a half-turn rotation is the same as a reflection through the axis of rotation, then there is no symbol at all. Just words and physical manipulation. It’s a simple idea.

If I am explaining to a high school student (or a well prepared 11 year old) what complex numbers mean, then we are going to start by doing a bunch of discussion of vectors in the context of geometry and mechanics.

If I am explaining how logarithms work, we are going to attack them from many directions: iterated multiplication and compound interest, exponential growth/decay, velocity proportional to current position, uniform circular motion, ...

There are many subtle and interesting concepts involved here. It’s worth taking them slowly and spending a few years building up fluency.

There is indeed an insight that the logarithm of a rotation is proportional to arclength. That’s really the key insight we are talking about here. Why does that happen? That part is pretty interesting and worth exploring (but not nearly so difficult or mind blowing as it is made out to be).

Iterated multiplication of rotations behaves similarly to iterated multiplication of scalars, and we can take logarithms of rotations and then add them just like we could do with logarithms of scalars. Uniform circular motion turns out to be a type of exponential growth.

Now if you take a rotation R and look at the arclength from some vector x to Rx on the circle (an abstract circle in the space of displacement vectors) of squared radius x^2, and then you iterate R, you get proportional arclengths.

That is, arclength(arc of origin-centered circle from x to RRx) = 2 × arclength(arc from x to Rx). And the extends to any other number of iterations of R. More generally if we compose two rotations then we add their arclengths.

So arclength of the circular arc is proportional in general to the logarithm of rotation, and we can therefore use arclength as a model of the logarithm. (This is called “angle measure”, and we can physically measure it with a protractor).

> concepts and rules created for a different domain

The history of the understanding of astronomy, complex numbers, etc. is not really relevant at the introductory level.

Slide rules and protractors are however useful tools to teach about.

Leading off with power series, classical trigonometry, complex numbers defined as purely abstract formal objects invented for finding roots of polynomials, etc. is not “insightful”, it is just obfuscatory. It is a product of our current anachronistic approach to teaching which is based on training human computers even though we no longer need them and neglecting problem solving, and focusing almost entirely on algebra and a symbol-heavy framing of “trigonometry” and calculus at the expense of geometric understanding.

At some later point it’s all fine and dandy to talk about the history of astronomy and chord/sine tables, the development of differential equations and the understanding of 2D and 3D rotation: Hipparchus, Archimedes, Ptolemy, Madhava, Al Kashi, Bürgi, Napier, Mercator, Newton, the Bernoullis, Euler, Argand, Rodrigues, Gauss, Cauchy, Riemann, Hamilton, Grassmann, Cayley, Maxwell, Gibbs, Clifford, Möbius, Klein, Lie, and all the rest. But not as an introduction.