I suggest to look at the literature listings of any undergrad math curriculum to get suggestions on what to start with. There are reasons 'linear algebra' is usually taught before 'Applications of the representation theory of Lie groups in special relativity'.
Or maybe I'm too conservative in my view on math education?
I would possibly throw this out as an interesting way to learn a bit (not really a whole lot, but a fun start) about abstract algebra: https://www.crypto101.io/
A ring/field-theoretic construction of complex numbers as a quotient R[X]/(X² + 1), an extension of the field R by the roots of the polynomial, is interesting/neat, and can be generalized in various interesting ways for number theory, algebraic geometry, etc., but is not an effective/intuitive early understanding for someone who wants to use them in geometry, physics, general data processing, etc., because most of its content is purely formal/abstract.
Thinking of the complex numbers as quotients of vectors in the Euclidean plane (under the geometric product) is a much, much more fruitful and enlightening interpretation, and makes the main theorems of complex analysis make clear geometric sense.
Most importantly, the basics of geometric algebra are very accessible to e.g. high school students. It could be used to completely replace typical courses in trigonometry, partially replace & supplement a Euclidean geometry course, partially replace & supplement instruction in solving systems of equations and basic matrix algebra, etc. A curriculum which was wholesale infused with GA tools and reasoning would give students more powerful tools and more vocabulary earlier, would help them solve harder problems in most of their math/physics courses, and would ultimately unify and simplify a lot of the mathematical modeling tools used throughout any undergraduate technical education.
To learn group theory, ring theory, etc., you want to start with a nice stable of explicit concrete examples, explored deeply (I would recommend Nathan Carter’s book Visual Group Theory as a start for someone with a high school math or undergraduate engineering background). Otherwise the theorems and proofs are just symbol twiddling and pattern-matching to some assignment handed down from above. Ideally students would start in on some examples in primary/secondary school (tessellations, transformation geometry, modular arithmetic, some basic number theory, polynomials treated as vectors, ...), and be pretty well prepared for an abstract treatment by the time they are undergraduates. Unfortunately there’s not much time for this in a typical undergraduate mathematics sequence, so most classes in those subjects are done entirely abstractly from axioms, with sparse examples only examined superficially. Many if not most students leave these courses very confused about the meaning of what they have been doing. GA provides some great examples for an abstract algebra course.
If you care about powerful abstract tools, (both finite and Lie-type) groups can be embedded in a geometric algebra and studied using GA tools in a similar way they can be embedded in linear algebra and studied as matrix groups (“representation theory”). In many cases the geometric algebra representation is both easier to understand and gives more tools to use. Both projective and affine geometry can be done with geometric algebra, and differential geometry can be recast as “geometric calculus” which reveals some structure usually hidden in the typical differential form version. See the papers at http://geocalc.clas.asu.edu/html/GeoAlg.html http://geocalc.clas.asu.edu/html/GeoCalc.html
On point! Like I feel my understanding is grounded in something I can reason about myself as opposed to trying to see the relationships that someone else came up with.
I also feel like most commenters misunderstood what I meant when I said “what complex numbers are about”.
What other math books can you recommend?