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If anyone needs this expanded analysis for anything else, send a PM.
Why did you make use of the commutativity property of the identity element in addition? Such an action made your computation unnecessarily specific to a sub-field of algebra and negated the usefulness of the procedure for vectors, matrices, and integration with analysis.
OK, can you solve the same problem with tensors, manifolds, elliptical functions, Legendre Polynomials and directional derivatives and make it easy for laypeople to understand?
Should be doable with number theory and ring theory...you may be able to use group theory instead of ring theory, leading to a more generalized solution, but I'm not 100% certain, I'd have to look at the problem closer.
We could get rid of algebra altogether and use graph theory, of course, but the real question is whether we could dispense with number theory, the Queen of Mathematics? Perhaps an approach using set theory would work.
What do you think 'laypeople' would find easier? Algebra was never very intuitive for me, maybe the use of number theory and graph theory, minimizing with an epsilon-delta argument from analysis?