Mathematicians get enamored with particular ways of looking at things, and fall into believing this is gospel. I should know: I am one, and I fight this tendency at every turn.

On one hand, "rational" and "algebraic" are far more pervasive concepts than mathematicians are ever taught to believe. The key here is formal power series in non-commuting variables, as pioneered by Marcel-Paul Schützenberger. "Rational" corresponds to finite state machines, and "Algebraic" corresponds to pushdown automata, the context-free grammars that describe most programming languages.

On the other hand, "Concrete Mathematics" by Donald Knuth, Oren Patashnik, and Ronald Graham (I never met Oren) popularizes another way to organize numbers: The "endpoints" of positive reals are 0/1 and 1/0. Subdivide this interval (any such interval) by taking the center of a/b and c/d as (a+c)/(b+d). Here, the first center is 1/1 = 1. Iterate. Given any number, its coordinates in this system is the sequence of L, R symbols to locate it in successive subdivisions.

Any computer scientist should be chomping at the bit here: What is the complexity of the L, R sequence that locates a given number?

From this perspective, the natural number "e" is one of the simpler numbers known, not lost in the unwashed multitude of "transcendental" numbers.

Most mathematicians don't know this. The idea generalizes to barycentric subdivision in any dimension, but the real line is already interesting.