Abstract concept tying real numbers to elementary functions? Real numbers can be broken into two categories: rational vs. irrational. Irrational numbers can be approximated, but never fully represented by rational numbers.
Analytic functions have Taylor polynomial representations. I know that polynomials have finite taylor expansions (trivially), while transcendental functions have infinite taylor expansions, which we truncate by necessity. So it seems again we have an easy-to-represent function (polynomials) with can be added infinitely to represent (or finitely-many times to approximate) an analytic function.
Similarly, other functions have Laurent expansions, and I can categorize them as having finite vs. infinite expansions…
I sense there’s a deeper abstraction here. Having read the first two chapters of an Abstract Algebra textbook, the word “isomorphism” comes to mind. From Analysis 1 I also know that these objects can be used to form metric spaces. Is there a deeper pattern here?
For example: real numbers can be added or multiplied to obtain other reals, and polynomials can be added or multiplied to obtain other polynomials (and from there, transcendentals). This reminds me of the field properties…
 A: You're asking some interesting questions, but there are a few misconceptions in your question that are worth clearing up.


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*"If I represent them using decimal notation, rational real numers have finitely many digits, while irrational numbers have infinitely many"
This is incorrect; just look at e.g., $1/7 = .1428571428\ldots$ . I strongly encourage you to consider anything around decimal or base expansions as largely a 'red herring' - it's not that there's nothing interesting there, but thinking of numbers in terms of base expansions tends to lead people rapidly astray. A rational number is 'just' a number that can be represented as a ratio of integers, and that's by far the most consistent way of thinking about it. The fact that rationals have eventually-repeating decimal expansions (and that any eventually-repeating decimal expansion represents a rational number) is then a theorem, not a definition.
"Polynomial functions have finite Taylor expansions (trivially), while transcendental functions have infinite Taylor expansions"
It's not entirely clear what you mean by 'transcendental function' here, but in fact the set of finite Taylor series is exactly the set of polynomials; you mentioned the triviality one direction, but it goes both ways trivially.  Note that this has some interesting consequences; for instance, this is one way of seeing that the $n$th derivatives of any given polynomial will be identically zero for all $n\geq$ some $n_0$.
You're also making a subtle analog of the same red herring that you made around numbers, because Taylor expansions are a poor way of thinking about many ('most', in some sense) functions; they work well for analytic functions, but that's almost by definition, and many functions (including some elementary ones) aren't analytic; look at, e.g., the 'Taylor expansion' of $e^{-1/x^2}$ about $x=0$.
That said, there are some analogies here that you could dive more deeply into if you wanted; for instance, Taylor series for rational functions are special in a way that corresponds loosely to the way that the expansion of rational numbers; they're 'rationally generated' in a sense, as the coefficients of any rational function satisfy a linear recurrence relation, and there are some connections here to regular expressions, context-free languages, etc.  You could also ask about cardinalities of some of the relevant sets: for instance, there are only countably many rational functions (over $\mathbb{Z}$), in the same way that there are only countably many rational numbers, and while there are uncountably many rational functions over $\mathbb{R}$, or elementary functions, or even continuous functions, in each case we can show that these functions are a 'small' subset of the set of all functions, in that they have cardinality $\mathfrak{c}$, whereas the cardinality of the set of all functions is $2^{\mathfrak{c}}$.  This starts to approach some of the topological aspects of the questions you're asking. Etc, etc...
