At the end of the day, a real number can be viewed simply as a function over the integers —> the naturals which tells you the digit as that ten’s place (assuming base ten)? You could augment this and say it’s a tuple (sign, function) to add +/- information.
This is the most general definition of a real number, right? I think I find confusion sometimes in wondering how numbers like pi and e are just numbers; but in reality they are just their definitions, and to pull out a digit at a given place you need to consult the definition. So in short these numbers contain more information than one might guess at first.
When I view real numbers this way, I feel I really understand what a number is.
I hope this makes sense. I only have a minor in math.
EDIT: My real goal is understanding: what information does a real number provide? What information "is" a number, crudely put? I first started thinking of this formalism when considering numbers like e and pi, which "just happen" to be 2.7... but really contain "infinite" information and are more clearly characterized by their expression as summations/whatever than as decimal numbers.