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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.

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    $\begingroup$ I don't think this gives you the real numbers. According to that definition, $0.1999\cdots \neq 0.2$. Established constructions solve that problem by not having decimal expansion be fundamental, but just a different way to refer to well-defined elements of $\mathbb R$ or $\mathbb Q$. $\endgroup$
    – silver
    Jun 29, 2021 at 18:51
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    $\begingroup$ Relying on any specific representation (decimal or otherwise) is perhaps contrary to really understanding what a real number is ... $\endgroup$
    – Hagen von Eitzen
    Jun 29, 2021 at 18:54
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    $\begingroup$ You have the problem that terminating decimals have two representations, but the functions are different. $\endgroup$
    – saulspatz
    Jun 29, 2021 at 18:54
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    $\begingroup$ Maybe the right question isn't whether real numbers can be viewed this way, but what's the value in viewing real numbers this way. Synergizing with intuition is indeed valuable! But many important aspects of real numbers are hard to express in this formalism (just adding two together, for example, much less the least upper bound property). $\endgroup$
    – Greg Martin
    Jun 29, 2021 at 18:55
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    $\begingroup$ Good question. Possible a duplicate of this one, which is clearly relevant. What is so wrong with thinking of real numbers as infinite decimals? $\endgroup$
    – Ethan Bolker
    Jun 29, 2021 at 19:18

3 Answers 3

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There are a couple of definitions of real numbers (using the rational numbers as a starting point) that are pretty standard. They are also necessarily equivalent to each other, because it's a theorem that there is a unique complete ordered field.

One definition defines an equivalence relationship between Cauchy sequences of rational numbers, and then defines each equivalence class to be a real number. You can think of this as the equivalence class of Cauchy sequences that converge to the real number you care about. The other definition (and the one I find more intuitive) uses "Dedekind cuts" -- a set of rational numbers that is bounded above and that is closed downward, which you can think of as the set of rational numbers less than the real number you care about.

These definitions are standard because they are useful. Using Dedekind cuts, for instance, it's quite easy to prove that the reals are a complete ordered set. It's a little more tedious to define the field operations but it's pretty straightforward to do so. So I guess the question becomes, once you correct your definition to account for terminating fractions, what makes it useful for you? Perhaps you'd find it a useful exercise to prove rigorously that your proposed definition is in fact equivalent to one of the standard definitions. One challenge you'll face is defining field operations in such a way that they are well defined; in other words, so that the two different equivalent representations of a terminating decimal give you equivalent results.

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    $\begingroup$ Thank you. This was very informative. $\endgroup$
    – Alex
    Jun 29, 2021 at 21:29
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There is an axiomatic approach to the real numbers which you can find in some advanced calculus textbooks (for example Fitzpatrick's "Advanced Calculus"). In those books, theorems about real numbers are proved from just the axioms alone, much as theorems about plane geometry can be proved starting from (a modern version of) Euclid's axioms alone.

One of those theorems says that every real number has exactly one or two decimal expansions, with the case of two decimal expansions always following a very specific pattern that generalizes the (in)famous equation $1.00000... = .99999.....$ I find this an interesting theorem, although to be honest it's really not that important in developing the theory, it's more of a "reality check" in which one compares the theory with our intuition for decimal expansions, in order to make sure that the theory and intuition are in accord with each other. For that reason I somehow never get around to proving that theorem in my Advanced Calculus course.

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You are correct that real numbers can be represented in this way as long as you take care to deal with the $1 = 0.9\dots$ issue. It is also not that difficult to show that such a function based representation can be transformed into Cauchy sequences or Dedekind cuts. Normally this representation isn't that useful. However it does have one useful property. Cantor's diagonal proof of the uncountablity of the real numbers is particularly clean in this model.

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  • $\begingroup$ The equivalence relation could be defined to take care of . ... n_1 n_2 ... 9 ...; i.e., a decimal representation with a tail of infinite 9s, perhaps best by first defining a "normal" form decimal representation: a real number f_1 is normal iff there does not exist some n in Z such that f_1(x) = 9 for all x more negative than n. To normalize some f_1, find the least negative n for which the above holds, and create a new function which is f_1(x) for all x less negative than n, f_1(x) + 1 for x = n, and 0 for x more negative than n. $\endgroup$
    – Alex
    Jul 27, 2021 at 1:40
  • $\begingroup$ Then, define a relation "equivalence of normal form real numbers" as follows: two normal (as defined in the previous comment) real numbers f_1, f_2 are equal iff for all n in Z, f_1(n) = f_2(n). Then, define the general equivalence relation as: two real numbers f_1, f_2 are equal if their normal forms are equal as previously defined. Thus, the .999... issue is handled. $\endgroup$
    – Alex
    Jul 27, 2021 at 1:45

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