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Can we take the decimal (or any arbitrary base) representation of a real number and just append some more digits beyond it? Is there a theory that covers this, maybe some kind of non-standard analysis?

E.g.: $$x=0.x_1x_2x_3\ldots\in[0,1]$$ Let's just append some more digits: $$x^*=0.\{x_1x_2x_3\ldots\}\{y_1y_2\ldots\}\{\ldots\}\{\ldots\}\ldots\in \text{ a non-standard version of }[0,1] $$ where $\{y_1y_2\ldots\}$ and each subsequent $\{\ldots\}$ is just another infinite string of digits.

So $y=0.\{\overline0\}\{y_1y_2\ldots\}$ would be like an infinitesimal, and we could say something like $0<y<\epsilon$ for every real number $\epsilon>0$. The $y_i$ digit string could be a non-standard infinitesimal base expansion.

Of course, I haven't claimed this is well-defined or defined at all. I'm not at all certain if such an idea can even be made sensible. I'm asking if there is some similar mathematical structure that is tractable and has been studied.

jdods
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    Wouldn't this kind of be like $\mathbb{R}^n$, or $\mathbb{R}^{\infty}$? – Adam Rubinson Apr 20 '21 at 19:07
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    Good idea. You can actually do this as far as I can tell. Essentially $,\mathbb{R}[x].$ – Somos Apr 20 '21 at 19:07
  • @AdamRubinson I suppose it could be like $\mathbb R^n$ or $\mathbb R^\infty$ depending on what kind of structure you want to superimpose on it, e.g. the type of operations, relations, and ordering, etc. I was looking for something that would preserve as much structure of $\mathbb R$ as possible which is why I was thinking of it like the non-standard reals. – jdods Apr 20 '21 at 20:27
  • @Somos I don't know much about rings, so that might be a relevant structure, but again though, I don't know if it has the kind of arithmetical/ordering structure I was going for though. That could have easily been unclear from my post. – jdods Apr 20 '21 at 20:30

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Yes, you can - in fact, it's forced upon us in nonstandard analysis, and leads to a version of "$0.9999...=1$" remaining true despite the existence of infinitesimals.

Specifically, let's recall the definition of decimal expansions: something like $$0.a_1a_2a_3...$$ is really shorthand for $$\sum_{i\in\mathbb{N}}a_i\cdot 10^{-i},$$ which in turn is short for $$\lim_{n\rightarrow\infty}\sum_{i\in\mathbb{N}_{<n}}a_i\cdot 10^{-i}.$$ (Note the slightly odd summation notation - that's deliberate and will come up below.)

Now in nonstandard analysis we embed the usual reals $\mathbb{R}$ (= the unique-up-to-isomorphism connected ordered field) inside a larger structure ${}^*\mathbb{R}$ we call the hyperreals.

  • Actually, there's no such thing as "the" hyperreals. We instead have a type of thing called a hyperreal field, of which there are many (even up to isomorphism). Almost always, the particular hyperreal field doesn't matter and so we just tacitly pick one, but this is technically an abuse of terminology.

But a particular case of this is that we also wind up embedding $\mathbb{N}$ inside its "nonstandard analogue" ${}^*\mathbb{N}$, and as a consequence just as $\mathbb{N}$-indexed sums make sense in $\mathbb{R}$ so too do ${}^*\mathbb{N}$-indexed sums make sense in ${}^*\mathbb{R}$. These are exactly your "long" decimal expansions. And we get a lot of similarity: just as we have $$\sum_{i\in\mathbb{N}}9\cdot 10^{-i}=1\quad\mbox{in $\mathbb{R}$},$$ so too do we have $$\sum_{i\in{}^*\mathbb{N}}9\cdot 10^{-i}=1\quad\mbox{in ${}^*\mathbb{R}$}.$$

Stepping back a bit, the existence of ${}^*\mathbb{R}$-analogues of $\mathbb{R}$-flavored notions in general - in a technical sense - is one of the defining properties of a hyperreal field! Hyperreal fields are not just non-Archimedean ordered fields containing $\mathbb{R}$, they have to "form good analogies" in a precise sense.

Meanwhile, $\mathbb{N}$-indexed sums don't work at all in ${}^*\mathbb{R}$: they (at best) point to infinitesimal neighborhoods rather than specific elements. So there's actually a trade-off: we gain ${}^*\mathbb{N}$-indexed sums but lose $\mathbb{N}$-indexed sums. I've said a bit more about this here. (Perhaps surprisingly this turns out to be a consequence of the "good analogies" principle mentioned in the previous paragraph, but this gets a bit involved; if you're interested, the key term is "overspill.")

Noah Schweber
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  • This idea isn't unusual in that it is still just "summing" over the reciprocal "natural number" powers, but it's just the set of nonstandard natural numbers. So we can have a "finite-transfinite decimal expansion" that has some nonzero digits at a finite number of transfinite locations and zeroes everywhere else, and this would be like an infinitesimal in $^\mathbb R$. So I presume exponentiation with nonstandard natural number powers $10^{-i}$ for $i\in {^\mathbb N}$ is well-defined, yes? I've long been confused about basic arithmetic operations with nonstandard numbers. – jdods Apr 20 '21 at 20:23
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    @jdods "exponentiation with nonstandard natural number powers ... is well-defined" Yes, it is. Indeed, everything you can do with real numbers has a nonstandard analogue which satisfies the same basic properties (technically: the same properties which are expressible using first-order logic). This is what the transfer principle demands. Now given the strength of transfer, it may not seem plausible that hyperreal fields exist at all; the standard proof uses ultrapowers, with transfer being a consequence of Los' theorem. – Noah Schweber Apr 20 '21 at 20:45
  • I'll pretend they "exist" until someone shows there is a logical error in their "construction"... ha! I know nothing of transfer principle and ultrapowers though have seen the terms tossed around. One day I will get there with set theory though... Many thanks for the answer! – jdods Apr 20 '21 at 20:55
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    @jdods It's a difficult construction but absolutely beautiful and worth your time. Incidentally, if I have the history right ultrapowers actually emerged in the context of functional analysis rather than logic - specifically, we can learn a lot by think about (minor modifications of) ultrapowers of Banach spaces. But this goes outside my area. – Noah Schweber Apr 20 '21 at 21:02