5

This, I hope, is not a duplicate; I am exercising my critical thinking here and I want to understand what going on, and the available content I have found online on this so far has not helped.

I'm getting conflicting information regarding whether $1=0.\overline{9}$ (i.e., "$0$ point $9$ recurring") holds in nonstandard analysis.

On one hand, we have this comment:

Simply speaking, NSA does not lead to conclusions about $\Bbb R$ that differ from those of standard analysis. Giving alternate definitions of $0.99…$ is not really the concern of most researchers working in NSA: we roll with the usual definition (as in Rudin above), and so $0.999⋯=1$.

This was found here:

Reference request: How is $0.99\cdots$ defined in nonstandard analysis?

On the other hand, we have:

About 0.999... = 1

In that question and its answers, the number

$$0.9_N:=\sum_{i=1}^N 9\cdot 10^{-i} $$

where $N\in{}^*\mathbb{N}\setminus\mathbb{N}$ is an infinite nonstandard natural number

is defined and it is shown that $1-0.9_N$ is a positive infinitesimal number.

Both seem reasonable to me, so what gives?

The Question:

Does $1=0.\overline{9}$ in nonstandard analysis? Does the question make sense; that is, is there an issue of interpretation or something that leads to these two seemingly opposing responses? Please would someone settle the matter with references?

Thoughts:

My best and only guess is that $0.9_N$ is not the standard definition of $0.\overline{9}$. I don't understand how though.

It is my understanding that

$$0.\overline{9}:=\sum_{n=1}^\infty \frac{9}{10^n}.$$

Edit: As explained in this comment of mine below, I suppose the problem is that I don't get why

$$\sum_{n=1}^\infty \frac{9}{10^n}$$

is not implicit in

$$\sum_{i=1}^N 9\cdot 10^{-i} $$

for infinite $N$.

Shaun
  • 42,617
  • 18
  • 62
  • 167
  • 2
    You wrote "I don't understand how". Is there a book/source you have that defines $0.\overline{9}$ in a way similar to $0.9_N$? If not, then that's how - no one defined them similarly, so there's no reason to expect they're the same. If so, then adding a quote from that source to your post would greatly clarify where your confusion comes from. – Mark S. Jun 03 '23 at 14:58
  • I have added what I believe is the standard definition of $0.\overline{9}$, @MarkS. – Shaun Jun 03 '23 at 15:02
  • 2
    You've written two different (but similar) definitions. And I see nowhere in your question or links that anyone has claimed they're the same/both definitions of $0.\overline 9$. So I'm not sure where your question lies. Is there a sentence somewhere in your links that you read as suggesting that the definition of one could be a definition of the other? Is there a question you have about either definition and/or conventions about $0.\overline 9$ not addressed by the other questions on this site? I'm not trying to be difficult, just trying the pinpoint the confusion so I might help. – Mark S. Jun 03 '23 at 15:10
  • 1
    I'm not sure, @MarkS., but, like I said, I think that's what the problem is: they're different definitions. But how is $$\sum_{n=1}^\infty \frac{9}{10^n}$$ not implicit by $$\sum_{i=1}^N 9\times 10^{-i}$$ for infinite $N$? – Shaun Jun 03 '23 at 15:17
  • 1
    That question about those two different expressions is the core of your question, and I would recommend highlighting it in an edit. It seems the distinction between $\infty$ as used in calculus and nonstandard infinite integers is the source of your confusion. I'm not at my computer right now, but can write an answer later if no one has beaten me to it. – Mark S. Jun 03 '23 at 15:21
  • I have made an edit, @MarkS. Please do write an answer, even if someone beats you to it. – Shaun Jun 03 '23 at 15:28
  • 1
    My comment (that your question cites) is followed immediately by *That said, "simply speaking" is necessarily inaccurate: questions of mathematical culture tend to be subtle rather than from simple. There are many possible alternative semantics for decimal notation that nonstandard analysis can formalize, and some good (social) reasons for adopting them. The reference you are looking for is the article [When is .999... less than 1?](https://arxiv.org/abs/1007.3018) by K. U. Katz and M. Katz...* Have you read the linked article? You should, because I think it thoroughly answers your question. – Z. A. K. Jun 03 '23 at 16:06
  • Yes, I saw that @Z.A.K. I guess it didn't register to me as an answer to my question, but instead as further clarification on the question it is under; I'm sorry. I haven't read the article yet. – Shaun Jun 03 '23 at 16:26
  • 1
    I think GEdgar's answer is good, but you also asked for a reference, so I will recommend Rob Goldblatt's _Lectures on the Hyperreals_, which is very readable. In particular it will give you a good sense of the transfer principle, which is the key notion you seem to be missing. – MJD Jun 03 '23 at 23:15

1 Answers1

4

First, $$ \sum_{n=1}^\infty \frac{9}{10^n} = 1 . \tag1$$ Non-standard analysis does not change this (standard) fact.
In a hyperreal field ${}^*\mathbb R$, we see that if $N$ is an infinite natural number, then $$ 1- \sum_{n=1}^N \frac{9}{10^n}\text{ is positive but infinitesimal.} $$ What we do know is: the following are eqivalent: $$ (a)\quad\sum_{n=1}^\infty \frac{9}{10^n} = 1 $$ and $$ (b)\quad\text{for all infinite natural numbers $N$, }\left|1- \sum_{n=1}^N \frac{9}{10^n}\right| \text{ is infinitesimal.} $$

What do we mean by $$ \sum_{n=1}^N \frac{9}{10^n}\qquad? \tag2$$ It seems to be a sum with infinitely many terms, which is nonsense. Here is Robinson's explanation. Take the (standard) function $S : \mathbb N \to \mathbb R$ defined by $$ S(n) = \sum_{n=1}^n \frac{9}{10^n} . $$ There is a corresponding non-standard object, say ${}^*S$ with ${}^*S : {}^*\mathbb N \to {}^*\mathbb R$. (The main usfulness of non-standard analysis is this "transfer principle".) Then $(2)$ is by definition: ${}^*S\; ( N)$.

If we merely have a non-archimedian ordered field $F$, then $(2)$ is undefined. To make $(2)$ meaningful, we require this particular field ${}^*\mathbb R$, with its corresponding transfer principle.

Reference
Robinson, Abraham, Non-standard analysis, Princeton, NJ: Princeton Univ. Press. xix, 293 p. (1996). ZBL0843.26012.

GEdgar
  • 104,283
  • 7
  • 103
  • 245