NOTE: I am not a "Math Guy", so my question may not use correct terminology.

I was helping my daughter with her math homework and I realized I did not recall the correct way to write something.

If I want to write a number that is infinitly close to 1, approaching from "less than" (0.99999999999999999999 on and on forever) I write: $0.\bar9$

What if I was coming from "greater than"? How would I write that? Something like $1.\bar01$? (I don't think that is right.)


closed as off-topic by Adam Hughes, Chris Brooks, hardmath, JonMark Perry, Zain Patel Mar 3 '17 at 5:26

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  • "This question is not about mathematics, within the scope defined in the help center." – Adam Hughes, Chris Brooks, hardmath, JonMark Perry, Zain Patel
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  • 4
    $\begingroup$ There is no such thing, so that's basically the issue: it doesn't exist. $\endgroup$ – Adam Hughes Mar 3 '17 at 0:39
  • 2
    $\begingroup$ $0.999\ldots = 1$. Not "infinitely close to $1$", simply $1$. $\endgroup$ – Robert Israel Mar 3 '17 at 0:41
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    $\begingroup$ Not a bad illustration of why the $0.99999...$ really is one. As far as literature, I recommend the book The Phantom Tollbooth, the section with the Mathemagician and infinity. $\endgroup$ – Will Jagy Mar 3 '17 at 0:43
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    $\begingroup$ Although he could meaningfully write $\lim_{n \to \infty} 1 + \frac{1}{10^n}$, which might be similar to his thinking. $\endgroup$ – Chris Mar 3 '17 at 0:43
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    $\begingroup$ We don't need a way to say $1.000000....01$ because that is just $1$. $\endgroup$ – Kaynex Mar 3 '17 at 0:47

I'm afraid the online preview stops before the quote I want. Milo keeps asking about the biggest number there is. On page 191, we get

"Just follow that line forever," said the Mathemagician, "and when you reach the end, turn left. There you'll find the land of Infinity, where the tallest, the shortest, the smallest, and the most and least of everything are kept."

enter image description here

Alright, they do show the end of that line:

enter image description here


It's subtle. Unless you are doing non-standard analysis, where "infinitesimals" are given real meanings, the notion of a number that is "greater than $1$ but 'infinitely close' to $1$" is not really immediately available.

The idea of $0.\overline{9}$ is meaningful because it represents $$ \sum_{n=1}^\infty \frac9{10^n} $$ which you can explain to your daughter as $$ \frac9{10}+\frac9{100}+\frac9{1000} + \cdots $$ But there is no corresponding concept for "coming from greater than" (other than $1.\overline{0}$ which is exactly $1$ with no fanfare).

I guess you could get at what you want by describing it as $$ 2.0 - 0.\overline{9} = 2 - \frac9{10}-\frac9{100}-\frac9{1000} - \cdots $$

So unless you are helping your daughter with an advanced upper class college math course, that is about the best you are going to do.


The usual notations are $1^-$ and $1^+$.

Generally you will see it used in limits expressions like $\lim\limits_{x\to1^+}$ it means $x\to 1$ and $x\gt 1$.

In the same way $1^-$ means $x\to 1$ and $x<1$.

So this is equivalent to say we are going to $1$ from the left side or from the right side, and actually we speak about a left side limit and a right side limit.

Since I'm downvoted, here is some defense :

First, thanks for explaining your reason for downvoting, many don't.

Yet I think the OP question was about how to express something infinitesimally close to $1$ but less than $1$, and the only valid answers in my opinion are $1^-$ and $1-\epsilon\ $ with $\epsilon$ being the one of non-standard analysis (or any extension of the reals like hyperreals, surreals, and co.)

All other answers that consider $0.99999...$ and $1.0000...1$ as stated by Jeff are achieved limits and in that sense their value is simply $1$, not infinitesimals in any way.

So yes $1^-$ and $1^+$ are not real numbers, and it is the way it has to be, it is a notation for something that carries a meaning greater than a number.

In the same way that we do some calculation with some $o(x)$ for instance, while $o(\cdot)$ not being a function, this is a notation of a concept, and consequently I believe I'm answering the request of Vaccano.

  • $\begingroup$ I've downvoted this because those are not numbers, and your answer implies that they are. $\endgroup$ – Deusovi Mar 3 '17 at 5:38
  • $\begingroup$ @Deusovi I disagree. Take the number $e$ for example. It is defined as a limit of a function. $\endgroup$ – Timothy Cho Mar 4 '17 at 0:19
  • $\begingroup$ @jeff: Yes, limits of functions are numbers. But $1^-$ and $1^+$ are not numbers (nor are they limits of functions). $\endgroup$ – Deusovi Mar 4 '17 at 1:07
  • $\begingroup$ @Deusovi I understand what you're trying to say :) $\endgroup$ – Timothy Cho Mar 4 '17 at 1:09

Two things: $0.\bar9$ is actually $1$. If you approach infinitely close, you will get one.

Second, there really isn't a way to represent something if you are coming from "greater than". Take $1.\bar01$ for example. Write out the first few digits: $1.000000000000000000000000000...$ The bar means that there are infinite zeros in the decimal representations. This implies that you will never get to the final "$1$" you put at the end.

  • $\begingroup$ Certainly one can imagine a world where one could write such a thing down. The second paragraph is no reason at all to reject the concept. It's true that $0.\overline 9=1$, but $1.\overline 01$ doesn't represent a real number, so there's no reason to conclude that it is $1$. $\endgroup$ – Matt Samuel Mar 3 '17 at 1:18
  • $\begingroup$ @MattSamuel There isn't a world where someone would have the time to write down infinite numbers. But your second point is valid though $\endgroup$ – Timothy Cho Mar 4 '17 at 0:17
  • $\begingroup$ You might as well say you can't take the sum of an infinite series because there's not enough time. Yet people happily find their sums all the time, in smarter ways. $1.\overline 01$ expresses it without literally writing them all down. $\endgroup$ – Matt Samuel Mar 4 '17 at 0:19
  • $\begingroup$ You'll still never get to the final $1$ though, by definition of the bar notation. $\endgroup$ – Timothy Cho Mar 4 '17 at 0:21
  • $\begingroup$ See @DanielV's comment above $\endgroup$ – Timothy Cho Mar 4 '17 at 0:22

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