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Just like what the title says.

Does: $$n = n.\overline 01?$$ For example, $1.\overline 01 = 1$?

Similar to $(n-1).\overline 9 = n$, for example, $0.\overline 9 = 1$.

The last statement is true but intuitively I also feel the first is true as well but I look for a proof online and couldn't find one.

I am oversimplifying the notation here obviously just to make it easier to understand on what I mean. The idea came form if $3.\overline 9 = 4$ then does coming form the other direction of 4 i.e. $5$ still equals $4$ which in this case would be $4.\overline 01$?

MathCubes
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    Putting a bar above a digit (or group of digits) means that those digits repeat forever. In particular, nothing can come after those digits. So having a bar over a digit and then another digit after that one does not have any mathematical meaning. – Greg Martin Sep 29 '23 at 22:22
  • @GregMartin I just thought it meant that the number repeats. – MathCubes Sep 29 '23 at 22:23
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    It makes no sense to have infinite many zeros FOLLOWED BY a one. Also, $\frac{\infty}{1}$ is not a valid expression. But the examples with $\bar 9$ are correct. – Peter Sep 29 '23 at 22:23
  • Oh I just thought it was just an shorter way of writing a infinitesimal. Oh I had it flip thanks. – MathCubes Sep 29 '23 at 22:26
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    $1-0.\bar 9$ is not $0.\bar 01$ , it is $0$. – Peter Sep 29 '23 at 22:27
  • @Peter Good point. – MathCubes Sep 29 '23 at 22:27
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    Ideally, you would attempt to work on the notation and use MathJax to improve readability. – cjferes Sep 29 '23 at 22:28
  • @cjferes just an basic idea question I had. I never heard of MathJax before. I tried the basic ~~ to create the overline, didn't work and so I just used unicode. – MathCubes Sep 29 '23 at 22:29
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    @cjferes I just thought that basic language was easy enough to convey what I was trying to ask. – MathCubes Sep 29 '23 at 22:30
  • $\frac{1}{\infty}$ is not valid either. We have $$\lim_{x\to \infty} \frac{1}{x}=0$$ but not $\frac{1}{\infty}=0$ – Peter Sep 29 '23 at 22:35
  • @Peter I always thought it was more or less the symbol for a lack of a better word since I read it somewhere and it implied the opposite.

    Thanks for clarification.

    I guess in a pure sense an infinitesimal would be x.0...2 or (n-1).9...8

     – MathCubes Sep 29 '23 at 22:37
  • So, 1.0000...01 is not standard notation for a number. But in math you're always allowed to make your own definitions!! And then we can look at what happens with these new definitions. So for instance, one possibility (but far from the only possibility), you could define 1.0000...01 to mean "the limit of the convergent sequence 1.1, 1.01, 1.001, 1.0001, 1.00001, 1.000001, ...". With this definition, can you prove that 1.000...01 = 1? – Stef Sep 29 '23 at 22:44
  • @Stef Then if the over line is not the correct way than what is the simplest way to write it? I do not think it's wise to write a few sentences for something that can sum up within a few 'words' if you know what I mean. – MathCubes Sep 29 '23 at 22:47
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    I'm guessing your in an undergraduate Calculus course. I mean no disrespect. Let $\epsilon$ = an infinitesimal. This could be informally defined as the closest possibly number to zero, but $\neq$ 0. $$.$$

    The catch is there any attempt to write out any possible for any possible $\epsilon$, there exists $\epsilon /2$ which is even closer to zero. $$.$$

    Or in proofs we often write $\epsilon > 0$ Being actually close to zero in that context may not be required.

    For most contexts in an undergraduate Calculus course, this understanding should be sufficient.

     – nickalh Sep 29 '23 at 22:54
  • I also do not really care about the 'correct' way to properly convey the phenomenon just the shortest, simplest and easiest way to understand the phenomenon to someone else which I am referring to. There are no correct way because in reality it's subjective and is man made just like difference languages that do change. There are no correct spellings etc... Just the one that are use the most. There are different dialects. – MathCubes Sep 29 '23 at 22:55
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    I do not argue over labels. I might argue what phenomenon fits the noumenon correctly. My gosh, I never study math professionally as in school I could never gasp it since how it's taught. They do not get you to think base on a scientist would, they do not teach the methods to come up with it. No, they just teach the sky is cyan because it's just cyan and just learn all of these facts. Never the method. I basically know the foundations of math and I self taught myself matrix and linear algebra. I thought what I wrote was good enough to convey what I meant. I guess not. – MathCubes Sep 29 '23 at 22:55
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    In cases like yours unfortunately this community can be excessively strict. The disconnect here has probably more to do with mathematical sophistication. Each class one takes in math takes previous ideas, refines them and defines them more strictly than previous classes. Most of this community is Master's, PhD and upper level Math major courses.

    Me personally, I encourage Calculus students to use $\frac1\infty$ but I tell them it's "mathematical slang". A few professors here and there will ding them points for it, but some will accept it.

     – nickalh Sep 29 '23 at 22:56
  • If something is not right then why can someone just edit it or ask if the person really do not know what I meant. Like I used to edit questions and answers with other stuff all of the time on Ask Ubuntu. And I do contribute to Wikipedia. Logically on a forum like this. I would expect that.

    ...

    I always had an interest in math since the youngest of age but the problem was I wanted to think of a mathematician to understand it completely and they again never taught the method and so that left me confused.

     – MathCubes Sep 29 '23 at 22:56
  • @nickalh Thanks, I do mean that. – MathCubes Sep 29 '23 at 22:56
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    On behalf of the community, I apologize. I realize it's very unwelcoming to I'm guessing students on a university freshman level. It's been discussed on math.meta.SE, but too many Master's level students rely on this as a resource to research or want this to be like an encyclopedia for it to change. Being a bit like an encyclopedia is one of the stated goals of this site. – nickalh Sep 29 '23 at 23:00
  • @nickalh Thanks. The basic idea came form if 3.99 = 4 then does 4.01 = 4? It's the smallest number/digit to 4 coming form the other way which is 5 and not 3.

    I just view it as a mathematical notation error as some of the information is lost when manipulated. So in a sense I am asking does it still hold true when approaching form the other side of the number.

     – MathCubes Sep 29 '23 at 23:11
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    Since we already have a definition for 0.999... but we don't yet have a definition for 1.000...01, we could use the former to define the latter. So a possible definition would be: 0.000...01 = 1 - 0.999... ; and of course 1.000...01 = 1 + 0.000...01. With this definition it should be easy to prove that 1.000...01 = 1. – Stef Sep 29 '23 at 23:11
  • @Stef Thanks alot!!! – MathCubes Sep 29 '23 at 23:15
  • @Stef Do you believe it would still hold true if you are coming form a direction from within the complex plane? – MathCubes Sep 29 '23 at 23:26
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    You mean, if instead of x=1-0.000...1 or x=1+0.000...1 we had $x=1+(e^{i \theta} \times 0.000...1)$ for some angle $\theta$? As long as $0.000...1 = 0$, we should be okay :) – Stef Sep 29 '23 at 23:33
  • @Stef

    Just a quick question for both of you. Do you actually believe most math teachers actually understand math? I never had a math teacher which could intuitively and fundamentally understand math as they always state it as a series of facts that you just need to know.

     – MathCubes Oct 01 '23 at 19:42
  • For example take two negatives. They make a positive. No math teacher ever in my life explain why, they always stated it is because it is type of thing. I never got an answer on the lines of it's simply a inverse of an inverse and so they cancel each other out. Example on what I mean, since they are not real their true meaning depends on the context but when we used them it's typically is an inverse. So going form the other way from the origin form 2 etc.. would be -2 but again we take the inverse of that and so we come back to 2. – MathCubes Oct 01 '23 at 19:43
  • Another way (-n)-(-n) = 0. Because if we go in the reverse of a direction. Then go in reverse of that after a certain distance, we would end up where we began.

    Thoughts? Again, I never had anyone explain that intuitively to me other than reading an article by an mathematician when I was younger when I got super curious into numbers and math.

     – MathCubes Oct 01 '23 at 19:44
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    There are good math teachers and bad math teachers. Unfortunately, a huge number of people today say they "hate maths" and when asked a few questions to understand where this hate comes from, it can be traced down to a particularly bad math teacher they have had. So it looks like there are a plethora of bad math teachers. – Stef Oct 01 '23 at 20:59
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    There's actually quite a few books about this issue of math being taught as a boring series of facts and procedures, notably the well-documented "The teaching gap" by James Stigler, as well as "The elephant in the classroom" and "What's math got to do with it?" by Jo Boaler. If you want to ask more questions on this topic then I suggest asking them at https://matheducators.stackexchange.com where they should be well received – Stef Oct 01 '23 at 21:03
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    @Stef Thanks for that. To teach the concept wouldn't even required math at all just basic concept, you know? What would we call an double negative such as non-counterfeit money? That should be good enough to get the point across.

    If someone is still confused let them play around with the concept and eventually they will understand once everything doesn't work form within math nor can it be applied to the real world. I guess it comes form it being an alien language to most and it's never explained other number sets don't behave like the ℝ numbers do and so it's not intuitive for them.

     – MathCubes Oct 01 '23 at 22:52

1 Answers1

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Some people have argued in the comments that the notation 1.000...01 cannot represent a number, because it makes no sense to have a remaining 1 at the end of an unending sequence of repeating zeroes.

I will not answer the question "does this notation represent a number?", but I will attempt to prove that if this notation corresponds to a number, and if the notation plays nice with arithmetic manipulations, then this number must be equal to the number 1.

Let $$x = 1.000...01$$ (whatever that means).

Then: \begin{align*} x & = \,\,\, 1.000...01 \\ 10 x & = 10.000...10 \\ 10 x & = 10.000...010 \\ 10 x & = \,\,\,9 + x \\ 9 x & = \,\,\,9 \\ x & = \,\,\,1. \end{align*}

A word of caution: If you can give a more formal definition of the meaning of the notation 1.000...01, then you'd need to verify that my calculations above are still correct with your definition.

In particular, between the second and third lines I appeared to have used something along the lines of "an unending sequence of zeroes followed by a 10 is the same thing as an unending sequence of zeroes followed by a 1". Is this really true?

Stef
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