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This question already has an answer here:

It's easy to represent (via fractions) numbers like $2,34$ or $2,\overline{34}$ and even $2,3\overline{4}$. But what about $2,\overline{3}4$?

$2,3333333333333333333333333333333333....$ and I'll never be able to write the number $4$!!

Does it exist with this notation?

Is a rational number?

Notation: The top bar represents periodic numbers.

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marked as duplicate by Asaf Karagila, Alex M., Ian Miller, user26857 abstract-algebra Aug 10 '16 at 19:23

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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    $\begingroup$ I think I know what you're asking and the answer is "no". No notation exists for that. It's an interesting question to ask because you would think that fractions with decimal places that never end are irrational. But the definition of a rational number is $p/q$ for $q\neq 0$ so something like $1/3$ is indeed rational. $\endgroup$ – KingDuken Aug 10 '16 at 14:47
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    $\begingroup$ The position of the $4$ is not defined, therefore a number cannot be defined this way. $\endgroup$ – Peter Aug 10 '16 at 14:49
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    $\begingroup$ @KingDuken: "fractions with decimal places that never end are irrational"??? $\endgroup$ – barak manos Aug 10 '16 at 14:49
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    $\begingroup$ @barakmanos I was explaining that his assumption for never ending decimals are irrational is interesting, even though it's not true. $\endgroup$ – KingDuken Aug 10 '16 at 14:51
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    $\begingroup$ As an aside, there are uses for generalizations of the notion of 'sequence' where you can have an infinite list of places, and then more places after all of those. However, none of those are decimals, since, by definition, all of the places in a decimal numeral are indexed by integers. $\endgroup$ – user14972 Aug 10 '16 at 15:19
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No such number exists. There is no infinitieth digit in the decimal representation of real numbers, because the notation $$0.A_1A_2A_3\cdots$$ is a shorthand for $$\sum_{k=1}^\infty A_k10^{-k}=\sup_{n\in\Bbb N}\left(\frac{A_1}{10}+\frac{A_2}{10^2}+\cdots+\frac{A_n}{10^n}\right)$$

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  • $\begingroup$ But the OP said nothing about real numbers in his question. Try to be open-minded. $\endgroup$ – Mikhail Katz Aug 10 '16 at 15:25
  • $\begingroup$ @mikhailkatz I deemed helpful stating clearly the meaning of the notation he was working with and why it does not allow what the post suggests. $\endgroup$ – user228113 Aug 10 '16 at 19:34
  • $\begingroup$ G., what you stated clearly is your interpretation of what the OP asked. I provided a different interpretation. $\endgroup$ – Mikhail Katz Aug 11 '16 at 7:53
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If you interpret (or define) $2.\overline{3}4$ as the limit of the sequence $2.34,2.334,2.3334,2.33334,\ldots$, then

$$2.\overline{3}4=2.\overline{3}={7\over3}$$

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  • $\begingroup$ The limit corresponds to taking the standard part. The number 7/3 is indeed the standard part of the hyperrational I mentioned in my answer. $\endgroup$ – Mikhail Katz Aug 10 '16 at 15:24
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    $\begingroup$ @MikhailKatz, my non-standard analysis is pretty rusty, but I think you're right. $\endgroup$ – Barry Cipra Aug 10 '16 at 15:28
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I'd say: yes the number does exist. It is however equal to $2.\overline{3}$. The final "4" is a limit of $\frac{4}{10^n}$ for $n \rightarrow \infty$, which is zero. The number exists the same way $0.\overline{9}$ exists.

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Nice question! Such a number exists in the hyperreals. Simply place the digit 4 at a rank represented by an infinite hyperinteger $H$. See Elementary Calculus. The number in question is a hyperrational number.

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  • $\begingroup$ But that would interpret $2.\bar{3}$ as notating a terminating decimal, not a periodic one! $\endgroup$ – user14972 Aug 10 '16 at 15:21
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    $\begingroup$ Your suggestion is not well-defined. There are infinitely many possible numbers, all of which distinct, which can be interpreted as your suggestion. Not to mention that sending someone asking such a question to the hyperreals is like giving a thirsty man in the desert a Klein bottle of water. $\endgroup$ – Asaf Karagila Aug 10 '16 at 15:30
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    $\begingroup$ (1) You are imagining you know my opinions, and you are wrong. I do, however, think that everything has time and place, and shoving infinitesimals and the hyperreals everywhere is plain wrong. (2) Am I also wrong about your interpretation not being well defined? $\endgroup$ – Asaf Karagila Aug 11 '16 at 8:25
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    $\begingroup$ @AsafKaragila, I think the language of "shoving" should be avoided in this type of venue if minimal norms of conduct are, for a change, to be observed, though perhaps my hope is vain. $\endgroup$ – Mikhail Katz Aug 11 '16 at 8:28
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    $\begingroup$ Irrespectively of the venial/grave/mortal sins @Asaf's point (1) may or may not be a proof of, what about his point (2)? You know, sticking to the maths and all... $\endgroup$ – Did Aug 12 '16 at 11:56