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That $...$ in title means the $...$ used at the end of example in question body. So my question is you must have numbers/patterns like

$$ W={1,2,3 ...} $$

Here $ .. $ makes sense. Also

$$ \frac{1}{3}=0.33... $$

Here also $...$ makes sense. However when we take an irrational numbers we write it as

$ 1.43857358357385... $

Now if we asked to find the fifth term of first $2$ examples we can easily tell this that is why we use point. However we don't know the next digit which will we used. Then why do we use $...$?

Background- Today I was learning set theory and there $...$ came which made me think and ask this question. I have tried fully to add context but sorry if it lacks context.

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    $\begingroup$ I do not know which answer you expect. I do not have an answer, but it should be clear that this indicates that the digits are "going on" and there are infinite many. Also theoretically it is possible to approximate every irrational up to a sufficient point. When you want to know the n-th digit of $\pi$, then you can compute it. How difficult this is, is an other question. $\endgroup$
    – Cornman
    Jun 13 at 16:10
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    $\begingroup$ Decimal expansion is just one way of expressing the value of a number - sometimes it is the most convenient. Other times numbers arise as the ratios of lengths (eg $\pi$) or the roots of equations or continued fractions or sums of convergent series - the form in which a number is expressed is not the same as the number itself, and can be chosen for convenience. $\endgroup$ Jun 13 at 16:11
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    $\begingroup$ The ellipsis just signifies continuation in this case. It is not indicating a pattern. $\endgroup$
    – John Douma
    Jun 13 at 16:12
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    $\begingroup$ The question is simple. "$...$" refers to simply something that "continues on." It doesn't need a pattern. Sometimes, a pattern is convenient, but not necessary to use this notation. $\endgroup$
    – Luna145
    Jun 13 at 16:15
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    $\begingroup$ @Cornman - It is absolutely not the case that, for any real number, there is an algorithm to compute the number up to arbitrary precision. That's what it means for a real to be computable, and there are many non-computable reals. $\endgroup$ Jun 13 at 18:22
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$\begingroup$

In general, the construct $\dots$ is used to mark an intentional omission, it is called an ellipsis. This usage is not limited or particular to mathematics.

It does not necessarily indicate that we know enough to fill in the blank.

There can by different reasons to omit something. One reason is that it is more or less redundant, this is the usage you have in mind. Others are that it is not very relevant to give all the details or that it is impossible to include everything.

The usage in the examples you mention are along the lines of these later cases. On the one hand we cannot possibly give all the decimal digits on the other hand to know the first few might be sufficient for our use-case.

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    $\begingroup$ That wikipedia link and "It does not necessarily indicate that we know enough to fill in the blank" helped me the most. Thanks for help $\endgroup$
    – user876009
    Jun 13 at 16:31
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$a = 1.23 \ldots $ is just a shorthand for $0 \leq a - 1.23 \leq 0.01$

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  • $\begingroup$ That's absolutely right. $\endgroup$
    – user21820
    Jun 13 at 16:18
  • $\begingroup$ (+1), but technically shouldn't it be $0 \le a - 1.23 \color{red}{\le}0.01$ because it might be that $a=1.239999\dots$, in which case $a=1.24$? $\endgroup$
    – Joe
    Jun 20 at 14:04
  • $\begingroup$ @Joe I guess you are right, or it depends on the convention... Let me rewrite it as you say, to be on the safe side. $\endgroup$
    – Sasha
    Jun 20 at 14:06
  • $\begingroup$ @Sasha: Thanks. Ironically, when I wrote $a=1.239999\dots$, that was an instance in which the ellipsis indicated a repeating pattern. $\endgroup$
    – Joe
    Jun 20 at 14:12

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