Why do we use $...$ with irrational numbers?

Question

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.

Answer 1

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.

Answer 2

$a = 1.23 \ldots $ is just a shorthand for $0 \leq a - 1.23 \leq 0.01$