Existence of this number

Question

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.

Answer 1

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 $$\sum_{k=1}^\infty A_k10^{-k}=\sup_{n\in\Bbb N}\left(\frac{A_1}{10}+\frac{A_2}{10^2}+\cdots+\frac{A_n}{10^n}\right)$$

Answer 2

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.

Answer 3

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

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

Answer 4

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.