So I would like to know whether I could have a repeating whole number, such as 9999... repeating or if that's a mathematical fallacy that breaks things. I don't really want to know if it would be natural or not (I assume not), I'm just wondering if its possible to have a number fitting my specifications. Also I know that repeating decimals are a thing, I'm wondering if the same concept can apply to whole numbers.
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3Does this answer your question? A "number" with an infinite number of digits is a natural number? – VIVID Dec 15 '20 at 05:27
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$0.999\cdots = 1$ – Doug M Dec 15 '20 at 05:30
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1In the $11$-adics, that number is $-9/10$. – Gerry Myerson Dec 15 '20 at 08:19
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Just observe that any natural number $m$ with $n+1$ number of digits can be uniquely written as $$m = \sum_{k= 0}^{n } a_k 10^k$$ where each $0 \leq a_k \leq 9$ and $a_n \ne 0$. So it is clear that $m > 10^n$.
Now if the number has infinite digits then the series $\sum_{k=0}^{\infty} a_k 10^k$ should diverge since $10^n \to \infty$ as $n \to \infty$. So there is no possibility of natural number whose number of digits are infinite.
If $m$ is a negative integer then consider $-m$ and apply the above argument
Infinity_hunter
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