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I just learned about modular numbers on wikipedia, such as $17 \equiv 3\pmod{7}$.

So what is infinity $\pmod{n}$? It can't very well be all the numbers at once, so what happens?

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When we talk about modular arithmetic, we restrict ourselves to the integers, the set $\mathbb{Z}$. (see: en.wikipedia.org/wiki/Integer) But "infinity" is not an integer, so what you are asking doesn't really make sense. We cannot take infinity $\pmod{n}$. – Eric♦ Mar 2 '11 at 5:15

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When we say $a (\bmod n)$, we need $a \in \mathbb{Z}$ and $n \in \mathbb{Z} \backslash \{0\}$. So your question "$\infty (\bmod n)$" doesn't make sense in the first place. It is like asking "What is $\text{apple} (\bmod n)$?"

What your probably mean and want to know is "What is $\displaystyle \lim_{x \in \mathbb{Z}, x \rightarrow \infty} x (\bmod n)$?".

If $n \neq \pm 1$, then the answer is "It doesn't exist". If $n = \pm 1$, then the answer is $0$.

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I see. Perhaps I should read up on this whole infinity business. Thanks – Joseph Mar 2 '11 at 5:27
+1, "Apple mod $n$" is really funny. – Eric♦ Mar 2 '11 at 5:29
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@Joseph: Yes. The take home message is that $\infty$ is not a number. It is a short-form to say that something is unbounded. – user17762 Mar 2 '11 at 5:30

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