# Modular numbers

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 Naslund Mar 2 '11 at 5:15

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

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

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 – Daenerys Naharis Mar 2 '11 at 5:27
+1, "Apple mod $n$" is really funny. – Eric Naslund Mar 2 '11 at 5:29
@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

What do you mean by "infinity"?

Suppose you mean the projective numbers (specifically the projective integers, which is the same as the projective rational numbers), which adds a new point $\infty$ with algebraic properties such as $1/\infty = 0$, $1/0 = \infty$, $\infty + 1 = \infty$, and the follwoing are undefined: $\infty + \infty$ and $\infty / \infty$.

If $n$ is prime, then we can also extend the integers modulo $n$ to the projective integers modulo $n$. Let's call the added element of that $\omega$.

In this case, we have $\infty \equiv \omega \pmod{n}$. We have other useful features too, such as

$$\frac{1}{n} \equiv \omega \pmod{n}$$

so we can reduce any rational number modulo $n$ (if the denominator is relatively prime to $n$, we can define the reduction to be division modulo $n$ as usual).

If $n$ is not a prime, then things become trickier.

If you mean something else by "infinity"; e.g. the extended real numbers you see in calculus (i.e. $\pm \infty$), or the infinite cardinalities you see in set theory, or some other sort of thing, then asking to reduce it modulo $n$ is going to be nonsensical.

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