Is infinity a number? Why or why not?

Some commentary:
I've found that this is an incredibly simple question to ask — where I grew up, it was a popular argument starter in elementary school — but a difficult one to answer in an intelligent manner. I'm hoping to see a combination of strong citations and solid reasoning in the answers.

It comes down to the definition of "number," as well as the definition of "infinity." Personally I don't think it's worth having an opinion on this subject; there are more precise words than "number" and "infinity" in mathematics. Historically the word "number" has come to mean an increasingly general list of things:


a positive integer
an integer
a rational number
a constructible number (say, to the ancient Greeks)
a real number
a complex number
a cardinal number
an ordinal number
a surreal number...


The word "infinity" has also come to mean an increasingly general list of things: it might refer to


a countably infinite set
an uncountably infinite set
the point at infinity in projective geometry
one of the two new points in the two-point compactification of the real numbers
the new point in the one-point compactification of the complex numbers, also known as the Riemann sphere
an infinite cardinal number
an infinite ordinal number...


Some of these meanings are compatible, as the above list demonstrates. But again, there are more precise words than "number" and "infinity" in mathematics, and if you want to get anywhere you should learn what those words are instead. 

Here are some of those more precise words.


A set is a formalization of the intuitive notion of a bag of objects, and we can talk about finite or infinite sets. For example, $\{ 1, 2, 3 \}$ is a finite set, whereas the set of natural numbers is an infinite set. One can do arithmetic with sets in a way that leads to the arithmetic of the natural numbers: for example, taking the disjoint union corresponds to addition, and taking the Cartesian product corresponds to multiplication. These ideas lead to the arithmetic of the cardinal numbers, and similar ideas lead to the arithmetic of the ordinal numbers. 
A ring is a formalization of the intuitive notion of a set of things you can add and multiply, so in some sense one can regard elements of rings as "generalized numbers" (but note that not every generalization I listed above can be interpreted in this way). When certain people say that "infinity is not a number," what they mean is that you can't adjoin an element called $\infty$ to the ring $\mathbb{R}$ of real numbers such that addition and multiplication do what you want them to do, the basic problem being that $\infty - \infty$ can't be consistently defined to satisfy the other rules of arithmetic if you also want it to be true that $n + \infty = \infty$ for any finite $n$. 
A field is a commutative ring in which it's also possible to divide by nonzero elements. Some people would like to say that $\frac{1}{0} = \infty$, but by mathematical convention the element $0$ never has a multiplicative inverse in a field, the basic problem being that $0 \cdot \infty$ can't be consistently defined to satisfy the other rules of arithmetic. However, one can make sense of the expression $\frac{1}{0}$ in projective geometry; it describes the point at infinity on the projective line. 
A topological space is an abstract setting for ideas like nearness and taking limits. Sometimes we don't want to view $\mathbb{R}$ as a ring, but as a space (the real number line), and we can talk about embedding this space into a larger space where more limits exist: this is known as compactification, and is an extremely useful tool in mathematics and physics. For example, we would like to say that the sequence $1, 2, 3, ... $ has limit $\infty$ in some sense, and we can do this compactifying $\mathbb{R}$.

