The argument "Addition breaks" proves just as well that zero "isn't a number", since it breaks division rather badly. (Yes, division is a badly-behaved version of the more upstanding multiplication. The article uses subtraction in what is nominally a complaint about addition). It doesn't address ordinal numbers at all, since addition works just fine with infinite ordinals, exactly the way the article claims you'd expect (what have I missed?).

The immediately obvious uses of infinity in calculus (the topic of the wikibooks article) are, according to wikipedia, termed the "affinely extended real number system" (I learned to just refer to the "extended reals"), which heavily implies that the points within are considered extended real _numbers_. http://en.wikipedia.org/wiki/Extended_reals

The terms "cardinal number" and "ordinal number" both definitively include infinite quantities -- infinitely many, even.

The IEEE standard for floating point defines two infinite numbers.

Essentially, I'm in full agreement with tokenadult; the only relevant question is "what do you mean by number?". But we can easily observe that varying infinities, including the calculus uses of infinity, are referred to as numbers all up and down the chain, including in the most unimpeachably correct sources, and that it walks and quacks like a duck, even if it may not quack in the precise manner of Anas Platyrhynchos.