[drt1245]

> The argument "Addition breaks" proves just as well that zero "isn't a number", since it breaks division rather badly.

Mathematically, numbers (be it natural, rational, real, or complex) are defined as a field. Fields (or, more accurately, rings, which all fields are) are defined by addition and multiplication, not both. [1]

[1] https://en.wikipedia.org/wiki/Ring_(mathematics)

[mturmon]

Sometimes you may use a "number", but you just want ordering properties from it.

You may not want addition and multiplication in such a "number".

This may be the case in numbers used for ranking outcomes, or counting, or optimization. Using infinity in this context does not cause problems.