Infinity is not a natural number (as is in Peano arithmetic) but in many cases it can be treated as an ordinary number e.g., for some problems on a complex plane.

You are in a good company thinking that infinity cannot be ever treated as a number http://scienceblogs.com/goodmath/2008/10/infinity_is_not_a_n...

It is. Just depends which field. Surreals, superreals, hyperreals... People have been taking the reals and adding infinity all the time.

Also, there are the cardinal numbers, for which the interesting ones consist entirely of infinity.

I know, but there are more numbers than real things, or at least maybe they do. Perhaps numbers fail only in extremely large regions that we don't perceive in everyday life. (Unlikely I know, just nitpicking really).