Quote by Archosaur
But that's basic algebra!

Indeed, the cost of including infinity in a number system is that certain basic expressions must remain undefined, when they include infinity. But so what? They're still defined as usual for every finite number, and the expressions where infinity is sensible (like [itex]1/\infty=0[/itex]) are still well defined.
There is also a finite number that does not work in many basic algebraic manipulations, but is nevertheless included in many standard number systems.