"Robert J. Kolker" <email@example.com> wrote in message news://3C0B423F.65E40D55@mediaone.net... > > Which gets me back to my original point. The augmented reals > are not algebraically closed under common arithmetic operations, > so why bother adding the infinities.
The unaugmented reals aren't closed under common operations anyway (specifically division).
You could equally well ask why we bother having irrationals; the rationals are closed under common arithmetic operations anyway, what more do we need? The only thing we gain is the completeness of the real line...
> The only thing gained is the > topological compactness of the augmented reals.