extended real numbers

The extended real numbers are the ordinary reals with two new endpoints bolted on, plus infinity at the far right and minus infinity at the far left. Adding these ideal points lets statements about limits and bounds avoid awkward exceptions.

Formally one adjoins two symbols, plus infinity and minus infinity, declaring minus infinity less than every real less than plus infinity. Many arithmetic rules extend naturally, such as x plus infinity equals infinity for finite x, and a positive number times infinity equals infinity. A great convenience is that in this system every set has a supremum and an infimum: an unbounded-above set has supremum plus infinity, and the empty set has supremum minus infinity.

The payoff is in limit language. A sequence diverging to infinity now has a limit, namely plus infinity, in the extended reals; the limit superior and limit inferior of any sequence always exist there. Measure theory and integration lean heavily on this convention, since measures and integrals are routinely allowed to be plus infinity.

The price is that the extended reals are not a field, because some combinations are simply undefined. Infinity minus infinity and zero times infinity have no consistent value and must be treated as indeterminate, exactly the situation L'Hôpital's rule exists to resolve. One uses the extended reals for their order and limit structure, not for unrestricted arithmetic.