An extended real number is usually a real number, but it might be . We can also think of an extended real number as a number of the form for (including ) or a number of the form for a real number (including an odd multiple of ).
There are actually two slightly different notions that both go by the name ‘extended real number’: one in which and are distinct (the logarithm), and one in which they are identified (the tangent?). The latter kind forms a quotient space of the former.
Classically, one simply defines an extended real number to be either a real number, , or (for the version in which these are distinguished) . However, one has to take more care for a definition that works in constructive mathematics. Such a definition may also be interesting to a classical mathematician for its own sake, of course.
One way to define extended real numbers is to adjust the notion of Dedekind cut. This naturally defines the sort of extended real number in which .
So let an extended Dedekind cut be a pair of sets of rational numbers such that:
(The usual slight variations available in the notion of Dedekind cut apply here as well, in the same ways.)
If is empty, then must be the set of all rational numbers; this extended cut represents . If is empty, then must be the set of all rational numbers; this extended cut represents . An extended Dedekind cut is bounded if instead and are both inhabited. The bounded cuts are the usual Dedekind cuts that represent real numbers. So the only change is that we no longer require this boundedness condition.
The space of extended real numbers in this sense is often denoted . As a topological space (or locale, constructively), it is the end compactification? of the locally compact space . Geometrically, this is a line segment?, the extended real number line segment.
Another way to define extended real numbers is to represent them as ratios of real numbers. This naturally defines the sort of extended real number in which .
So let a nontrivial ratio of real numbers be a pair of real numbers such that either or . (By we mean the usual apartness relation on real numbers.) We write the ratio as , and we consider and to be equivalent, written , if . This is an equivalence relation on nontrivial ratios; modulo this equivalence relation, the nontrivial ratios represent extended real numbers.
If , then the nontrivial ratio is equivalent to and represents the real number . The nontrivial ratios for are all equivalent and represent . (The ratio is considered trivial and does not represent any extended real number.)
This is a special case of forming the projective line? of a field, in this case the field of real numbers. Accordingly, the space of extended real numbers in this sense is often denoted . As a topological space (or locale, constructively), it is the one-point compactification of the locally compact space . Geometrically, this is a circle, the extended real number circle.
We can put a linear order on by setting for any real number . In terms of Dedekind cuts, this is the straightforward extension of the usual definition of on such cuts.
We can put a cyclic order on by setting true if any of the following hold:
(These orders can be defined constructively too.)
In , we take , , and to be (for a real number) and similarly , , and to be . Of course, and are opposites (additive inverses). These are all straightforward extensions of the usual definition of addition for Dedekind cuts. However, when we try or , the Dedekind cut calculation gives us , which is not a cut, even extended. (Although some sources define these sums, it is most common to leave them undefined, I think.)
In , we similarly take and to be (for a real number). Now is its own opposite. These all agree with the additive rule . If we apply this to , however, the result is , so again we take that to be undefined. (However, in this case, the definition is also widely seen, I think.)
Note that although we have blithely referred to ‘additive inverses’, addition does not form a group in either case. The reason is that the opposites of are not really additive inverses, since the result of the relevant addition is undefined rather than zero. (You can't fix this by any clever definition, either, since shows that the operation is not cancellative.)
In , we take to be when but for . (This continues to work even if is infinite too.) But now is undefined. (Many sources define this to be zero, although I think this is really because they are using in the lower reals, at least in applications to measure theory.)
In , we take to be as long as ; but is undefined. We may write and to define division, but again these are not really multiplicative inverses. (With , again multiplication is not cancellative.)
The definitions above give us located extended real numbers. We can of course consider unlocated extended real numbers, either one-sided or MacNeille?. The natural definitions of these are at least somewhat extended; the lower one-sided reals include , the upper one-sided reals include , and the MacNeille reals include both (with ). However, one can always restrict to bounded numbers or alternatively fully extend the one-sided numbers, so being extended and being located are technically independent. Still, whenever shows up constructively (especially if only one of them appears), then this is a sign that one may want to generalise to one-sided (or even MacNeille) reals.