This next topic makes me nervous, because it's bound to cause confusion, and that confusion will lead to controversy, and I prefer to avoid controversy. However, both the textbook and the lecture notes covered it, so I will too.

We are still looking at extending number systems, and now we're looking at extending the real numbers. Let's go back to the definition of the real numbers: they are a set of numbers that have the least upper bound property, and contain the rational numbers as a subset. So every set with an upper bound has a least upper bound, which is pretty great. Wouldn't it be greater if we could extend the real numbers so that every set has an upper bound?

We can do it. The extended real number system includes all of the real numbers, but it also includes the numbers +∞ and −∞. Positive infinity is defined as a number greater than any real number, so any set of numbers in the extended real numbers has an upper bound. Likewise, negative infinity is less than any real number, and so is always a lower bound of any set in the extended real numbers.

The bad news that this totally breaks addition and multiplication. Any real number added to infinity is still infinity, which easily leads to serious math errors. 1+∞=∞=2+∞, which implies that 1=2, for example. The only way to make things work at all is to make lots of computations involving infinity undefined. For example, ∞−∞, ∞/∞, and 0×∞ are all undefined. (We encountered the undefined value 0^{0} previously. Saying that the result is undefined means that it must take on different values in different contexts, so if you end up encountering that expression, you need to find a different way of solving the problem that avoids it.)

As a consequence, the extended real numbers are no longer a field. There's an exception in one of the multiplication properties for 0, but there aren't any exceptions for infinity. And it's really easy to accidentally compute something using one of the undefined expressions and not notice, so most of the time the extended real numbers aren't worth the effort. We stick with the real numbers, where numbers can move toward infinity, but they can never equal infinity.