>Further, I googled for "system of extended real numbers" and achieved >only one result, unrelated to this. So I will assume that this is not >the name I should be looking for when attempting to learn. What should I >search for in order to get more useful results?
You can start with Cantor's theory of transfinite numbers
you can try the HYPER-REAL number system.
Please note, that while both are mathematical systems that allow reasonable investigations into properties of "infinity", they are not the same. In a simplistic sense, Cantorian transfinites are more like integers, while the hyper-reals are more like the reals. Thus while many things are the same in both systems, some things are different.
I would suggest that to get a "handle on infinity", the Cantorian transfinites are easier to deal with rigorously. However, if you want to apply to understanding real-world idealizations, e.g., Dirac delta functions, the hyper-reals are more useful, however maintaining rigor requires more effort.