What are hyperreal numbers? (Clarifying an already answered question) This question already has an answer here. That answer is abstract. Could you help me with some not-so-abstract examples of what the answerer is talking about? For example, give examples of hyperreal numbers which are written as numbers, if that is possible. 
Another examples that I would like to understand are these statements:
Hyperreal numbers extend the reals. As well, real numbers form a subset of the hyperreal numbers.
I've not yet studied mathematics at university level.

Unfortunately, there is no "concrete" description of the hyperreals. For instance, there is no way to give a concrete description of any specific infinitesimal: the infinitesimals tend to be "indistinguishable" from each other. (It takes a bit of work to make this claim precise, but in general, distinct infinitesimals may share all the same definable properties. Contrast that with real numbers, which we can always "tell apart" by finding some rational - which is easy to describe! - in between them; actually, that just amounts to looking at their decimal expansions, and noticing a place where they differ!)
Similarly, the whole object "the field of hyperreals" is a pretty mysterious object: it's not unique in any good sense (so speaking of "the hyperreals" is really not correct), and it takes some serious mathematics to show that it even exists, much more than is required for constructing the reals.
While the hyperreals yield much more intuitive proofs of many theorems of analysis, as a structure they are much less intuitive in my opinion, largely for the reasons above.

To answer your other question, yes, the reals are (isomorphic to) a subset of (any version of) the hyperreals; that's what's meant by saying that the hyperreals extend the reals.
