What are superreal numbers? I am trying to understand the extensions of the real numbers. Nothing too serious like a course or research. Especially since the wikipedia article on superreal numbers is pretty empty. (https://en.wikipedia.org/wiki/Superreal_number).
Are the surreal numbers just the union of reals, hyperreal, and superreal numbers?
I am sorry if this question is too vague; I do not need a super rigorous explanation. I am looking for a more "naive" (or intuitive) explanation.
 A: Unfortunately this Wikipedia page is misleading. The relevant notion is that of super-real fields, introduced by Dales and Woodin in their monograph Super-real fields. 
Super-real fields are up to isomorphism fraction fields of quotients by prime ideals of algebras of continuous real valued functions on completely regular spaces. They include what they call hyper-real fields, which are quotients of the same algebras by maximal ideals, which themselves include ultrapowers of $\mathbb{R}$ for some ultrafilter on a cardinal.
Dales and Woodin proved that those fields are real closed and gave conditions on the prime ideals for the unique underlying field order to be an $\eta_1$-set.
In informal terms, super-real fields form a class of ordered field extensions of $\mathbb{R}$, and may be small or big. The surreal numbers of birthdate $<\kappa$ for some cardinals $\kappa$ are super-real and even hyper-real fields.
Thus the phrasing "superreal numbers" is bad, just like it would be bad phrasing to say that a "ring number" is an element of a ring. Each super-real field embeds in the field of surreals, but this is nothing specific to super-real fields.

There is a nice short presentation of those fields and their purpose which you can find here.
