> I think nonstandard analysis interesting,but don't know well about where > belong infinitesimals and unlimiteds in a system of real numbers. > Don't they belong to real number? Are there another number on real line > besids real number?
You may informally think of infinitessimals are "fluff" surrounding each real number. Technically, they are called "hyper-reals", as "infinitessimals" are the hyper-reals near 0. The inverses of infinitessimals are called "unlimited" hyper-reals (think infinite). Limited hyper-reals are those "close" to standard finite reals.
A few points to remember about hyper-reals:
1. The are infinitely dense. Just as with standard reals, you have the property that between any two distinct hyper-reals, there is a hyper-real between them.
2. A limited hyper-real r is infinitessimally close to exactly one standard real. That is to say, there are no hyper-reals "straddling the fence" between the reals. This standard real is called the "shadow of r", denoted sh(r).
3. The set of all hyper-reals infinitely close to some standard real r is called the "halo of r", denoted hal(r). Halos do not intersect, they are separate. So if hyper-real s is in both hal(r1) and hal(r2), you know that r1 = r2.