Hyperreal Number - Properties of Infinitesimal and Infinite Numbers

Properties of Infinitesimal and Infinite Numbers

The finite elements F of *R form a local ring, and in fact a valuation ring, with the unique maximal ideal S being the infinitesimals; the quotient F/S is isomorphic to the reals. Hence we have a homomorphic mapping, st(x), from F to R whose kernel consists of the infinitesimals and which sends every element x of F to a unique real number whose difference from x is in S; which is to say, is infinitesimal. Put another way, every finite nonstandard real number is "very close" to a unique real number, in the sense that if x is a finite nonstandard real, then there exists one and only one real number st(x) such that x – st(x) is infinitesimal. This number st(x) is called the standard part of x, conceptually the same as x to the nearest real number. This operation is an order-preserving homomorphism and hence is well-behaved both algebraically and order theoretically. It is order-preserving though not isotonic, i.e. implies, but does not imply .

• We have, if both x and y are finite,

• If x is finite and not infinitesimal.

• x is real if and only if

The map st is continuous with respect to the order topology on the finite hyperreals; in fact it is locally constant.