Editing Hyperreal number (section)

== 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 [[ring homomorphism|homomorphic]] mapping, st(''x''), from '''F''' to '''R''' whose [[kernel (algebra)|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''&nbsp;&ndash;&nbsp;st(''x'') is infinitesimal. This number st(''x'') is called the [[standard part function|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. <math> x \le y</math> implies <math>\operatorname{st}(x) \le \operatorname{st}(y)</math>, but <math>x < y </math> does not imply <math>\operatorname{st}(x) < \operatorname{st}(y)</math>.

* We have, if both ''x'' and ''y'' are finite, <math display="block"> \operatorname{st}(x + y) = \operatorname{st}(x) + \operatorname{st}(y) </math> <math display="block"> \operatorname{st}(x y) = \operatorname{st}(x) \operatorname{st}(y) </math>
* If ''x'' is finite and not infinitesimal. <math display="block"> \operatorname{st}(1/x)  = 1 / \operatorname{st}(x) </math>
* ''x'' is real if and only if <math display="block"> \operatorname{st}(x) = x </math>
The map st is [[continuous function (topology)|continuous]] with respect to the order topology on the finite hyperreals; in fact it is [[locally constant function|locally constant]].