# real number system

Definition from Wiktionary, the free dictionary

## English[edit]

### Noun[edit]

**real number system** (*plural* **real number systems**)

- (mathematics) The complete Archimedean ordered field (of real numbers).
**2008**, Ambar N. Sengupta, “Notes in Introductory Real Analysis”, www.math.lsu.edu, accessed on 2012-07-13:- As we have mentioned before, the structure of Euclidean geometry, as formalized through the axioms of Hilbert, produces an archimedean ordered field. To com- plete the story, one can add to these axioms the further requirement that this field is maximal in the sense that it cannot be embedded inside any larger archimedean ordered field. It turns out then that any such ordered field is isomorphic to any other, and thus there is essentially one such ordered field. This ordered field is the
**real number system**.**R**

- As we have mentioned before, the structure of Euclidean geometry, as formalized through the axioms of Hilbert, produces an archimedean ordered field. To com- plete the story, one can add to these axioms the further requirement that this field is maximal in the sense that it cannot be embedded inside any larger archimedean ordered field. It turns out then that any such ordered field is isomorphic to any other, and thus there is essentially one such ordered field. This ordered field is the
**2010**April 16, Lindsay Keegan, “The Ghosts of Departed Quantities”, www.math.ufl.edu, accessed on 2012-11-13:- The real number system is a complete, linearly ordered field .