# real number system

## English

### Noun

real number system (plural real number systems)

1. The complete Archimedean ordered field (of real numbers).
• 2008 November, Ambar N. Sengupta, “Notes in Introductory Real Analysis”, in www.math.lsu.edu[1], retrieved 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.
• 2010 April 16, Lindsay Keegan, “The Ghosts of Departed Quantities”, in www.math.ufl.edu[2], archived from the original on 30 September 2011:
The real number system is a complete, linearly ordered field ${\displaystyle {\mathcal {R}}=(\mathbb {R} ,+,\cdot ,<)}$.