▁are ▁also ▁many ▁ways ▁to ▁construct ▁" the " ▁real ▁number ▁system , ▁for ▁example , ▁starting ▁from ▁natural ▁numbers , ▁then ▁defining ▁rational ▁numbers ▁algebra ically , ▁and ▁finally ▁defining ▁real ▁numbers ▁as ▁equivalence ▁classes ▁of ▁their ▁C auch y ▁sequences ▁or ▁as ▁D ed ek ind ▁cut s , ▁which ▁are ▁certain ▁subsets ▁of ▁rational ▁numbers . ▁Another ▁possibility ▁is ▁to ▁start ▁from ▁some ▁rig orous ▁ax iom at ization ▁of ▁E uclidean ▁geometry ▁( H il bert , ▁Tar ski , ▁etc .) ▁and ▁then ▁define ▁the ▁real ▁number ▁system ▁geomet r ically . ▁All ▁these ▁constru ctions ▁of ▁the ▁real ▁numbers ▁have ▁been ▁shown ▁to ▁be ▁equivalent , ▁that ▁is ▁the ▁resulting ▁number ▁systems ▁are ▁is omorphic . ▁ ▁A xi om atic ▁approach ▁▁ ▁Let ▁R ▁denote ▁the ▁set ▁of ▁all ▁real ▁numbers . ▁Then : ▁ ▁The ▁set ▁R ▁ ▁is ▁a ▁field , ▁meaning ▁that ▁addition ▁and ▁multiplication ▁are ▁defined ▁and ▁have ▁the ▁usual ▁properties . ▁ ▁The ▁field ▁R ▁ ▁is ▁ordered , ▁meaning ▁that ▁there ▁is ▁a ▁total ▁order ▁ ≥ ▁such ▁that , ▁for ▁all ▁real ▁numbers ▁x , ▁y ▁and ▁z : ▁ ▁if ▁x ▁ ≥ ▁y ▁then ▁x ▁+ ▁z ▁ ≥ ▁y ▁+ ▁z ; ▁ ▁if ▁x ▁ ≥ ▁ 0 ▁and ▁y ▁ ≥ ▁ 0 ▁then ▁x y ▁ ≥ ▁ 0 . ▁ ▁The ▁order ▁is ▁D ed ek ind - complete ; ▁that ▁is : ▁every ▁non - empty ▁subset ▁S ▁of ▁R ▁