▁another ▁ax iom at ization ▁of ▁ ℝ , ▁see ▁Tar ski ' s ▁ax iom at ization ▁of ▁the ▁re als . ▁ ▁Const ruction ▁from ▁the ▁rational ▁numbers ▁▁ ▁The ▁real ▁numbers ▁can ▁be ▁constructed ▁as ▁a ▁completion ▁of ▁the ▁rational ▁numbers ▁in ▁such ▁a ▁way ▁that ▁a ▁sequence ▁defined ▁by ▁a ▁decimal ▁or ▁binary ▁expansion ▁like ▁( 3 ; ▁ 3 . 1 ; ▁ 3 . 1 4 ; ▁ 3 . 1 4 1 ; ▁ 3 . 1 4 1 5 ; ▁...) ▁converges ▁to ▁a ▁unique ▁real ▁number , ▁in ▁this ▁case ▁. ▁For ▁details ▁and ▁other ▁constru ctions ▁of ▁real ▁numbers , ▁see ▁construction ▁of ▁the ▁real ▁numbers . ▁ ▁Properties ▁ ▁Basic ▁properties ▁▁ ▁Any ▁non - zero ▁real ▁number ▁is ▁either ▁negative ▁or ▁positive . ▁ ▁The ▁sum ▁and ▁the ▁product ▁of ▁two ▁non - negative ▁real ▁numbers ▁is ▁again ▁a ▁non - negative ▁real ▁number , ▁i . e ., ▁they ▁are ▁closed ▁under ▁these ▁operations , ▁and ▁form ▁a ▁positive ▁cone , ▁thereby ▁giving ▁rise ▁to ▁a ▁linear ▁order ▁of ▁the ▁real ▁numbers ▁along ▁a ▁number ▁line . ▁ ▁The ▁real ▁numbers ▁make ▁up ▁an ▁infinite ▁set ▁of ▁numbers ▁that ▁cannot ▁be ▁inject ively ▁mapped ▁to ▁the ▁infinite ▁set ▁of ▁natural ▁numbers , ▁i . e ., ▁there ▁are ▁un count ably ▁infinitely ▁many ▁real ▁numbers , ▁whereas ▁the ▁natural ▁numbers ▁are ▁called ▁count ably ▁infinite . ▁This ▁establish es ▁that ▁in ▁some ▁sense , ▁there ▁are ▁more ▁real ▁numbers ▁than