▁real ▁numbers ; ▁this ▁vector ▁space ▁may ▁be ▁identified ▁to ▁the ▁n - dimensional ▁space ▁of ▁E uclidean ▁geometry ▁as ▁soon ▁as ▁a ▁coordinate ▁system ▁has ▁been ▁chosen ▁in ▁the ▁latter . ▁For ▁example , ▁a ▁value ▁from ▁R 3 ▁consists ▁of ▁three ▁real ▁numbers ▁and ▁spec ifies ▁the ▁coordinates ▁of ▁a ▁point ▁in ▁ 3 ‑ dimensional ▁space . ▁ ▁In ▁mathematics , ▁real ▁is ▁used ▁as ▁an ▁ad jective , ▁meaning ▁that ▁the ▁underlying ▁field ▁is ▁the ▁field ▁of ▁the ▁real ▁numbers ▁( or ▁the ▁real ▁field ). ▁For ▁example , ▁real ▁matrix , ▁real ▁polynomial ▁and ▁real ▁Lie ▁algebra . ▁The ▁word ▁is ▁also ▁used ▁as ▁a ▁n oun , ▁meaning ▁a ▁real ▁number ▁( as ▁in ▁" the ▁set ▁of ▁all ▁re als "). ▁ ▁General izations ▁and ▁extensions ▁▁ ▁The ▁real ▁numbers ▁can ▁be ▁generalized ▁and ▁extended ▁in ▁several ▁different ▁directions : ▁ ▁The ▁complex ▁numbers ▁contain ▁solutions ▁to ▁all ▁polynomial ▁equations ▁and ▁hence ▁are ▁an ▁algebra ically ▁closed ▁field ▁unlike ▁the ▁real ▁numbers . ▁However , ▁the ▁complex ▁numbers ▁are ▁not ▁an ▁ordered ▁field . ▁ ▁The ▁aff in ely ▁extended ▁real ▁number ▁system ▁adds ▁two ▁elements ▁+ ∞ ▁and ▁− ∞ . ▁It ▁is ▁a ▁compact ▁space . ▁It ▁is ▁no ▁longer ▁a ▁field , ▁or ▁even ▁an ▁add itive ▁group , ▁but ▁it ▁still ▁has ▁a ▁total ▁order ; ▁moreover , ▁it ▁is ▁a ▁complete ▁lattice . ▁ ▁The ▁real ▁project ive ▁line ▁adds ▁only ▁one ▁value ▁ ∞ . ▁It ▁is ▁also ▁a ▁compact