▁space . ▁Again , ▁it ▁is ▁no ▁longer ▁a ▁field , ▁or ▁even ▁an ▁add itive ▁group . ▁However , ▁it ▁allows ▁division ▁of ▁a ▁non - zero ▁element ▁by ▁zero . ▁It ▁has ▁cy clic ▁order ▁described ▁by ▁a ▁separation ▁relation . ▁ ▁The ▁long ▁real ▁line ▁past es ▁together ▁ <0xE2> <0x84> <0xB5> 1 * ▁+ ▁ <0xE2> <0x84> <0xB5> 1 ▁copies ▁of ▁the ▁real ▁line ▁plus ▁a ▁single ▁point ▁( here ▁ <0xE2> <0x84> <0xB5> 1 * ▁denotes ▁the ▁revers ed ▁ordering ▁of ▁ <0xE2> <0x84> <0xB5> 1 ) ▁to ▁create ▁an ▁ordered ▁set ▁that ▁is ▁" loc ally " ▁identical ▁to ▁the ▁real ▁numbers , ▁but ▁somehow ▁longer ; ▁for ▁instance , ▁there ▁is ▁an ▁order - pres erving ▁embedding ▁of ▁ <0xE2> <0x84> <0xB5> 1 ▁in ▁the ▁long ▁real ▁line ▁but ▁not ▁in ▁the ▁real ▁numbers . ▁The ▁long ▁real ▁line ▁is ▁the ▁largest ▁ordered ▁set ▁that ▁is ▁complete ▁and ▁locally ▁Arch im ede an . ▁As ▁with ▁the ▁previous ▁two ▁examples , ▁this ▁set ▁is ▁no ▁longer ▁a ▁field ▁or ▁add itive ▁group . ▁ ▁Order ed ▁fields ▁extending ▁the ▁re als ▁are ▁the ▁hyper real ▁numbers ▁and ▁the ▁sur real ▁numbers ; ▁both ▁of ▁them ▁contain ▁inf init es imal ▁and ▁infinitely ▁large ▁numbers ▁and ▁are ▁therefore ▁non - Arch im ede an ▁ordered ▁fields . ▁ ▁Self - ad joint ▁operators ▁on ▁a ▁Hil bert ▁space ▁( for ▁example , ▁self - ad joint ▁square ▁complex ▁matrices ) ▁general ize ▁the ▁re als ▁in ▁many