▁infinitely ▁long ▁line ▁called ▁the ▁number ▁line ▁or ▁real ▁line , ▁where ▁the ▁points ▁corresponding ▁to ▁integers ▁are ▁equally ▁spac ed . ▁Any ▁real ▁number ▁can ▁be ▁determined ▁by ▁a ▁possibly ▁infinite ▁decimal ▁representation , ▁such ▁as ▁that ▁of ▁ 8 . 6 3 2 , ▁where ▁each ▁consecutive ▁digit ▁is ▁measured ▁in ▁units ▁one ▁t enth ▁the ▁size ▁of ▁the ▁previous ▁one . ▁The ▁real ▁line ▁can ▁be ▁thought ▁of ▁as ▁a ▁part ▁of ▁the ▁complex ▁plane , ▁and ▁complex ▁numbers ▁include ▁real ▁numbers . ▁ ▁These ▁descri ptions ▁of ▁the ▁real ▁numbers ▁are ▁not ▁sufficiently ▁rig orous ▁by ▁the ▁modern ▁standards ▁of ▁pure ▁mathematics . ▁The ▁discovery ▁of ▁a ▁suit ably ▁rig orous ▁definition ▁of ▁the ▁real ▁numbers — inde ed , ▁the ▁real ization ▁that ▁a ▁better ▁definition ▁was ▁needed — was ▁one ▁of ▁the ▁most ▁important ▁develop ments ▁of ▁ 1 9 th - century ▁mathematics . ▁The ▁current ▁standard ▁ax iom atic ▁definition ▁is ▁that ▁real ▁numbers ▁form ▁the ▁unique ▁D ed ek ind - complete ▁ordered ▁field ▁ ▁up ▁to ▁an ▁isomorphism , ▁whereas ▁popular ▁construct ive ▁definitions ▁of ▁real ▁numbers ▁include ▁declaring ▁them ▁as ▁equivalence ▁classes ▁of ▁C auch y ▁sequences ▁of ▁rational ▁numbers , ▁D ed ek ind ▁cut s , ▁or ▁infinite ▁decimal ▁representations , ▁together ▁with ▁precise ▁interpret ations ▁for ▁the ▁arithmetic ▁operations ▁and ▁the ▁order ▁relation . ▁All ▁these ▁definitions ▁satisfy ▁the ▁ax iom atic ▁definition ▁and ▁are ▁thus ▁equivalent . ▁ ▁The ▁set ▁of ▁all ▁real ▁numbers ▁is ▁un count able