▁Real ▁numbers ▁satisfy ▁the ▁usual ▁rules ▁of ▁arithmetic , ▁but ▁floating - point ▁numbers ▁do ▁not . ▁ ▁Comput ers ▁cannot ▁directly ▁store ▁arbitrary ▁real ▁numbers ▁with ▁infinitely ▁many ▁digits . ▁The ▁achiev able ▁precision ▁is ▁limited ▁by ▁the ▁number ▁of ▁bits ▁allocated ▁to ▁store ▁a ▁number , ▁whether ▁as ▁floating - point ▁numbers ▁or ▁arbitrary - prec ision ▁numbers . ▁However , ▁computer ▁algebra ▁systems ▁can ▁operate ▁on ▁ir r ational ▁quantities ▁exactly ▁by ▁manip ulating ▁formulas ▁for ▁them ▁( such ▁as ▁▁ ▁or ▁) ▁rather ▁than ▁their ▁rational ▁or ▁decimal ▁approximation . ▁It ▁is ▁not ▁in ▁general ▁possible ▁to ▁determine ▁whether ▁two ▁such ▁expressions ▁are ▁equal ▁( the ▁constant ▁problem ). ▁ ▁A ▁real ▁number ▁is ▁called ▁comput able ▁if ▁there ▁exists ▁an ▁algorithm ▁that ▁yields ▁its ▁digits . ▁Because ▁there ▁are ▁only ▁count ably ▁many ▁algorithms , ▁but ▁an ▁un count able ▁number ▁of ▁re als , ▁almost ▁all ▁real ▁numbers ▁fail ▁to ▁be ▁comput able . ▁Moreover , ▁the ▁equality ▁of ▁two ▁comput able ▁numbers ▁is ▁an ▁und ec id able ▁problem . ▁Some ▁construct iv ists ▁accept ▁the ▁existence ▁of ▁only ▁those ▁re als ▁that ▁are ▁comput able . ▁The ▁set ▁of ▁defin able ▁numbers ▁is ▁bro ader , ▁but ▁still ▁only ▁count able . ▁ ▁" Re als " ▁in ▁set ▁theory ▁▁ ▁In ▁set ▁theory , ▁specifically ▁descript ive ▁set ▁theory , ▁the ▁B aire ▁space ▁is ▁used ▁as ▁a ▁sur rog ate ▁for ▁the ▁real ▁numbers ▁since ▁the ▁latter ▁have ▁some ▁topological ▁properties ▁( connected