▁there ▁are ▁elements ▁in ▁any ▁count able ▁set . ▁ ▁There ▁is ▁a ▁hierarchy ▁of ▁count ably ▁infinite ▁subsets ▁of ▁the ▁real ▁numbers , ▁e . g ., ▁the ▁integers , ▁the ▁rational s , ▁the ▁algebraic ▁numbers ▁and ▁the ▁comput able ▁numbers , ▁each ▁set ▁being ▁a ▁proper ▁subset ▁of ▁the ▁next ▁in ▁the ▁sequence . ▁The ▁comple ments ▁of ▁all ▁these ▁sets ▁( ir r ational , ▁trans c endent al , ▁and ▁non - comput able ▁real ▁numbers ) ▁with ▁respect ▁to ▁the ▁re als , ▁are ▁all ▁un count ably ▁infinite ▁sets . ▁ ▁Real ▁numbers ▁can ▁be ▁used ▁to ▁express ▁measurements ▁of ▁continuous ▁quantities . ▁They ▁may ▁be ▁expressed ▁by ▁decimal ▁representations , ▁most ▁of ▁them ▁having ▁an ▁infinite ▁sequence ▁of ▁digits ▁to ▁the ▁right ▁of ▁the ▁decimal ▁point ; ▁these ▁are ▁often ▁represented ▁like ▁ 3 2 4 . 8 2 3 1 2 2 1 4 7 ..., ▁where ▁the ▁elli ps is ▁( three ▁d ots ) ▁indicates ▁that ▁there ▁would ▁still ▁be ▁more ▁digits ▁to ▁come . ▁This ▁hints ▁to ▁the ▁fact ▁that ▁we ▁can ▁precisely ▁denote ▁only ▁a ▁few , ▁selected ▁real ▁numbers ▁with ▁finit ely ▁many ▁symbols . ▁ ▁More ▁formally , ▁the ▁real ▁numbers ▁have ▁the ▁two ▁basic ▁properties ▁of ▁being ▁an ▁ordered ▁field , ▁and ▁having ▁the ▁least ▁upper ▁bound ▁property . ▁The ▁first ▁says ▁that ▁real ▁numbers ▁compr ise ▁a ▁field , ▁with ▁addition ▁and ▁multiplication ▁as ▁well ▁as ▁division ▁by ▁non - zero ▁numbers , ▁which ▁can ▁be