▁modes ▁and ▁is ▁proved . ▁ ▁There ▁exist ▁comput able ▁numbers ▁for ▁which ▁a ▁rounded ▁value ▁can ▁never ▁be ▁determined ▁no ▁matter ▁how ▁many ▁digits ▁are ▁calculated . ▁Specific ▁instances ▁cannot ▁be ▁given ▁but ▁this ▁follows ▁from ▁the ▁und ec id ability ▁of ▁the ▁hal ting ▁problem . ▁For ▁instance , ▁if ▁Gold bach ' s ▁conject ure ▁is ▁true ▁but ▁un prov able , ▁then ▁the ▁result ▁of ▁round ing ▁the ▁following ▁value ▁up ▁to ▁the ▁next ▁integer ▁cannot ▁be ▁determined : ▁ 1 + 1 0 − ▁where ▁ ▁is ▁the ▁first ▁even ▁number ▁greater ▁than ▁ 4 ▁which ▁is ▁not ▁the ▁sum ▁of ▁two ▁pr imes , ▁or ▁ 1 ▁if ▁there ▁is ▁no ▁such ▁number . ▁The ▁rounded ▁result ▁is ▁ 2 ▁if ▁such ▁a ▁number ▁ ▁exists ▁and ▁ 1 ▁otherwise . ▁The ▁value ▁before ▁round ing ▁can ▁however ▁be ▁approxim ated ▁to ▁any ▁given ▁precision ▁even ▁if ▁the ▁conject ure ▁is ▁un prov able . ▁ ▁Inter action ▁with ▁string ▁searches ▁R ounding ▁can ▁advers ely ▁affect ▁a ▁string ▁search ▁for ▁a ▁number . ▁For ▁example , ▁ ▁rounded ▁to ▁four ▁digits ▁is ▁" 3 . 1 4 1 6 " ▁but ▁a ▁simple ▁search ▁for ▁this ▁string ▁will ▁not ▁discover ▁" 3 . 1 4 1 5 9 " ▁or ▁any ▁other ▁value ▁of ▁ ▁rounded ▁to ▁more ▁than ▁four ▁digits . ▁In ▁contrast , ▁trunc ation ▁does ▁not ▁suffer ▁from ▁this ▁problem ; ▁for ▁example , ▁a ▁simple ▁string ▁search ▁for ▁" 3 . 1