, ▁the ▁nearest ▁fraction ▁  =   / ▁whose ▁numer ator ▁ ▁and ▁denomin ator ▁ ▁do ▁not ▁exceed ▁a ▁given ▁maximum . ▁This ▁problem ▁is ▁fairly ▁distinct ▁from ▁that ▁of ▁round ing ▁a ▁value ▁to ▁a ▁fixed ▁number ▁of ▁decimal ▁or ▁binary ▁digits , ▁or ▁to ▁a ▁multiple ▁of ▁a ▁given ▁unit ▁. ▁This ▁problem ▁is ▁related ▁to ▁F are y ▁sequences , ▁the ▁Stern – B roc ot ▁tree , ▁and ▁continued ▁fra ctions . ▁ ▁R ounding ▁to ▁an ▁available ▁value ▁Fin ished ▁l umber , ▁writing ▁paper , ▁capac itors , ▁and ▁many ▁other ▁products ▁are ▁usually ▁sold ▁in ▁only ▁a ▁few ▁standard ▁sizes . ▁ ▁Many ▁design ▁procedures ▁describe ▁how ▁to ▁calculate ▁an ▁approximate ▁value , ▁and ▁then ▁" round " ▁to ▁some ▁standard ▁size ▁using ▁phr ases ▁such ▁as ▁" round ▁down ▁to ▁nearest ▁standard ▁value ", ▁" round ▁up ▁to ▁nearest ▁standard ▁value ", ▁or ▁" round ▁to ▁nearest ▁standard ▁value ". ▁ ▁When ▁a ▁set ▁of ▁preferred ▁values ▁is ▁equally ▁spac ed ▁on ▁a ▁log arith mic ▁scale , ▁choosing ▁the ▁closest ▁preferred ▁value ▁to ▁any ▁given ▁value ▁can ▁be ▁seen ▁as ▁a ▁form ▁of ▁scale d ▁round ing . ▁Such ▁rounded ▁values ▁can ▁be ▁directly ▁calculated . ▁ ▁R ounding ▁in ▁other ▁context s ▁ ▁D ither ing ▁and ▁error ▁diffusion ▁When ▁digit izing ▁continuous ▁signals , ▁such ▁as ▁sound ▁waves , ▁the ▁overall ▁effect ▁of ▁a ▁number ▁of ▁measurements ▁is ▁more ▁important ▁than ▁the ▁accuracy ▁of ▁each ▁individual ▁measurement . ▁In ▁these ▁circumstances