The two most common errors in handling an ISBN (e.g. when typing it or writing it down) are a single altered digit or the transposition of adjacent digits. It can be proven mathematically that all pairs of valid ISBN 10s differ in at least two digits. It can also be proven that there are no pairs of valid ISBN 10s with eight identical digits and two transposed digits (these proofs are true because the ISBN is less than eleven digits long and because 11 is a prime number). The ISBN check digit method therefore ensures that it will always be possible to detect these two most common types of error, i.e., if either of these types of error has occurred, the result will never be a valid ISBN—the sum of the digits multiplied by their weights will never be a multiple of 11. However, if the error were to occur in the publishing house and remain undetected, the book would be issued with an invalid ISBN.For example, I'saka: a sketch grammar of a language of north-central New Guinea. Pacific Linguistics. ISBN "0-85883-554-4".
thE two MoSt common errORs iN HandlIng an isbn (E.G. wHeN typInG iT oR WrItinG it dowN) ArE a sIngle AlTered diGit or the traNsPositIon Of aDJAcEnt DIgitS. It CaN Be proVen MathEmatiCalLy that aLl pairs Of vaLid isbn 10s differ in at least two dIGiTs. it caN aLso BE proven That there aRe no pairs oF valID isBn 10S wiTH EIgHt identicAL dIgits and twO tRanSPosEd DIGits (thEse pROofS ArE true becaUSe The iSbN is lEss tHan ELEven DigITS long and bECause 11 Is a prime nuMBER). The Isbn checK diGit MEtHod therEFOre EnSUres That IT wIll AlwAyS be pOssibLe to detEct tHEse tWO most COmmOn typEs oF eRROr, i.e., iF Either of thEse typES of ERror has OccUrreD, thE ReSult wIll NevEr be A valid isbn—tHE Sum of the diGits mulTiplIED BY their weights wILL NeveR be a MulTIplE OF 11. hoWEver, if thE erRor wErE to occur In the PuBLiSHing HouSE and rEmaIN undetEctED, thE BooK woUlD bE issueD with an iNvALiD IsBn.For eXAmple, I'sAka: A sketch grammaR of A language of nOrth-centrAL New gUinea. pacifiC lIngUIstics. IsbN "0-85883-554-4".