Two schemes, two different sums
ISBN-10 weights the first nine digits by 10, 9, 8 down to 2, adds them up, and picks a check value that brings the total to a multiple of 11. Because the remainder can be 10 and a single digit cannot express that, the character X stands in — which is why an ISBN-10 is not a number in any arithmetic sense and should be stored as a string. Mod 11 is the stronger of the two schemes: it catches every single-digit error and every transposition of any two digits, adjacent or not.
ISBN-13 is an EAN-13 barcode number, so it uses the barcode world's rule instead: weights alternating 1 and 3 across the first twelve digits, check digit chosen to reach a multiple of 10. That catches all single-digit errors but misses transpositions where the two digits differ by five — 27 and 72 produce the same sum. The trade was deliberate. Being an ordinary EAN meant books could go through the same scanners and the same retail systems as everything else on a shelf, and that was worth more than a slightly tighter check.
Why 978 converts and 979 does not
When the format moved to 13 digits at the start of 2007, the whole existing ISBN-10 space was lifted wholesale into a reserved EAN block by prefixing 978. The middle digits — group, registrant, publication — stayed exactly as they were; only the check digit was recomputed under the new rule. That is why the conversion is mechanical and lossless in both directions for anything starting 978.
The 979 prefix is new space, opened once 978 began to fill. Nothing under it has a 10-digit ancestor, so there is nothing to convert back to. 979-8 in particular is the block issued to US self-publishing, which is why so many recent print-on-demand titles have no ISBN-10 at all. Software that assumes it can always derive a 10-digit form will silently corrupt those records, and it is a common enough bug that library systems have written about it.
Hyphens, and why they are ignored here
The hyphens in 978-0-306-40615-7 are not decoration: they separate the prefix, the registration group, the registrant and the publication element. But the split points are variable-width and depend on a range table published by the International ISBN Agency, because a group with many small publishers allocates short registrant blocks and a group with a few large ones allocates long blocks. You cannot infer the positions from the digits alone without that table.
Validation does not need them. The checksum runs over the bare digits, so this tool strips hyphens and spaces and works with what is left. If you need correctly hyphenated output — for a bibliographic record, say — you need the range table, and that is a data problem rather than an arithmetic one.
Where the check stops
A clean check digit means the digits are mutually consistent. It does not mean a publisher registered that number, that a book was printed under it, or that the number in your hand belongs to the book in front of you. Publishers reuse numbers by mistake, self-publishing platforms assign their own, and a title that appears in hardback, paperback, ebook and audio carries four different ISBNs that are all equally valid. If the question you are actually asking is "what book is this", the answer lives in a catalogue — WorldCat, the Library of Congress, the national agency for the registration group — and not in the check digit.
Questions people ask
What is the X at the end of some ISBN-10s?
The ISBN-10 check value is a remainder mod 11, so it ranges from 0 to 10. Ten needs two characters as a digit, so the Roman numeral X is used instead. It only ever appears in the last position, and only on ISBN-10 — ISBN-13 works mod 10, where every possible check value is a single digit. Practically, this is why an ISBN belongs in a string column and not an integer one.
Why will this not convert my 979 number to an ISBN-10?
Because no such number exists. ISBN-10 could only ever address the range that later became the 978 block, and 979 was opened afterwards precisely because 978 was running out. Anything issued under 979, including the 979-8 range used heavily by US self-publishing, has only ever had a 13-digit form.
The check digit is valid — does the book exist?
Not necessarily. The checksum only proves the digits are consistent with each other. Any nine digits plus the right tenth will pass. To find out whether a real title sits behind the number, search a library catalogue or the ISBN agency for that registration group; this page makes no network requests and looks nothing up.
The barcode and the copyright page disagree.
It happens, usually on reprints where the barcode film was updated and the interior was not, or the other way round. Run both through this check: often one of them fails outright, which settles it. If both pass, trust the barcode, since that is the number retail and distribution systems will have been scanning, and note the discrepancy if you are creating a catalogue record.