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ISBN-10 to ISBN-13

ISBN: How the Check Digit Validates a Book's Code

An ISBN is not just a catalog number: its final digit exists to prove the whole code was typed or printed correctly. This page shows how the check digit is calculated for both ISBN-10 and ISBN-13, and converts a real code from one format to the other, recalculating the digit by hand, before you look one up in the tool.

Two moduli, two formats

ISBN-10 has 9 content digits plus 1 check digit, calculated in modulo 11: each of the 10 digits is multiplied by a weight that descends from 10 to 1, left to right, and the sum of the products must be a multiple of 11. Since that remainder can land on 10, the standard allows the letter X in the check-digit position to represent that value.
ISBN-13 dropped modulo 11 for modulo 10 with alternating weights of 1 and 3, exactly the same algorithm as an EAN-13 barcode, which is what lets a store scan a book at checkout with the same reader used for any other product. Every ISBN-13 starts with the prefix 978 or 979, reserved by GS1 for publishing within the EAN system.
The trap is assuming you can just paste 978 in front of an ISBN-10 to convert it: the check digit changes because the underlying algorithm changes, so it has to be recalculated from scratch. Take ISBN-10 0-306-40615-2: the weighted sum from 10 down to 1 comes to 132, an exact multiple of 11, so the code is valid. Prefixing 978 to its first 9 digits and summing with alternating weights of 1 and 3 gives 93, whose remainder on division by 10 is 3; the check digit is 10 minus that remainder, which is 7. The correct ISBN-13 is 978-0-306-40615-7, not 978-0-306-40615-2.

Three check-digit verifications

Valid ISBN-10

Input: 0-306-40615-2
Expected output: Soma ponderada 132, múltiplo exato de 11: válido

The weights run from 10 down to 1, left to right, and the sum lands exactly on a multiple of 11, with no remainder.

Conversion to ISBN-13

Input: 978-0-306-40615-?
Expected output: Soma alternada 1/3 dá resto 3; dígito verificador recalculado é 7

The 978 prefix switches the algorithm from modulo 11 to modulo 10 with weights of 1 and 3, so the final digit changes too: the correct ISBN-13 ends in 7, not 2.

Wrong check digit

Input: 0-306-40615-3
Expected output: Soma ponderada 133, não é múltiplo de 11: rejeitado

Changing only the last digit shifts the remainder on division by 11 and invalidates the entire code, even though the other 9 digits are correct.

Full tool FAQ

Does the tool accept ISBN-10 and ISBN-13?

Yes, that is the expected behavior. Since the publishing market has worked with both numbering formats, a lookup usually accepts both as long as the informed code is recognized by the underlying source.

Does every ISBN lookup return a cover image and complete metadata?

What is an ISBN?

What is the difference between ISBN-10 and ISBN-13?

Can two different books have the same ISBN?

Do ebooks have ISBN?

Does a new edition or new translation need a new ISBN?

Are ISBN and ISSN the same thing?

Frequently asked questions

Why does ISBN-10 accept the letter X?

Because ISBN-10's modulo 11 can produce a remainder of 10, a value that doesn't fit in a single decimal digit; the standard solves this by allowing the letter X in the check-digit position only.

Does every book have both an ISBN-10 and an ISBN-13?

Books published before the industry moved to the 13-digit standard in 2007 originally only had an ISBN-10, but any ISBN-10 can be mathematically converted to ISBN-13 by prefixing 978 and recalculating the check digit; books released after 2007 are issued with an ISBN-13 from the start.

What does each block of an ISBN represent?

Prefix (978 or 979 in ISBN-13), registration group (identifies the country or language area), registrant (the publisher), publication number (the specific title), and check digit; the size of each block varies with how much a publisher or country has already registered.

Is the ISBN the same as the book's barcode?

Yes for ISBN-13: it follows the EAN-13 algorithm exactly, so the barcode printed on a book's back cover is the ISBN-13 itself encoded, with no additional conversion.

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ISBN-10 to ISBN-13

ISBN: How the Check Digit Validates a Book's Code

Two moduli, two formats

ISBN-10 has 9 content digits plus 1 check digit, calculated in modulo 11: each of the 10 digits is multiplied by a weight that descends from 10 to 1, left to right, and the sum of the products must be a multiple of 11. Since that remainder can land on 10, the standard allows the letter X in the check-digit position to represent that value.

ISBN-13 dropped modulo 11 for modulo 10 with alternating weights of 1 and 3, exactly the same algorithm as an EAN-13 barcode, which is what lets a store scan a book at checkout with the same reader used for any other product. Every ISBN-13 starts with the prefix 978 or 979, reserved by GS1 for publishing within the EAN system.

The trap is assuming you can just paste 978 in front of an ISBN-10 to convert it: the check digit changes because the underlying algorithm changes, so it has to be recalculated from scratch. Take ISBN-10 0-306-40615-2: the weighted sum from 10 down to 1 comes to 132, an exact multiple of 11, so the code is valid. Prefixing 978 to its first 9 digits and summing with alternating weights of 1 and 3 gives 93, whose remainder on division by 10 is 3; the check digit is 10 minus that remainder, which is 7. The correct ISBN-13 is 978-0-306-40615-7, not 978-0-306-40615-2.

Three check-digit verifications

Valid ISBN-10

Input: 0-306-40615-2
Expected output: Soma ponderada 132, múltiplo exato de 11: válido

The weights run from 10 down to 1, left to right, and the sum lands exactly on a multiple of 11, with no remainder.

Conversion to ISBN-13

Input: 978-0-306-40615-?
Expected output: Soma alternada 1/3 dá resto 3; dígito verificador recalculado é 7

The 978 prefix switches the algorithm from modulo 11 to modulo 10 with weights of 1 and 3, so the final digit changes too: the correct ISBN-13 ends in 7, not 2.

Wrong check digit

Input: 0-306-40615-3
Expected output: Soma ponderada 133, não é múltiplo de 11: rejeitado

Changing only the last digit shifts the remainder on division by 11 and invalidates the entire code, even though the other 9 digits are correct.

Full tool FAQ

Does the tool accept ISBN-10 and ISBN-13?

Does every ISBN lookup return a cover image and complete metadata?

What is an ISBN?

What is the difference between ISBN-10 and ISBN-13?

Can two different books have the same ISBN?

Do ebooks have ISBN?

Does a new edition or new translation need a new ISBN?

Are ISBN and ISSN the same thing?

Frequently asked questions

Why does ISBN-10 accept the letter X?

Because ISBN-10's modulo 11 can produce a remainder of 10, a value that doesn't fit in a single decimal digit; the standard solves this by allowing the letter X in the check-digit position only.

Does every book have both an ISBN-10 and an ISBN-13?

What does each block of an ISBN represent?

Is the ISBN the same as the book's barcode?

Yes for ISBN-13: it follows the EAN-13 algorithm exactly, so the barcode printed on a book's back cover is the ISBN-13 itself encoded, with no additional conversion.