Editing ISBN (section)

===ISBN-10 check digits===
According to the 2001 edition of the International ISBN Agency's official user manual,<ref>{{cite web|url=http://www.isbn.org/standards/home/isbn/international/html/usm4.htm |title=ISBN Users' Manual (2001 edition) – 4. Structure of ISBN |publisher=International ISBN Agency |url-status=dead |archive-url=https://web.archive.org/web/20130522043458/http://www.isbn.org/standards/home/isbn/international/html/usm4.htm |archive-date=22 May 2013 }}</ref> the ISBN-10 check digit (which is the last digit of the 10-digit ISBN) must range from 0 to 10 (the symbol 'X' is used for 10), and must be such that the sum of the ten digits, each multiplied by its (integer) weight, descending from 10 to 1, is a multiple of 11. That is, if {{var|x}}{{sub|{{var|i}}}} is the {{var|i}}th digit, then {{var|x}}{{sub|10}} must be chosen such that:
{{block indent|<math>\sum_{i = 1}^{10} (11-i)x_i \equiv 0 \pmod{11}.</math>}}
For example, for an ISBN-10 of 0-306-40615-2:
{{block indent|<math>
\begin{align}
 s &= (0\times 10) + (3\times 9) + (0\times 8) + (6\times 7) + (4\times 6) + (0\times 5) + (6\times 4) + (1\times 3) + (5\times 2) + (2\times 1) \\
   &=    0 + 27 +   0 +  42 +  24 +   0 + 24  +   3 + 10 + 2\\
   &= 132 = 12\times 11.
\end{align}
</math>}}

Formally, using [[modular arithmetic]], this is rendered
{{block indent|<math>(10x_1+9x_2+8x_3+7x_4+6x_5+5x_6+4x_7+3x_8+2x_9+x_{10})\equiv 0 \pmod{11}.</math>}}

It is also true for ISBN 10s that the sum of all ten digits, each multiplied by its weight in ''ascending'' order from 1 to 10, is a multiple of 11. For this example:
{{block indent|<math>
\begin{align}
 s &= (0\times 1) + (3\times 2) + (0\times 3) + (6\times 4) + (4\times 5) + (0\times 6) + (6\times 7) + (1\times 8) + (5\times 9) + (2\times 10) \\
   &=    0 + 6 +   0 +  24 +  20 +   0 + 42  +   8 + 45 + 20\\
   &= 165 = 15\times 11.
\end{align}
</math>}}

Formally, this is rendered
{{block indent|<math>(x_1 + 2x_2 + 3x_3 + 4x_4 + 5x_5 + 6x_6 + 7x_7 + 8x_8 + 9x_9 + 10x_{10})\equiv 0 \pmod{11}.</math>}}

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.<ref>For example, ''I'saka: a sketch grammar of a language of north-central New Guinea.'' Pacific Linguistics. ISBN "0-85883-554-4".</ref>

In contrast, it is possible for other types of error, such as two altered non-transposed digits, or three altered digits, to result in a valid ISBN (although it is still unlikely).