# Definition:International Standard Book Number

## Definition

An international standard book number is an index number used to uniquely identify a published book.

### ISBN-$10$

The ISBN-$10$ format consists of a string of $9$ digits followed by a check digit in the set $\set {0, 1, \ldots, 9, X}$.

The check digit $d$ is calculated as follows.

Let $c$ be calculated as:

$c = \displaystyle \sum_{k \mathop = 1}^9 k d_k \pmod {11}$

where $d_k$ denotes the $k$th digit in the ISBN-$10$.

Then the check digit $d$ is defined as:

$d = \begin{cases} c & : c \le 9 \\ X &: c = 10 \end{cases}$

### ISBN-$13$

The ISBN-$13$ format consists of a string of $12$ digits followed by a check digit.

The check digit $d$ is calculated as:

$d = \displaystyle \sum_{k \mathop = 1}^{12} k d_k \pmod {10}$

## Also known as

An international standard book number is usually referred to by its abbreviation ISBN.

## Linguistic Note

When referring to an international standard book number by its abbreviation ISBN, beware not to vocalise it as "ISBN number".

The word number is already included in the abbreviation ISBN, so does not need to be repeated.