# Definition:International Standard Book Number/ISBN-10

## Definition

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

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:

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

Hyphens may be included in the string to separate out the sections for lexicographical convenience.

## Examples

### Example: $0$-$19$-$853453$-$1$

The book whose ISBN-$10$ code begins:

- $0$-$19$-$853453$

has a check digit of $1$.

## 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.

## Sources

- 1996: John F. Humphreys:
*A Course in Group Theory*... (previous) ... (next): Chapter $6$: Error-correcting codes