## Section3.6Application: ISBN

An ISBN (International Standard Book Number) is a number that uniquely identifies a book. Until 2007, the assigned ISBNs were 10 digits long, but the newly-assigned ISBNs are 13 digits long. As a practical application of the operation $$\fmod\text{,}$$ we discuss the check digit of 10-digit ISBNs.

ISBNs are example of an application of an area of mathematics called coding theory. The code used for the ISBN-10 is an error detecting code. That means it can be detect whether a common error, such as a wrong digit or two swapped digits, was made in handling the number.

We write a 10-digit ISBN-10 as

\begin{equation*} x_1\,\texttt{-}\,x_2\,x_3\,x_4\,\texttt{-}\,x_5\,x_6\,x_7\,x_8\,x_9\,\texttt{-}\,x_{10}\text{.} \end{equation*}

Here $$x_1\text{,}$$ $$x_2\text{,}$$ $$x_3\text{,}$$ $$x_4\text{,}$$ $$x_5\text{,}$$ $$x_6\text{,}$$ $$x_7\text{,}$$ $$x_9\text{,}$$ and $$x_{10}$$ are variables, that is, placeholders for numbers. The purpose of the numbers in the subscript is to distinguish the variables. The choice of numbering also hints on the order. The dashes “$$\mathtt{-}$$” do not mean subtraction but are there to group the numbers.

Each of the first 9 digits $$x_1$$ to $$x_9$$ is an integer from 0 to 9. The 10th digit $$x_{10}$$ is called a check digit, used to detect certain errors in ISBN-10s. The possible values for the last digit $$x_{10}$$ are integers from $$\mathtt{0}$$ to $$\mathtt{9}$$ and the letter $$\mathtt{X}\text{,}$$ interpreted as the value 10.

As there would be too many parentheses in the following computations, we amend our conventions concerning the order of operations. We evaluate multiplication before addition, so that instead of $$(a\cdot b)+c$$ we can write $$a\cdot b+c\text{.}$$ When $$a$$ is an integer we sometimes also leave out the dot ($$\cdot$$) to save horizontal space. For example, instead of $$1\cdot b$$ we write $$1b$$ and instead of $$9\cdot b$$ we write $$9b\text{.}$$

### Definition3.6.1.

The digits

\begin{equation*} x_1\,\texttt{-}\,x_2\,x_3\,x_4\,\texttt{-}\,x_5\,x_6\,x_7\,x_8\,x_9\,\texttt{-}\,x_{10}\text{.} \end{equation*}

form a valid ISBN-10 if

\begin{equation*} x_{10}=(1x_1+2x_2+3x_3+4 x_4+5 x_5+6 x_6+7 x_7+8 x_8+9 x_9) \fmod 11\text{.} \end{equation*}

In the video in Figure 3.6.2 we introduce ISBNs and give first examples.

The locations of the dashes in ISBNs vary. They, for example, depend on the language in which a book was written and the publisher. The locations of the dashes have no implication on the validity of ISBNs.

Determine whether $$\mathtt{0-345-45374-3}$$ is a valid ISBN-10.

Solution.

We use the notation from Definition 3.6.1. We have $$x_1=0\text{,}$$ $$x_2=3\text{,}$$ $$x_3=4\text{,}$$ $$x_4=5\text{,}$$ $$x_5=4\text{,}$$ $$x_6=5\text{,}$$ $$x_7=3\text{,}$$ $$x_8=7\text{,}$$ and $$x_9=4\text{.}$$ By Definition 3.6.1

\begin{equation*} x_1\,\texttt{-}\,x_2\,x_3\,x_4\,\texttt{-}\,x_5\,x_6\,x_7\,x_8\,x_9\,\texttt{-}\,x_{10} \end{equation*}

is an ISBN-10 when

\begin{equation*} x_{10}=(1x_1+2x_2+3x_3+4 x_4+5 x_5+6 x_6+7 x_7+8 x_8+9 x_9) \fmod 11\text{.} \end{equation*}

We have

\begin{align*} x_{10}\amp=(1x_1+2x_2+3x_3+4 x_4+5 x_5+6 x_6+7 x_7+8 x_8+9 x_9) \fmod 11\\ \amp =(1\cdot 0+2\cdot 3+3\cdot 4+4\cdot 5+5\cdot 4+6\cdot 5+7\cdot 3+8\cdot 7+9\cdot 4)\fmod 11\\ \amp =(0+6+12+20+20+30+21+56+36)\fmod 11\\ \amp =(0+6+1+9+9+8+10+1+3)\fmod 11 =47\fmod 11=3\text{.} \end{align*}

As the last digit $$\mathtt{0-345-45374-3}$$ is $$\mathtt{3}$$ and we have computed that we should have $$x_{10}=3$$ we conclude that $$\mathtt{0-345-45374-3}$$ is a valid ISBN-10.

Determine whether $$\mathtt{0-475-02548-7}$$ is a valid ISBN-10.

Solution.

We use the notation from Definition 3.6.1. We have $$x_1=0\text{,}$$ $$x_2=4\text{,}$$ $$x_3=7\text{,}$$ $$x_4=5\text{,}$$ $$x_5=0\text{,}$$ $$x_6=2\text{,}$$ $$x_7=5\text{,}$$ $$x_8=4\text{,}$$ and $$x_9=8\text{.}$$ So the last digit of the ISBN-10 $$\mathtt{0-475-02548-}x_{10}$$ is

\begin{align*} x_{10}\amp=(1x_1+2x_2+3x_3+4 x_4+5 x_5+6 x_6+7 x_7+8 x_8+9 x_9) \fmod 11\\ \amp =(1\cdot 0+2\cdot 4+3\cdot 7+4\cdot 5+5\cdot 0+6\cdot 2+7\cdot 5+8\cdot 4+9\cdot 8)\fmod 11\\ \amp =(0+8+21+20+0+12+35+32+72)\fmod 11\\ \amp =(0+8+10+9+0+1+2+10+6)\fmod 11 =46\fmod 11=2 \end{align*}

As the last digit of $$\mathtt{0-475-02548-7}$$ is $$\mathtt{7}$$ and we have computed that $$x_{10}=2$$ we conclude that $$\mathtt{0-475-02548-7}$$ is not a valid ISBN-10.

In Checkpoint 3.6.5 go through the calculations from the problem yourself to determine the validity of an ISBN.

You are given

\begin{equation*} x_1-x_2 x_3 x_4 x_5 x_6 - x_7 x_8 x_9 - x_{10} \end{equation*}

as

\begin{equation*} 0 - 000 09 - 060 - 3. \end{equation*}

Enter the digits

$$x_1=$$ $$x_2=$$ $$x_3=$$ $$x_4=$$ $$x_5=$$ $$x_6=$$ $$x_7=$$ $$x_8=$$ $$x_9=$$ $$x_{10}=$$

Compute

\begin{equation*} C:=(x_1+2\cdot x_2+3\cdot x_3+4\cdot x_4+5\cdot x_5+6\cdot x_6+7\cdot x_7+8 x_8+9 x_9) \text{ mod } 11 \end{equation*}

$$C=$$

If $$C=x_{10}$$ then we have a valid ISBN-10.

Is $$0 - 00009 - 060 - 3$$ a valid ISBN-10 ?

• No

• Yes

Similarly we can also determine the tenth digit of an ISBN when it is missing.

The ISBN-10 of Euclid's The Thirteen Books of the Elements, Vol. 1: Books 1-2 is $$\mathtt{0-486-60088-}$$$$x_{10}\text{,}$$ where the check digit $$x_{10}$$ is missing. Determine the complete ISBN-10.

Solution.

We use the formula from Definition 3.6.1

\begin{equation*} x_{10}=(1x_1+2x_2+3x_3+4 x_4+5 x_5+6 x_6+7 x_7+8 x_8+9 x_9) \fmod 11 \end{equation*}

to compute the missing check digit $$x_{10}\text{.}$$

\begin{align*} x_{10}\amp =(1\cdot 0+2\cdot 4+3\cdot 8+4\cdot 6+5\cdot 6+6\cdot 0+7\cdot 0+8\cdot 8+9\cdot 8)\fmod 11\\ \amp =(8+24+24+30+64+72)\fmod 11\\ \amp=(8+2+2+8+9+6)\fmod 11=2 \end{align*}

So, the complete ISBN-10 is $$\mathtt{0-486-60088-2}\text{.}$$

In Checkpoint 3.6.7 go through the calculations from the problem yourself to determine the validity of an ISBN.

If the first nine digits of an ISBN-10 are

\begin{equation*} 3 - 628 - 58069 \end{equation*}

then the tenth digit is:

The ISBN is constructed in such a way that certain common errors that occur can be detected. For example, multiplying the first digit by 10, the second by 9, and so on, makes it possible to detect whether two digits an ISBN-10 have been inadvertently swapped. The final example of this section demonstrates the detection of such an error. Additionally, notice that we apply Theorem 3.4.10 to help us reduce each entry in the sum prior to adding so that we can work with smaller numbers.

Marion tried to enter the ISBN-10 of a book she wanted to purchase on a booksellers web page. Instead of entering the number $$\mathtt{3-540-13140-X}$$ she entered $$\mathtt{3-450-13140-X}\text{.}$$ The booksellers web server computes

\begin{align*} x_{10}\amp =(1\cdot 3 + 2\cdot 4+3\cdot 5+4\cdot 0+5\cdot 1+6\cdot 3+7\cdot 1+8\cdot 4+9\cdot 0)\fmod 11\\ \amp =(3+8+15+0+5+18+7+32+0)\fmod 11\\ \amp = (3+8+4+0+5+7+7+10)\fmod 11 =33\fmod 11 =0\text{.} \end{align*}

As the tenth digit entered by Marion is $$\mathtt{X}$$ which stands for $$10$$ and $$10\ne 0$$ the web server sends the message, that there is no book with the requested ISBN-10 number.

We have seen that the operation $$\fmod$$ is used in the authentication of ISBN numbers. Similar methods are used in the authentication of other numbers such as serial numbers of banknotes and credit card numbers,.