Integers and Modern Applications for the Uninitiated

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 an example of an application of an area of mathematics called coding theory. The code used for ISBN-10 is an error-detecting code. This means it can detect whether a common error, such as a wrong digit or two swapped digits, occurred while handling the number.

Here \(x_1\text{,}\) \(x_2\text{,}\) \(x_3\text{,}\) \(x_4\text{,}\) \(x_5\text{,}\) \(x_6\text{,}\) \(x_7\text{,}\) \(x_8\text{,}\) \(x_9\text{,}\) 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.

To avoid too many parentheses in the next computations, we change our rules for the order of operations. We evaluate multiplication before addition. Therefore, instead of \((a\cdot b)+c\) we can write \(a\cdot b+c\text{.}\) When \(a\) is an integer, we sometimes 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 \(9b\text{.}\)

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

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

The ISBN is built 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 by mistake swapped. The final example of this section shows the detection of such an error. Additionally, notice that we apply Theorem 3.4.10 to help us reduce each entry in the sum before adding so that we can work with smaller numbers.

We saw that \(\fmod\) is used to authenticate ISBN numbers. Similar methods authenticate other numbers, such as banknote serial numbers and credit card numbers.