MAT 112 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 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.

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 \(9b\text{.}\)

In the video in Figure 3.74 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.

In Checkpoint 3.77 go through the calculations from the problem yourself to determine the validity of an ISBN.

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

In Checkpoint 3.79 go through the calculations from the problem yourself to determine the validity of an ISBN.

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 3.46 to help us reduce each entry in the sum prior to adding so that we can work with smaller numbers.

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 bank notes and credit card numbers,.