An ISBN is a 10 digit "structured" number - different parts of the number have different meanings (similar to the ZIP codes). The parts of the number are separated by spaces or hyphens (hyphens are preferred, but not required.) The ISBN is broken up into four parts, the sizes of the first three parts are variable, but the total number of digits used in these parts must add up to nine. The last digit is a check digit which is calculated from the previous nine digits. The ISBN's are usually printed on the back cover of a book and look like these examples:

The first part of the ISBN identifies a country, area or language area participating in the ISBN system. Some members form language areas (e.g. group number 3 = German language group) or regional units (e.g. South Pacific = group number 982). A group identifier may consist of up to 5 digits. Group number 0 is an English language group which includes the United States, the United Kingdom, Australia, South Africa and other countries. Group number 87 is Denmark, and group number 99942 is Sudan (Africa).

The second part of the ISBN identifies a particular publisher within a group. The publisher identifier usually indicates the exact identification of the publishing house and its address. If publishers use up their initial collection of title numbers, they may be given an additional publisher identifier. The publisher identifier consists of up to seven digits.

The third part of the ISBN identifies a specific edition of a publication of a specific publisher. A title identifier may consist of up to six digits. As an ISBN must always have ten digits, leading zeros are used to fill up the number in the title section.

The check digit of an ISBN is calculated in a more complex way than the check digit of a PostNET code. For an ISBN check digit, multiply the first digit by 10, the second digit by 9, the third digit by 8, ..., the ninth digit by 2 and add up all these numbers. The check digit is the number you have to add to this total to get up to a multiple of 11. So, for the second example given above, we would calculate:

80 + 63 + 8 + 7 + 0 + 35 + 20 + 15 + 18 = 246.

The calculation for the first example is:

0 + 36 + 56 + 7 + 6 + 45 + 0 + 12 + 14 = 176.

**When the check digit turns out to be "10" an "X" is written in the ISBN check digit place ["X" is the Roman numeral for 10] so that there are exactly 10 symbols used in an ISBN.**

Find the check digits for the following:

- ISBN 0 - 7195 - 4400 - ?
- ISBN 3 - 12 - 565751 - ?
- ISBN 3 - 04 - 013341 - ?

In these exercises you had to work with multiples of 11 to find the check digit. There are some easy rules that can help you find these check digits quickly which work because 11 is a special number. There is a quick trick for multiplying a number by 11 because 11 = 10 + 1. Consider multiplying the two digit number ab by 11. Now, ab × 11 = ab × (10 + 1) = ab × 10 + ab = ab0 + ab = a (a+b) b. That is, to multiply a 2 digit number like 35 by 11, the first digit of the answer will be 3 and the last digit will be 5, the middle digit is the sum of 3 and 5, i.e., 8, so 35 × 11 = 385. You can see this pattern in these examples, 25 × 11 = 275, and 36 × 11 = 396. But, what happens if the sum a + b is ten or more (that is, more than 1 digit long)? In this case, the middle digit of the answer is the last digit of the sum, and the first digit of the answer is 1 more than a (this is just the "carry" of addition). Thus, 48 × 11 = 528 (since 4 + 8 = 12, we put a 2 in the middle and increase the 4 in the beginning by 1), and 68 × 11 = 748 (since 6 + 8 = 14). Knowing these rules, it is very easy to multiply two digit numbers by 11 in your head. There is a similar rule for multiplying larger numbers by 11, but we will not discuss it here. Can you figure out how this rule works? [Hint: Look at this example, which can be done in your head: 74123454321 × 11 = 815357997531 and work from the back of the number to the front].

Do these multiplications in your head:

- 26 × 11 =
- 84 × 11 =
- 66 × 11 =

Knowing this trick for multiplying by 11, it is easy to see when a 3 digit number is a multiple of 11. Either the middle digit is the sum of the first and last, or the middle digit + 11 is the sum of the first and last digits. So, 792 is a multiple of 11 since 9 = 7 + 2, and 924 is a multiple of 11 since 11 + 2 = 9 + 4. This second rule comes about because when ab is multiplied by 11 and a + b is 10 or more, the three digits of the product are (a + 1) (a + b -10) (b), so we see that the sum of the first and last digits (a + 1) + b = a + b + 1 is the same as 11 + the second digit, 11 + a + b - 10 = 1 + a + b.

Decide which of these numbers is a multiple of 11 without using a calculator or writing anything on paper:

- 473
- 560
- 899
- 858

Now we can look at the question of how to quickly find the check digit of an ISBN. When we calculate the sum, the answer is almost always a 3 digit number (it can be smaller, but never larger). Suppose the 3 digits in this number are x, y and z in that order. The check digit is y - (x + z) mod 11, that is, subtract the sum of the first and last digit from the middle digit, and then use clock arithmetic mod 11 to write the answer as a number between 0 and 10. The clock arithmetic means that if we get a negative number, we just add enough 11's to make sure that the answer is not negative. Here are a couple of examples to show how this rule works:

**395** : 9 - (3 + 5) = 9 - 8 = 1, and since this is not negative we stop, the check digit is 1. Notice that 395 + 1 = 396 a multiple of 11.

**428** : 2 - (4 + 8) = 2 - 12 = -10, since this is negative add 11, 11 - 10 = 1 and we stop, the check digit is 1. Notice that 428 + 1 = 429 = 39 × 11.

**628** : 2 - (6 + 8) = 2 - 14 = -12, so we add 11 to get 11 - 12 = -1, so we add another 11 and finally get 11 - 1 = 10 the check digit. Again, notice that 628 + 10 = 638 = 58 × 11.

**858** : 5 - (8 + 8) = 5 - 16 = -11, so we add 11 to get 11 - 11 = 0, and 0 is the check digit. We could have written this immediately by noticing that 858 is a multiple of 11 since 11 + 5 = 8 + 8.

Find the check digits for these 3 digit sums:

- 476
- 560
- 859
- 616

A question that you may have thought of is - why should the ISBN check digit be calculated in this complicated way? The answer is more practical than mathematical in nature. At the time the book publishers were working out this system, computers were just starting to be used for business purposes, and the system was supposed to make it easy to store all this information about books on computers. They realized that when these numbers were put into the computers, the humans doing this job would make many mistakes in typing (that's just human nature) and these would be very hard to discover. The idea of using a check digit came about to help find such errors easily. The most common typing mistake that humans make is switching two digits in a number, such as typing "12354" when the real number is "12345". This type of typing error is called a ** transposition error**. In the PostNET scheme, an error of this kind would not be detected because none of the digits in the ZIP code would change and that method for calculating a check digit would not be very good in this situation. The chief advantage of the ISBN scheme is that any transposition error will be detected by the check digit. This comes about because each digit of the ISBN is multiplied by a different number, and none of these multipliers divide 11. Say that a book with number ISBN 91 - 21 - 06534 - 9 was mistyped as ISBN 91 - 21 - 06354 - 9. As a check on the typing accuracy, the check digit would be recalculated (by computer): 9×10 + 1×9 + 2×8 + 1×7 + 0×6 + 6×5 + 3×4 + 5×3 + 4×2 = 90 + 9 + 16 + 7 + 0 + 30 + 12 + 15 + 8 = 187, and the check digit for 187 is 0 and not 9. So, this typing error is detected and the typist would be asked to retype the number.

Which of the following are correct numbers? :

- ISBN 0 - 7195 - 4402 - 9
- ISBN 3 - 13 - 565760 - 1
- ISBN 3 - 04 - 013341 - X

The ISBN system has been used for over 30 years and we have gotten to the point where we will soon be running out of numbers (in part because ISBN's are now used for more than just books). By January 1, 2007 all ISBN's will be replaced by a new system, ISBN-13. In the conversion process (which is taking place right now) all items with an ISBN are changed in the following way: A 13-digit number is created by starting the number with 978 followed by the first 9 digits of the current ISBN and finishing with a new check digit. The new check digit is calculated in a different way than the old ISBN check digit that we have been talking about. After the conversion process is completed, when the old ISBN's have run out, new ones will start with the digits 979. Another reason for converting to the new system is that it permits the use of a popular bar code (EAN-13) to code the ISBN's. You can see the converted numbers in new books which have the ISBN given in a bar code (look at the little numbers below the bar code) - [not all books, in particular popular paperbacks, use the ISBN bar code, but school books usually do].

Answers to Exercise 1

- 2
- 9
- X (since the check digit should be 10)

Answers to Exercise 2

- 286
- 924
- 726

Answers to Exercise 3

- yes, 473 = 43 × 11 ( 7 = 4 + 3)
- no (6 is not 5 + 0 and 17 is not 5 + 0)
- no (9 is not 8 + 9 and 20 is not 8 + 9)
- yes, 858 = 78 × 11 (16 = 8 + 8)

Answers to Exercise 4

- 8
- 1
- X ( = 10)
- 0

Answers to Exercise 5

- Not correct.
- Not correct.
- Correct.