# ISBN Information

## ISBN Check Digit For The 13-Digit ISBN

#### n.b. this page shows the check digit calculation for the modern 13-digit ISBN. It is a different calculation to that for the old 10-digit ISBN.

Every ISBN number contains a check digit. It is the final single character of the ISBN. In the case of the old 10-digit ISBN the check digit can be any number from zero to nine, and it can also be a letter "x", like a Roman ten. In the case of the current 13-digit ISBN, then it could be any single digit from zero to nine.

## What The ISBN Check Digit Is For

The check digit in an ISBN number is there to help prevent errors in transmission. When an ISBN number is read and entered into any system that is used to deal with ISBNs, such as a book cataloguing system, whether the ISBN is entered by hand or by means of a bar code scanner, the check digit as read from the entered number is compared to a check digit calculated from the remainder of the ISBN that has been entered. If the two check digits are the same, then there is a good chance that the complete ISBN, all of its digits, has been read correctly.

As an example, if I enter the ISBN 9781861972712 into a system that uses ISBNs, it will take the first twelve digits (978186197271) of the ISBN and calculate the check digit for that series of twelve digits. It will come up with the number 2 as the correct digit and compare it to the number that I entered earlier, and find that I entered the number 2 as well. In that case, the chances are I entered the digits correctly. However, supposing I entered the ISBN incorrectly and made a mistake by transposing the fifth and sixth digits, entering the ISBN as 9781681972712. When the system calculate the check digit for 978168197271 it will come up with 8 as the check digit for that number. It will compare 8 with 2 and know that a mistake of some sort has been made. It won't know where the error is, only that something is wrong.

Another example. Perhaps this time I misread a digit and enter the ISBN as 9781861973712. The system would calculate the check digit for 978186197371 as the number 9. Again, 9 does equal 2 - something wrong.

The ISBN check digit system is not completely foolproof, and will not catch all such errors, but will trap many.

## How The ISBN Check Digit Is Calculated

In the previous section I calculate the check digit for a couple of example ISBNs, but didn't show my working! We'll put that right now and show the calculation for the correct number, and for both of those numbers that I entered with errors.

This is the process:

1. Take the first 12 digits of the 13-digit ISBN
2. Multiply each number in turn, from left to right by a number. The first digit is multiplied by 1, the second by 3, the third by 1 gain, the fourth by 3 again, and so on to the eleventh which is multiplied by 1 and the twelfth by 3.
4. Do a modulo 10 division on the result from step 2. (Don't know what a modulo 10 division is? It's easy. It's just the remainder from a whole number division by 10. I bet you learned to do that in junior school, before you even learned about decimal fractions.)
5. Take that remainder result from step 4.If it's a zero, then the check digit is zero. If the remainders isn't zero then subtract the remainder from 10. The answer to that is your check digit.

## First Example - correct ISBN - 9781861972712

First 12 digits x their multipliers = the results

```9 x 1 = 9
7 x 3 = 21
8 x 1 = 8
1 x 3 = 3
8 x 1 = 8
6 x 3 = 18
1 x 1 = 1
9 x 3 = 27
7 x 1 = 7
2 x 3 = 6
7 x 1 = 7
1 x 3 = 3```

`9 + 21 + 8 + 3 + 8 + 18 + 1 + 27 + 7 + 6 + 7 + 3 = 118`

Modulo 10 the result:

`118 modulo 10 = 110 remainder 8`

Work out the check digit:

`8 doesn't equal zero, so check digit = 10 - 8 = 2`

Compare the calculated check digit (2) with the one we first entered (2). They are the same, so looks like we have a good ISBN.

## Next example - transposed digits - 9781681972712

Notice how the fifth and sixth digits have been swapped around compared to the good ISBN in the first example.

First 12 digits x their multipliers = the results

```9 x 1 = 9
7 x 3 = 21
8 x 1 = 8
1 x 3 = 3
6 x 1 = 6
8 x 3 = 24
1 x 1 = 1
9 x 3 = 27
7 x 1 = 7
2 x 3 = 6
7 x 1 = 7
1 x 3 = 3```

`9 + 21 + 8 + 3 + 6 + 24 + 1 + 27 + 7 + 6 + 7 + 3 = 122`

Modulo 10 the result

`122 modulo 10 = 12 remainder 2`

Work out the check digit

`2 doesn't equal zero, so check digit = 10 - 2 = 8`

Compare the calculated check digit (8) with the one we first entered (2). They are not the same, so looks like we have a bad ISBN.

## Third example - misread digit - 9781861973712

Notice how the tenth digit has been read as a three rather than as a two.

First 12 digits x their multipliers = the results

```9 x 1 = 9
7 x 3 = 21
8 x 1 = 8
1 x 3 = 3
8 x 1 = 8
6 x 3 = 18
1 x 1 = 1
9 x 3 = 27
7 x 1 = 7
3 x 3 = 9
7 x 1 = 7
1 x 3 = 3```

`9 + 21 + 8 + 3 + 6 + 24 + 1 + 27 + 7 + 6 + 7 + 3 = 121`

Modulo 10 the result

`121 modulo 10 = 12 remainder 1`

Work out the check digit

`1 doesn't equal zero, so check digit = 10 - 1 = 9`

Compare the calculated check digit (9) with the one we first entered (2). They are not the same, so looks like we have a bad ISBN.

For further details about the ISBN-13, see the ISBN-13 page. The method of calculating the check digit for a 10-digit ISBN is different to that shown here for the 13 digit ISBN - for details see the ISBN-10 page.