ISBN numbers are created so that the first nine digits are information digits
and the last digit, the one you want to calculate, is a check digit. This last
number helps people notice and correct mistakes that might be made in recording
the information digits.
Here is an ISBN number: 0-205-12669-3. The digit 0 indicates the book is
written for English-speaking people. The number 205 and the number 12669
identify the publisher and the book published by that publisher.
The last digit, the 3, was chosen in the following way. Each digit is given
a weight by using the digits 1 to 10 and counting right to left.
These weights are added. The sum is:
0(10)+ 2(9)+ 0(8) + 5(7) + 1(6)
+ 2(5) + 6(4) + 6(3) + 9(2) + __(1)
(The digits of the ISBN information numbers are listed in order left to
right and weighted by counting down from 10 to 1.) The check digit is chosen to
be a number from 0 to 10 so that this weighted sum is a multiple of 11. That
is, the sum must be divisible by 11.
Let us see what sum we have so far with these particular nine
information digits:
0(10)+ 2(9)+ 0(8) + 5(7) + 1(6)
+ 2(5) + 6(4) + 6(3) + 9(2)
= 0 + 18 + 0 + 35 + 6 + 10 +
24 + 18 + 18
= 129
The check digit 3 was chosen because 129 + 3(1) = 132 and 132 is a
multiple of 11. 129 is greater than 11(11) = 121 and 14 less than
11(13) = 143. The only possible number from 0 to 10 to use as a check digit
is 3.
Sometimes the only choice for a
check number will be 10. Since this is a two digit number and cannot be used as
such, the Roman numeral 10, the symbol X, is used.
This system for assigning the last number of an ISBN number makes it possible
to detect errors made by writing a single digit incorrectly or by writing a
pair of digits in the wrong order.
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