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### Calculating the Last Digit of an ISBN Number

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Date: 07/15/98 at 09:28:36
From: Daymon Smith
Subject: How to Calculate the last number in an ISBN number

How do you calculate the last number in an ISBN number?
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Date: 08/01/98 at 13:44:21
From: Doctor Roberta
Subject: Re: How to Calculate the last number in an ISBN number

Hi, Daymon -

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

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.

Codes provide some interesting mathematics!

- Doctor Roberta, The Math Forum
Check out our web site! http://mathforum.org/dr.math/
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Associated Topics:
High School Number Theory

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