First a little about the difference between ISBN-10 and ISBN-13. ISBN-10 is an old standard and is no longer issued to new publications. ISBN-10 used 10 digits and was superseded by ISBN-13 on 1st January, 2007. Since that date, all new ISBN numbers issued comprise 13 digits. For a lot more detail about ISBN-10, see the ISBN-10 page. Likewise, comprehensive details about ISBN-13 may be found on the ISBN-13 page.
For the purposes of the information here, consider an ISBN to consist of just its 10 or 13 numbers - ignore any embedded spaces, hyphens, or the label consisting of the characters "ISBN" that you may sometimes see written as part of an ISBN as they are not relevant to the discussion here. The one exception is that some 10-digit ISBNs have the letter x (or X) as their final digit. (Why? Details on the ISBN-10 page.)
Here's what we're going to do in the seven steps below:
Here we go.
Example starting ISBN-10
1 Take all 10 digits of the ISBN-10 and drop the final digit - just forget it. The final digit is a check digit that was only good for the 10-digit ISBN - we need to calculate a new one for the new 13-digit ISBN in steps 3 to 6 below. (What's the check digit all about? Details on the ISBN Check Digit page.)
2 Prepend the digits "978" onto the front of the nine digits. (Why 978? See the ISBN-13 page.)
3 In this and the next three stages we have to calculate the new check digit to append to the twelve digits from step 2. We take each of the 12 digits from step 2, one by one, and multiply them by a number. We multiply the first digit by 1, the second digit by 3, the third digit by 1 again, the fourth digit by 3 again, and so on, alternating 1 and 3 until we get to the twelfth digit.
Here's what I got. Did you get the same?
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
4 Now add up all 12 of the answers you got in step 3.
9 + 21 + 8 + 3 + 8 + 18 + 1 + 27 + 7 + 6 + 7 + 3 = 118
5 Take the answer from step 4 and perform a modulo 10 division. A modulo 10 division is also called casting out 10s. It's just the remainder when you do a whole number division (as opposed to a decimal division where you expect a decimal fraction answer).
118 mod 10 = 8. 118 divided by 10 = 11 remainder 8. We're interested in just the remainder.
6 Final step of calculating the check digit for a ISBN-13. Take the result from step 5. If it is zero, then the check digit is zero. If the result from step 5 isn't zero, then subtract the result from step 5 from the number 10. That result is the check digit.
10 - 8 = 2.
7 Final step of converting our ISBN-13 from the ISBN-10 is to append the check digit arrived at in step 6 onto the end of the 12 digits we arrived at in step 2.
And that's it.
Here's what we did: