Use of congruence in ISBN: what is its necessity?

Question

Suppose we are considering printed books with $10$ digit ISBN

To each book, there is assigned, in a specific way, a $10$ digit number $x_1x_2\cdots x_{10}$, such as $$ 0-321-50031-8. $$ The numbers entered should satisfy condition $$ (10)x_1 + (9)x_2+\cdots + (1)x_1 \equiv 0\pmod{11}. $$ For above example, such sum becomes $99$, which is $0\pmod{11}$.

My question could be philosophical, but I did not understand the need of introducing this number; therefore posting here the problem, since I do not have any expert of this area in my institute.

Question: Suppose I have two copies of books - one is original, one is duplicate, and both have printed same ISBN. If I carry duplicate book with me, then at the time of checking ISBN, since the congruence relation is satisfied (since it was copy-paste from original book), so no one will feel that the book I carried is duplicate. Then, what is the aim of ISBN number assigning to a book?

(Sorry, the question could be irrelevant; but I am not understanding properly the need of ISBN; whether it assigns same number to thousand copies of the same book - including pirated copy?)

Answer 1

This is done to check if the numbers were entered/read correctly, see Wikipedia:

The two most common errors in handling an ISBN (e.g. when typing it or writing it down) are a single altered digit or the transposition of adjacent digits. It can be proven mathematically that all pairs of valid ISBN-10s differ in at least two digits. It can also be proven that there are no pairs of valid ISBN-10s with eight identical digits and two transposed digits. (These proofs are true because the ISBN is less than eleven digits long and because 11 is a prime number.) The ISBN check digit method therefore ensures that it will always be possible to detect these two most common types of error, i.e., if either of these types of error has occurred, the result will never be a valid ISBN – the sum of the digits multiplied by their weights will never be a multiple of 11. However, if the error were to occur in the publishing house and remain undetected, the book would be issued with an invalid ISBN.

Answer 2

There is a ISBN for each edition of each book. So, if you know the ISBN of a book, then, in princie, you know which book is that. But we are humans and humans make mistakes. That extra digit is there for that. If you make a mistake in one digit or if you swap two of them, then the rule that you have mentioned won't apply anymore.

Answer 3

Digits to check the accuracy of scanners and humans alike exist for a reason ( though in things like debit/credit cards they can be abused) . Algorithms like the luhn algorithm and the one used for UPC scanners help reduce error rates, not unlike medical double checks. In these codes the values identify things. The first digit out front in a UPC for example classifies it by use (coupons etc have historically had certain lead digits) the first batch underneath is a company related value, the second is the actual product, and the check digit is the right most outside, to check via algorithms that what's scanned is valid and makes sense.

In the case of ISBN 13 it's equal to UPC 12 but with the digits broken up to represent publisher etc. So not editing your ISBN, would almost give credit ( as would title pages, references , book covers, etc.) So now you have a reproduction , if it's not a good book,you spent more on the material and process to copy it than it's likely worth. Enjoy ...