(over)

Mathematics 5
Winter Term 2000
The World According to Mathematics

Dwight Lahr and Josh Laison

Class Discussion: Week #7

February 18, 2000

Today we are going to be discussing pure and applied mathematics—the differences,
similarities and interplay between them. We first will look at some mathematical examples,
and then we will discuss an excerpt from a book by G.H. Hardy, a twentieth century
mathematician.

1.

ISBN Error-correcting Algorithm

Here is an algorithm for correcting common errors in transmitted ISBN’s.

Theorem:

Suppose a transmitted ISBN is recorded as a-bc-defghi-jand its
check-sum reports an error Z≠0:
[10a+9b+8c+7d+6e+5f+4g+3h+2i+j](mod11)=Z.
Off-by-one errors: If one of the digits of the transmitted ISBN is one

(a)

more or less than it should be, then

one more:

the digit with the multiplier Zis wrong (we treat the check-

digit jas having multiplier 1).
one less:the digit with the multiplier congruent to -Z(mod11)is
wrong.

Transposition errors: If the adjacent transposed digits of the transmitted
ISBN are, in order from left to right,uand v, then Zis congruent to (u–
v)(mod 11).

(b)

Answer the questions below.

a.

Suppose the actual ISBN is 0-679-79171-X and it is transmitted as 0-679-79181-X.
Suppose further that the result of applying the ISBN algorithm to 0-679-79181-X
yields Z = 3. Verify that the error-correction theorem above gives the correct
information about where the error occurs.

b.

Suppose the actual ISBN is again 0-679-79171-X and it is transmitted as 0-579-
79171-X. Suppose further that the result of applying the ISBN algorithm to 0-579-
79171-X yields Z = 2. Verify that the error-correction theorem above gives the
correct information about where the error occurs.

c.

Suppose the actual ISBN is again 0-679-79171-X and it is transmitted as 0-679-
71971-X. Suppose further that the result of applying the ISBN algorithm to 0-679-
71971-X yields Z = 3. Verify that the error-correction theorem above gives the
correct information about where the error occurs.