Mathematics 5
Winter Term 2002
The World According to Mathematics
Dwight
Lahr and David Rudel
Class Discussion: Week #5
February 1, 2002
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 
and its check-sum reports an error 
:
.
(a) Off-by-one
errors: If one of the digits of the transmitted ISBN is one more or less than
it should be, then
one more: the digit with the multiplier 
is wrong (we treat the check-digit 
as having multiplier 1).
one less: the digit with the multiplier congruent to 
is wrong.
(b) Transposition
errors: If the adjacent transposed digits of the transmitted ISBN are, in order
from left to right, 
and 
,
then Z is congruent to (u – v)(mod
11).
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.
d. How could
you use the above theorem to correct errors in a transmitted ISBN? Explain.
e. What
mathematical ideas do you suppose were used to develop the above
error-correcting algorithm? (To get you started in your thinking, remember that
Gauss introduced the idea of congruence modulo 
.)
Is there an interplay between pure and applied mathematics? Explain.
2. The
Euler 
-function
a. Your groups will be calculating powers of
numbers in various mods. Each of you
will be given a number, and your group will be given a mod. For example, your group may be given the mod
14, and you may have the number 5. In
that case, you would compute powers of 5 modulo 14. Discuss with your group what types of patterns you see.
b. Recall that
two integers are relatively prime if
the only divisor they have in common is 
.
We then use this notion to define the Euler 
-function
(pronounced “fee-function”): If 
is a natural number, then 
equals the
number of integers from 
to 
that are relatively prime to 
.
After we discuss the above results in part a. as a class, calculate the following values:
Do you see a pattern?
Test it on:
Discuss your findings with your group. Then calculate:
3. A
Mathematician’s Apology
In
1940, the eminent mathematician G.H. Hardy published A Mathematician’s
Apology. The book was published subsequently in 1941, 1948, and 1967. We
have prepared a handout from this book for today’s discussion. The handout is a
Xerox of pages 59 and 60, from section 21 of the 1941 edition.
Read this selection and discuss it in light of the above mathematical examples and in light of what you know about pure and applied mathematics. The following quote is particularly pertinent: “The ‘real’ mathematics of the ‘real’ mathematicians, the mathematics of Fermat and Euler and Gauss and Abel and Riemann, is almost wholly ‘useless’...”