May 1997 Presentation Topic (continued)

Algebra over the integers is already an interesting subject. We
call an integer p a *prime* number if its only divisors are
+/- 1 and +/- p. (Exception: the numbers +/- 1 themselves
are not called prime; they are called *units* instead).

**Question 1**: Can you show that there are infinitely many
prime numbers?

There are many interesting questions about prime numbers. For instance, are there any primes with 100 digits? How many primes like this are there? These questions have become more important in recent years since constructing (and breaking!) many computer security systems depends on being able to factor huge numbers.

**Question 2**: Using a line segment (designated to be of
length one), a straight edge and a compass, construct the
diagonal of a square of side length one. That is, using these
tools (all available to the Greeks!), construct a line segment
that is the diagonal of a square of side length one. Can you show
that this side length is not a rational multiple of the unit
length? Phrased another way, can you show that sqrt(2) is not
a rational number?

When this fact was discovered, it caused quite a sensation. Not only was a mathematical idea proven incorrect, but a whole philosophy was discredited.

The positive integers have a nice property: there is a smallest positive
integer (namely, 1). In fact, even if you throw away the number 1,
or even if you throw away a whole collection of numbers (as long as
you don't throw them all away!), there will still be among the remaining
numbers one which is the smallest. This is called the
*well-ordering principle*.

However, the rational numbers with their natural ordering do not satisfy the well-ordering principle.

**Question 3**: Show that there is no smallest positive
rational number. This suggests that the Greek notion of an *atom*
(a smallest indivisible amount) might not exist.

It is a
surprising result from set theory that every set admits a well-ordering.
Therefore, we *can* order the positive rational
numbers in such a way that (a) for any
two distinct numbers, one is larger than the other;
and (b) in any collection of numbers (other than the empty set, of course),
there's always a smallest number.
However, this ordering
is not at all the same as the usual ordering!

**Question 4**: Try to find an ordering on the positive rational
numbers which has these properties.

**Question 5**: Show that 0.999... = 1. We say there are
two ways to represent the number 1 in decimal notation. Are
there any other ways to represent the number 1 in decimal
notation?

This suggests that real numbers are much more complicated than ordinary old rational numbers. It is quite hard to get a hold of a real number: we need to know all of its digits. This really disturbed me when I was first learning mathematics. If this is the case, in what sense do we know that the number pi = 3.14... exists?

**Question 6**: Two monkeys (Abe and Bonzo) come upon a
clearing containing two barrels (A and B) and an infinite number
of coconuts, conveniently labeled with the positive integers. Abe
suggests two ways of filling the barrels. The monkeys can do this
very fast, so that we can ask what will happen when ALL the
coconuts are in the barrels. Which method should Bonzo agree to?

- [(A)] Abe puts 100 coconuts in barrel A (his barrel) and 1 in
barrel B (Bonzo's barrel), 1000 coconuts in barrel A and 2 in
barrel B, 10000 in barrel A and 3 in barrel B,
*et cetera*. - [(B)] Abe puts 100 coconuts in barrel B and removes the smallest numbered coconut in barrel B and puts it in barrel A, then puts 100 more coconuts in barrel B and removes the smallest coconut (there are now 199 coconuts in barrel B) from barrel B and puts it in barrel A. He continues in this way.

Intuitively it seems obvious that there are more real numbers than rational numbers and more rational numbers than integers. What do you think about the relative sizes of such infinite collections of numbers? In what sense do we mean that there are more reals than integers? These are interesting questions to ponder.

We'll discuss this, and other similar problems, in terms of
formulas known to ancient Chinese mathematicians.
We'll also explain how modern mathematics studies these
problems using *modular arithmetic*.

In the integers modulo 7, there are only seven numbers: 0, 1, 2, 3, 4, 5, and 6. You can divide 3 by 2 in this number system: the quotient is 5, since 2 times 5 equals 3 in arithmetic modulo 7.

**Question 8**: Try to show that you can *always* divide
any number by any non-zero number, in the integers modulo 7. Can this
still be done in the integers modulo 12?

**Question 9**: In the integers modulo 7, compute
2^7, 3^7 and 5^7 (there is a clever way to
do this last one). What do you notice? Can you prove your
conjecture?

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