- You have a homework due today.
- You've got a new assignment, due next Wednesday.
- Logo day is coming up: Wednesday, 4/29. You will submit your logo
and logo paper to me at the time of class. You may not hand that in
late, so get cracking!
- Reading for next time: Loves Me, Loves Me Not (p. 155)
- I want to start with a few things from the New York Times. I was
pouring over some old copies, and I found several relevant news items
that I want to share:
- Embellished: Math Put to Good Use (Melissa Kay Jewelry)
- Performing Math and Mime, for Fun and Profit
- 'Finding Zero': A Long Quest for Naught
- The modern understanding of infinity dates to work done by
Georg Cantor, in the late 1800s.
- Our author quotes David Hilbert (himself a very famous and
brilliant mathematician), who said "No one shall expel us from the
Paradise that Cantor has Created."
Strogatz says that his objective is to "give you a glimpse of this paradise." (p. 253)
- Welcome to the Hilbert Hotel!
- there is a room for each natural number.
- Everyone can be asked to move (but only once per night).
- Notions of infinity rely on the idea of one-to-one correspondence, which you may recall from our early study of numbers.
- Can the Hilbert Hotel really be full?
- Hilbert Hotel and the infinite schoolbus
- Hilbert Hotel and the limitless stable of schoolbuses
- infinity is NAN: not a number (and is therefore neither
even nor odd, as Strogatz explains to Kim Forbes and her son Ben).
- infinity is NAS: not a size, either! the reason is that there
is more than one size of infinity! this is the most
important thing I want you to remember (this fact will allow
you to win on the playground, when someone tells you
that they hate you an infinite amount -- you can still
hate them more!;).
- The natural numbers are infinite in number (
).
- cardinality
- two sets have the same cardinality if there is a one-to-one correspondence between them.
- Here's a sensible rule you can be sure of: a subset is never
bigger in size than the set itself.
and if the sets are finite, the proper subset is always smaller....
but if the set is infinite, we may actually be able to throw away elements of a set and not change the size of the set!
- We'll assume that they're not bigger -- that there is a one-to-one correspondence -- then show that that is impossible.
- We'll need this fact: every real number can be expressed by an infinite decimal.
- We need to be aware, however, that some real numbers are expressed by more than one decimal (for example, 1=.99999.... -- did you know that?)
- We'll show that just the real numbers between 0 and 1 are too numerous to be put in one-to-one correspondence with the natural numbers.
- We show this using a procedure that we might call "dodge ball".
There are infinitely many sizes of infinity. It turns out that the power set of a set is always of larger cardinality than the set itself. Thus every infinite set is smaller than its power set, which is an infinite set, which is smaller than its power set, etc....
- One is the
loneliest number (one is not prime, but can't be expressed as a
product of primes in a unique way -- rats!).
- The sieve of Erotasthenes -- finding primes
- Twin primes: A love poem for lonely prime numbers
- The prime factorization theorem: every natural number (other than
1) is either prime, or can be expressed as a product of primes in
a unique way.
- In a way most series get lonelier and lonelier as the get farther
from zero. Think of these:
- Power of 2
- Fibonacci numbers
- So why not the primes? But it's that they're getting lonelier in a kind of regular way that was so interesting... to Gauss, anyway!:)