Day 43, MAT115

• Just FYI: your final is coming up soon! Check the home page for time.

• Only got four submissions of fractal photos, so I'll just give them each a pair of "Get out of quiz free" cards -- for submitting, and we'll just say that they won the contest.

• The final will come in two parts: new stuff and old stuff. The new stuff will be like an hour exam in its own right, and I'll grade it separately and treat it like an hour exam. Then I'll take the best three of four hour exams.

  Then the rest of the exam will be a review of old material. The best preparation for the final would be to review your old tests and check out the readings.

• This Friday is Logo day.

  You'll submit a one-page typed description of your logo, due at class time, and you will each be giving us a short description of your logo:

  • How and why you chose your logo, and
  • the mathematical content.
  You will provide me with a one- or two-page, typed paper summarizing those two points.

• What does this image have to do with our topic?

• Something similar, but not quite:

  Eternity is tied up with the concept of infinity, frequently.

• What does this line from Amazing Grace's sixth stanza have to do with this topic?
  When we've been here ten thousand years...
  bright shining as the sun,
  we've no less days to sing God's praise
  than when we've first begun.
  

  About another Newton (other than Sir Isaac): John Newton 1725-1807; ironically Newton was a slave trader (who ultimately saw the error of his ways).

• Let's look at some curious problems about the Hilbert Hotel, the subject of Chapter 30.
  • 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!
    • The building and rules:
      • There is a room for each natural number.
      • We can't force new clients into the occupied rooms.
      • Everyone can be asked to move (but only once per night).

    • Can the Hilbert Hotel really be full?
    • Hilbert Hotel and the infinite schoolbus
    • Hilbert Hotel and the limitless stable of schoolbuses
      • Using a diagonalization procedure (as described in the text, p. 253)
      • Using the prime factorization -- every natural number is either prime, or can be factored into a product of primes in a unique way.

• Here's what we need to know:
  • infinity is NAN: not a number

  • 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!;)

  • The natural numbers are infinite in number (but it's the smallest infinite number, "aleph null": ). A Hebrew letter is used to represent infinite numbers!

  • cardinality

    two sets have the same cardinality if there is a one-to-one correspondence between them.

    intuitively: two sets are the same size if they have the same cardinality.

  • 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!

    Example: the even natural numbers are the same size as the natural numbers themselves, by the one-to-one correspondence

    

• Are the rational numbers a bigger infinity than the natural numbers? (The natural numbers are a subset of the rational numbers.)

  We can re-use the idea of the example of p. 253 to show that the rational numbers -- ratios of natural numbers, and their negatives -- can be put into one-to-one correspondence with the natural numbers.

  The prime numbers, the Fibonacci numbers, perfect squares, etc. are all countably infinite sets, of size .

• The real numbers are bigger than the natural numbers, and we can prove it -- by contradiction (p. 255).
  • 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'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?)
  • In fact, it's the irrational numbers that are so numerous: the rational numbers are the same size as the natural numbers -- a countable infinity.

• The power set: showers of infinities!
  • Definition: The power set of a set is the set of all subsets.

    For example, if we consider the set , the power set is the set

    

    Even the empty set has one element in its power set -- itself! The set , the empty set, has power set

    

  • The power set is intimitely tied to Pascal's triangle:
    

  • 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....

We do a thought experiment lasting exactly 60 seconds. Start with a large empty barrel and a long line of ping-pong balls, numbered in order 1,2,3,. . . . Start the clock.

1. In the first 30 seconds, put the first 10 balls into the barrel (numbers 1-10), find number 1 and throw it out.
2. In half the remaining time (15 s), put the next 10 balls into the barrel (numbers 11-20), find number 2 and throw it out.
3. In half the remaining time (7.5 s), put the next 10 balls into the barrel (numbers 21-30), find number 3 and throw it out.
4. Continue in this way until 60 seconds has passed, then stop.