Day 28, MAT115

• Your quiz today is over infinity.

• Reminder: we're only days from our project and logo demonstrations (the last week of class).

  We'll do logos on Tuesday, and projects on Thursday. If you have any conflicts on those days, let me know in advance. You are expected to attend to support your classmates. Please be on time.

  I'll want your digital materials in advance. Please get those to me by Monday evening (logos), and by Wednesday evening (projects). Anything that we'll want to display, so that I can have it ready to go.

  If you want to go early in the presentations, let me know. Otherwise, your presentations will be ordered randomly (although I'll have prepared the order in advance, with the materials lined up).

  You'll have three minutes for your presentation. You should talk about the logo/project itself, your motivation or what it means to you, and about the particular mathematics involved. You should definitely have three minutes of material -- make me cut you off!

  At the end of class, you'll be asked to give me some positive comments about three of the presentations that resonated with you. These will be shared (anonymously) with the presenters at a later date.

• Our final:
  • Is comprehensive
  • 1/3 New material: knots, links, bands, infinity
  • 1/3 Exam 1 material
  • 1/3 Exam 2 material

• We discovered that the power set of the natural numbers is always bigger than the set of naturals, meaning that there are at least two sizes of infinity.

• Since the power set is always bigger than the set itself, we know that there are infinitely many infinities: \[ Card(N) \lt Card(P(N)) \lt Card(P(P(N))) \lt Card(P(P(P(N)))) \lt Card(P(P(P(P(N))))) \lt \ldots \]
• We added up an infinite number of things in various ways. Some of them make sense, like \[ \frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\frac{1}{64}+\ldots=1 \]

  Others were a little more problematic: \[ S_1 = 1-1+1-1+1-1+1-1+\ldots = \frac{1}{2} \] or \[ S_2 = 1-2+3-4+5-6+7-8+\ldots = \frac{1}{4} \] or \[ S_3 = 1-3+6-10+15-21+28+\ldots = \frac{1}{8} \]

• The reason for us to consider this problem seriously is contained in Pascal's triangle. We first added in all the invisible zeros: "Pascal's triangle floats on a sea of zeros."

  We then set out to figure out how to continue Pascal's triangle in the backwards sense. We know how to go forward, but how do you go backward? It turns out that we have infinitely many options, which can be good or bad. One needs to make a wise choice.

• Symmetry is a fundamental property of the triangle, and it seems like a useful thing to preserve. That leads us to just one choice for those two entries above 1 -- and once we make that choice, the entire row above the first one is complete.

• Recall what the entries of Pascal's triangle add up to: powers of 2. That pattern should continue. But we had been adding finite sums down below. We're now in the presence of an infinite sum -- or rather many infinite sums!

• But the big story was: \[ S = 1+2+3+4+5+6+7+8+\ldots \] What should that add up to? Turns out, using the same strategies, that it adds up to \(-\frac{1}{12}!\)

  "For those of you who are unfamiliar with this series, which has come to be known as the Ramanujan Summation after a famous Indian mathematician named Srinivasa Ramanujan, it states that if you add all the natural numbers, that is 1, 2, 3, 4, and so on, all the way to infinity, you will find that it is equal to -1/12. Yup, -0.08333333333." (source)

Question of the day:

• 150 years ago, Cantor was puzzling over his first notions about infinity: he was corresponding with another famous mathematician of the day, Dedekind when, "...on November 29, 1873, Cantor moves on to new ideas:

  Allow me to put a question to you. It has a certain theoretical interest for me, but I cannot answer it myself; perhaps you can, and would be so good as to write me about it. It is as follows.

  Take the totality of all positive whole-numbered individuals n and denote it by (n). And imagine, say, the totality of all positive real numerical quantities x and designate it by (x). The question is simply, can (n) be correlated to (x) in such a way that to each individual of the one totality there corresponds to one and only one of the other? At first glance one says to oneself no, it is not possible, for (n) consists of discrete parts while (x) forms a continuum. But nothing is gained by this objection, and although I incline to the view that (n) and (x) permit no one-to-one correlation, I cannot find the explanation which I seek; perhaps it is very easy."¹

• Today we consider a rather famous paradox, which we'll call "the ping-pong ball conundrum".

  Start with a barrel, and a lot of ping-pong balls, each one labelled with its own natural number: 1, 2, 3, 4, ....

  Conduct the following thought experiment in exactly one minute:

  • Begin with 60 seconds on the clock. Within the first half minute, the first 10 balls (labelled 1-10) fall into the barrel, and you have to rummage around in the barrel and pull out ball number 1. You throw it out the door.

  • The ping-pong balls keep coming. In the next quarter of a minute, you see the next 10 balls (11-20) drop into the barrel, and you rummage around and pull out ball number 2. Throw it out.

  • The ping-pong balls keep coming. In the next eighth of a minute, you see the next 10 balls (21-30) drop into the barrel, and you rummage around and pull out ball number 3. Throw it out.

  • And this continues! But each time you have only half the time to complete the task, and it's getting harder to find the balls amidst all those in the barrel! But you do it, because this is a thought experiment, and you are capable of deep thoughts....

  • The question is: how many ping-pong balls are left in the barrel at the end of the minute?

    Related: can you think of a formula for when we would have pulled out the \(n^{th}\) ball?1

  • What if we had pulled out the balls \(10, 20, 30,\ldots\) rather than 1, 2, 3, ...? How many balls would be left in the barrel?

  • What if we had simply poured all the ping-pong balls into the barrel to start, and then taken them out 1, 2, 3, 4, .... per the initial schedule? Would there be any ping-pong balls left in the barrel?

• Here are a few more posers:
  1. Suppose you have a set. If you remove some of the things from the set, will the collection of the remaining things always contain fewer items in it than the original set? If so, demonstrate why; if not, illustrate with an example.

  2. Describe a collection of numbers you could remove from the set of natural numbers so that the set of remaining numbers contains fewer number than the set of all natural numbers.

  3. Suppose we have two sets and we are able to pair elements from the first set with elements from the second set in such a way that all the elements from the first set are paired with elements from the second set; but there are elements from the second set that were never paired. Does this pairing imply that the two sets do not have the same cardinality?

  4. Suppose we have two sets and every possible pairing of elements from the first set with elements from the second set results in having elements from the second set never paired. Do these pairings imply that the two sets do not have the same cardinality?

Links:

• Ross-Littlewood paradox (the older name of "the ping-pong ball conundrum")

  Here was the earliest known reference to the conundrum:

  

  (from Littlewood's Miscellany)

• Was Cantor Surprised?