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Winners receive an extra Get-Out-Of-Quiz-Free card -- congratulations!
We'll do logos on Monday, and projects on Tuesday. 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 Sunday evening (logos), and by Monday 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.
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 \[ S_3 = 1-3+6-10+15-21+28+\ldots = \frac{1}{8} \]
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.
"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:
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 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
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:
Related: can you think of a formula for when we would have pulled out the \(n^{th}\) ball?
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