HCSSiM Workshop day 4
As usual, we started with the students showing us solutions to their problem sets. Today one of them showed a sharp lower bound on the Fibonacci numbers, although he hadn’t proved it was sharp.
Arithmetic modulo n
Then we talked more about how we can talk about addition, and now also multiplication, with a finite set of symbols . Then we wrote out the multiplication tables for and The students noticed and proved that there is a multiplicative inverse for modulo if and only if using what we did yesterday with the Edwinian Algorithm and the way we turned it around to express gcd’s as linear combinations. We defined some notation and the natural map:
Finally, we wrote down the subsets of which map to each element of
Posets and graphs
We went back to the idea of a partial ordering, and came up with a bunch of examples (including the set of integers under “divides evenly into”). We talked for a while about how to represent partial orderings, and finally settled on a graph. We talked a bit about poset chains and antichains, and then we formally defined a graph (we voted and decided to call it a “visual representation”).
The complex plane
The founder and director of the program is David Kelly. The program has been going for 40 years now and for maybe the first time ever Kelly himself isn’t teaching a workshop, so I’ve invited him to do some guest lectures in my workshop on complex geometry. It’s always a treat to watch him teach.
Kelly came in and built on the idea of “modding out by an integer” by definine which he described as “modding out by a polynomial”. He asked the students to investigate this idea and they eventually discovered that if it also must be true that which allowed them to write every polynomial with as a linear combination of 1 and so as . Then they thought about the addition law and multiplication law and decided they had the complex plane. So we decided to start calling the symbol ““.
We then defined to be the point on the unit circle discarding once and forever the notation (we justified this definition in last night’s problem set). We showed we could recover useful trigonometric identities that way (I skipped trigonometry myself and this is the only way I ever knew how to derive those identities) and that we could alternatively write any point on the complex plane in polar coordinates, so as . Finally, we noted that if we multiply anything by the number , we end up stretching it by and rotating it by .
We heard a funny story Kelly told us about taking a test to get his pilot’s license. He was given 30 minutes and lots of suggestions once to compute a heading which involved a calculation in polar coordinates. Since he was a mathematician he was too proud to accept the props they offered him, and finished with 29 minutes to spare. Once aloft though he quickly realized his calculation simply couldn’t be correct, but fudged the test by eyeballing it and following a highway. Turns out that pilots use due north as the axis along which the angle is zero, and then they go clockwise from there. I’m not sure what the moral of the story is, but it’s something like, “don’t be arrogant unless it’s a clear day and you have a backup plan.”