A few weeks back I came across Parable of the Polygons, a “playable post” from Vi Hart and Nicky Case. I’ve been pretty entranced by it since. This post uses simple, interactive cellular automata to demonstrate how preferences in each individual can affect the entire system. This is a metaphor, and a model, for how even slight individual biases can lead to segregation at the societal level.
The beauty of Hart and Case’s work is in it’s simplicity. Continue reading
One of my tutoring students is taking an Introduction to Mathematical Proof course. During one of our recent sessions, he had a problem that challenged him to come up with a way of enumerating the rational numbers, thus showing that they can be mapped to the natural numbers, and thus are a countably infinite set. I saw the proof that the rational numbers are countable way back in undergrad, but I couldn’t remember the process. So, I tried to come up with a means of counting the rational numbers. The enumeration I came up with is solid, I’m pretty sure, and I can’t seem to find it replicated elsewhere on the web.
So, I’m putting it here for others to comment on and to see if anyone can point me to where this method may have been used before. Here is my method for enumerating the rational numbers.
Let , and , such that and is irreducible.
Now let refer to the th prime number.
The number . This value is unique in the natural numbers.
Thus, each positive rational number–and zero–maps onto a unique natural number. This same process can be duplicated for the negative rational numbers. Since both the positive and negative rational numbers can be shown to be countable in this way, the union of the two sets is also countable.
Here’s something I don’t do very often: a pet-peeve blog. Let’s do this.
You know what really grinds my gears? Those math polls on Facebook. You know the ones like
Monday night I received an email from DC Teaching Fellows offering me a place in their summer 2011 cohort. Within a couple of months I will be moving to DC and, beginning this Fall, I will be teaching math.
Everyone I’ve told has been very excited for me, which helps me get excited. My excitement is tempered by the prospect of now finding a place to live and moving me and my life across the entire country. If anyone has recommendations for how to go about such a move, I would appreciate the feedback. Also, if you have any leads for housing make sure you contact me.
Ideally, I want to find an apartment or house with some roommates. It would be great to move into a house that is already established, with people familiar with the area. I am hoping there might be someplace that tends to cater to Fellows, where there will be others sharing my experience.
It is clear now that getting in was the easy part. I know that the challenge is just begun.
When I first opened the email I was really filled with a bit of terror. A moment where I felt myself drop into open space. Since then, I have asked myself over and over again “Is this something I really want to do? Will it be worth the expense, the difficulty, the challenge?” But when I think about what I will be doing every day–planning lessons, showing off math, breaking apart the ideas into manageable portions, grading homework, interacting with students–I know that it is all I want to do, something I must do.
This is going to be good. And keep an eye here, because I know I will want to share it.