I still wasn't feeling splendid when I woke up this morning. I'm hoping this goes away soon... It's getting pretty old. Anyhow, I got up a little early to get some studying in before I went to class. So I did. I usually aim for 5 to 10 problems, depending on how early I wake up.
So yeah. Class happened. We talked about duration matching of portfolios. This is a method of trying to ensure that a company's assets cover the long-term liabilities. It's a topic covered on the actuarial exam I took... so I already knew what was going on.
So during atrium time we had a brief discussion about how to represent a certain area of math to people who don't like math. This has always been of great interest to me because I personally believe that more people would like (and be good at) math if not for all the tedious computation. I'm going to give it a shot and summarize a one-semester course into a couple paragraphs. Just because I feel like it. Here we go. And I have purposely not said which course that is for the time being.
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In this class we talk about one thing: how things change. For instance, we can talk about how someone's location changes in terms of their velocity. For instance, I can walk from my apartment to the landlord's office at 1 meter per second. My location is changing and every second I am 1 meter closer to my destination. This is what we call the "rate of change": how fast something is changing. What is changing? My location. At what rate (or speed) is it changing? Well every second I am 1 meter closer to my destination, so we would say "1 meter per second." We can also talk about the rate at which someone's speed changes. This is called acceleration. For instance, on earth gravity has an acceleration of 9.8 meters per second per second. This is a bit confusing at first, so let's take it slowly. Let's say you were dropping a ball from a 100 meter platform. The first second it would be traveling at 9.8 meters per second. But during the next second the speed would increase by another 9.8 meters per second. So during second 2 the ball would be falling at 19.6 meters per second. Then during the next second it would move even faster: 29.4 meters per second. So every second the velocity is increasing by 9.8 meters per second. So that's pretty neat.
All of this was an application of the mathematics that deals with limits. You see, it is often the case (in mathematics) that we want to see what happens in very small time intervals. For instance, when you are running you aren't running at exactly the same pace all the time. You might run at 6 miles per hour, the 6.2 miles per hour, then 5.9 miles per hour and so forth. In fact, this changes continuously. It might be really really small, but there it's always changing. Let's say I wanted to know how fast you were running at exactly 5 minutes into your run. That would require impossibly accurate tools to measure something so precise! But we could easily estimate it by measuring how far you go in between time 4.5 and 5.5. This would give a very rough indicator. But we could get an even better idea if we measured between 4.75 and 5.25. We can keep getting closer and closer to 5, measuring between time 4.9 and 5.1, then 4.99 and 5.01. Each time we make the interval over which we are timing smaller, we are getting a more accurate picture of how fast you ran at exactly time 5. Ideally, we would want the length of the measuring interval to approach 0. That would mean that we're measuring a length so small that it is actually at exactly time 5. Wouldn't that be neat?
Well. It would be. In fact, that is precisely what we do in this area of math. We use tricky arithmetic to determine what happens in really small intervals. Because sometimes... weird things happen. In math, we usually like things to work nicely. And for the most part everything does actually work out nicely. But sometimes funny things happen. Take the picture below for instance.
There is a gap on the curve, right where the dotted line is. If we think of the running example, maybe at exactly time 5 you stubbed your toe and stopped... but only for the smallest fraction of a second. Well, if we wanted to see what your rate of change was at time 5, our previous method didn't account for this. Now, with the exception of that little gap the curve looks pretty normal. It just... curves. We can see that there is a gap at the red line. But from the way the line moves around that gap, you can easily guess how you could fill in that gap in a sensible way, right? Well this is a pretty useful tool, so they gave it a name: the limit. This is the idea of getting really really close to something without actually reaching it.
Do you know where else this is useful? Circles! I love circles. Did you ever wonder how people discovered the area of a circle? Well, it all started with finding the area of a square. Then the area of a pentagon. Then a hexagon. The a heptagon, octagon, nonagon, and so forth. The great mathematician Archemedes realized that if you kept adding sides, you eventually would get a circle. Tell me: if I drew a circle, then right next to it a polygon with 4 billion equal sides... do you think you could tell a difference? Unless you had a microscope of some serious magnitude, you would be unable to tell the difference. So in today's terms, we could refer to the limit of the polygon as the number of sides gets really really close to infinity*.
*In this branch of math, we say things like "getting close to infinity" even though we know that you can't actually ever get to infinity.
Anyhow. That's just a little thing we call calculus. It's really not as scary as people make it seem. The arithmetic is a little tedious, but the concepts are pretty cool and can be made pretty understandable. Now I have no idea whether I actually accomplished that or not... but I am confident that it can be done.
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So yeah. There's that. Then I went to life con and we had class and stuff. We got our quiz back. I think it's the first quiz I've ever gotten where I missed points for answering the correct question. See, Dr. Foley takes the quizzes straight from the homework. He does this instead of actually grading the homework. But on the last quiz he changed one of the questions (this is actually becoming increasingly common). When we asked him about the change of questions during the quiz he said "Oh, just leave it as a function of time." This is because he didn't tell us what time to use (but the problem in the homework did). So I left it as a function of time. Like he said. And we all missed 5 points because we left it as a function of time... like he said to do. When I evaluate it using the numbers he used for his bounds of integration I get the same answer... but since he told us not to evaluate it I did not. And missed 5 points. When I brought it up to him in class he remembered that he had asked us to do that... so he's giving us 3 points back. Still...confused.... oh well.
On the way back from class I got gas and stopped at Family Video. There was a Star Wars game that recently came out for XBox Kinect, so I thought I'd give it a try. When I got back I basically spent the rest of the day alternating between eating, studying, and playing a game. My goal is to get through 30 problems each day. And I did, so I was pleased.
So yeah. That was my day. And a bit of a math lesson. So now I'm going to bed. Night!
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