2 posts tagged “mathematics”

Last night, after my Calculus class, I turned in the extra credit problem I wrote about earlier.  I had actually done it wrong in that entry and had to do somewhat more complex Calculus to really vigorously support my conclusion.  But I got it right this time.  And then my teacher took me aside to talk to me.

At first I was kind of confused as to what he was talking about, because he started explaining to me that there are interesting questions about the issues we're covering in class right now that are way beyond the scope of our class.  For instance, you probably have learned that A+B=B+A.  But we are working on infinite series (∑) -- which are essentially adding up infinitely many numbers -- and there is the question of whether ∑f(x)+∑g(x)=∑g(x)+∑f(x). Stuff like that.

And I was thinking, "OK, fine, but what is he getting at?" And then he told me that he'd like to lend me an advanced Calculus text so I can read about this sort of thing and see if it interests me, because that is the sort of thing I would be studying if I continue with a Mathematics degree.

That's the closest thing to encouragement that I've ever received in a Math course.

Math seems to occupy a strange place in education these days.  You go through school and you have to take certain subjects, including English, Math, Science, Government/Civics, etc.  And it is my experience that you are expected to do well in every single one of those subjects except for Math.  Of course, you must pass them in order to graduate from your grade, but your teachers don't expect you to be good at Math.  The message is constantly, "it's OK that you're bad at Math because it is so very hard."  But you don't hear, "it's OK that you're bad at writing because it is so very hard" even though the rules of grammar they expect you to know are at least as difficult to learn as Algebra and quite a bit more complex, if only because they're so contradictory. 

The quintessential image of "student fighting back against the Man" is the bold young rebel standing up in Algebra and asking when they're ever going to use this in Real Life.  But you probably use Algebra more than you use your knowledge of the parts of a plant cell, but we never see or applaud young rebels questioning the validity of their Biology courses.

So you hardly ever seem to come across a teacher who really encourages their students to be good at math.  Instead, they try to make it as easy as possible for students to pass rather than trying to help them understand and really learn things.  They try to endear themselves to their students by saying that they know math is horribly boring and hard but it'll all be over soon. 

To me, this all adds up to discouraging students from being interested in math.  So it is a rare and, frankly, quite cool experience to have a teacher who expects his class to learn and understand, and is willing to actually encourage their students to continue.

Edit: Thinking about this problem later, the reasoning in this entry is, unfortunately, incorrect.  Well, it's valid for this particular case, but not for all problems of this sort so it's not good enough.  I'd write an entry explaining how the problem is actually solved, but it involves a lot of Calculus concepts that are way too difficult to explain on a blog entry.

I had a flash of brilliance driving home from my Calculus class.

We were given an equation to try to work out as extra credit last Wednesday at the end of class, and I wasn't there as I left half-way through.  Today everyone was talking about it, and nobody had worked it out, so I spent half the class vaguely working on it.  And then, driving home, of course, the answer occured to me in a flash of brilliance.  Or I guess I should say it was a moment of clarity where I was temporarily not-stupid.

You see, we are working on infinite series.  I'll try to explain briefly, if you care.  Which you should, because math is cool.

Anyway, an infinite series is basically adding up infinite amounts of numbers.

Here's a simple one: ∑n.  The ∑ is the sum notation, and normally there would be at the bottom the number you start at and at the top there'd be the number you end at.  So in this case, we'll go from n=1 to ∞ (infinity).  So:

So if we add up to infinity, our ultimate answer would be infinity.  But there are some infinite series that actually add up to a finite (i.e., not infinite) number, or rather, so close to that number that it may as well be that number.  An example is this one:

So that would be 1/2 + 1/4 + 1/8+ 1/16 + 1/32 + 1/64 + ... + 1/2^n.  If you put just 1/2 through 1/64 in your calculator, you'll get about 0.984.  Add the next few terms in the series and it just keeps getting closer and closer to 1.  So we would say that the series ∑1/2^n converges to 1.

There are a few things that distinguish series that converge to a real number from those that diverge (go to infinity) and they're pretty logical.  First, that the numbers have to be getting smaller as you go along the series.  That makes sense; you can't add up ever increasing numbers and expect to get anything other than infinity.  Another feature is that as you get really far along the series, the individual numbers you are adding have to eventually get so small that they're effectively zero.  For instance, if you're adding numbers that keep getting smaller, but never get smaller than, say, 1, then you're still going to end up with infinity.

So in my class we've learned various techniques to determine whether a given series goes to infinity or not and Monday we learned one called the root test, which involves a bunch of stuff you care about even less than this long, stupid math post.  And we were given the following extra credit problem:

And we have to find out whether it converges to a real number or diverges to infinity.  And we all puzzled over it and tried to do complicated integrations and did crazy tests and then driving home it occured to me that, duh, this series doesn't even past the first rule; this series has numbers that are getting bigger, not smaller.

Flashback to algebra.  E is a constant number, about 2.7 something, and it is raised to a negative exponent, which means that you make a fraction, and put it in the denominator and make the exponent positive.  Here's a picture because math doesn't mix with plain text:

Now, we know with fractions that the bigger the denominator is, the smaller the total value of the fraction is, right?  1/4 is smaller than 1/2.  And in our series, n is getting bigger and bigger.  So e^n is getting really big, really fast.  So the fraction 1/e^n is getting really small.

So back to the original equation we were working on.  We are subtracting 1/e^n from 1.  So we are subtracting ever smaller numbers from 1.  So therefore, the total value of 1 minus 1/e^n is increasing.  To illustrate in case you're lost (or even reading anymore), let's try some values of n.  Like, 1 and 2.  So if n=1 then we have (1-1/e^1)^1.  Plug that into my calculator and I get about .63.  Then with n=2, we have (1-1/e^2)^2 which is about .75. 

So the numbers are getting ever bigger, thus breaking the very first and most basic condition of infinite series converging to a real number.