If you're interested in learning more about what Sierpinski numbers are, you can read about them here.
I just learned that there is a type of number named after Stephen Colbert. It is a specific kind of megaprime (a megaprime is a prime with at least one million digits). Rather than explain exactly what a Colbert number is, I'll just let you read about it here. This site also includes several video clips of Colbert talking about math (though the folks at Comedy Central seem to be as confused as the rest of the population about the fact that science is not the same as math). Be sure to check out the second clip--it is by far my favorite, and it coincidentally has a lot to do with my next video!
If you're interested in learning more about what Sierpinski numbers are, you can read about them here.
If you've read many of my other posts, it's probably become obvious that I enjoy spatial reasoning. It's fun because it involves creating images in your head and knowing how things work together.
One of my favorite childhood toys was this set called Gearopolis. It was made by Discovery Toys, and it only seems to be available used now. But all is not lost for fellow gear-lovers who want to play with a toy like this. Gear Tech is a similar set (and it even includes belts to connect gears!), and it's available from Amazon.com.
Now, one of the things I like about spatial reasoning is figuring out how systems work before testing them. So let's play that game with my gear setup.
If the yellow gear with the handle turns counter-clockwise (as the arrow shows), which way will this green gear turn?
Okay, let's try another one. If we once again turn the yellow gear counter-clockwise, which way will this big purple gear turn?
If you want a real challenge, think about how fast each gear turns. If I turn the yellow gear at a constant rate, which will turn faster, the green gear or the big purple one? Or will they be the same? Will they be faster or slower than the gear with the handle?
Have you figured it all out? Now you can watch this video to see the gears in motion:
Several of the students I tutor have trouble when it comes to understanding graphs and charts. I have the feeling that they only think they have trouble because the graphs and charts they're reading are on the ACT, and no one feels calm and happy when reading charts on the ACT.
GraphJam is a hilarious website that has graphs and charts of all types--and none of the graphs are serious. If you think you don't understand graphs, check out this website, and you'll change your mind. I promise.
Here are a few more of my favorites:
The great thing about graphs is they don't have to have a traditional format for us to understand them:
Okay, so I can't pretend I haven't noticed that when most people hear I do math, they say, "Oh, I hate math!" And it doesn't escape my attention that most of the students I tutor don't really like math even if they're good at it. The fact is, most people don't think math is very important.
A student I worked with the other day said that her least favorite subject was math because she didn't like having to do problems and learn concepts that she would never use again.
Yes, I admit that most of the math we learn in high school seems pretty useless. I have never been a fan of "realistic" word problems like the oft parodied "two trains leave a station" problem. And most of the techniques we learn in high school are techniques we will never use again. So, where are we going with this? What redeeming qualities can be found in the math we learn in school?
Part of the problem is that we're looking at math too narrowly. What we learn in math class is not simply how to solve for x. We're learning how to look at a problem we've never seen before and figure out how to solve it. As it turns out, this is an extremely important skill in almost every profession.
So when you think it doesn't matter how you do on the math section of the SAT or the ACT because you're going to be an English major, you're wrong. Colleges care about your math score because it's an indication of your ability to solve problems, not an indication of your ability to do math.
Here's an interesting fact: math and physics majors score highest on the LSAT (the Law School Admission Test). On average, math and physics majors score 160 (the highest score is 180). In contrast, pre-law students rank next-to-last for LSAT scores, with an average of 148.3.
What do Abstract Algebra and Complex Analysis have to do with defending a client in court? Almost nothing. So why do math majors do so well on the LSAT? Math majors do well for the same reason that philosophy majors come in second with an average score of 157.4. Both majors train students to think critically and solve complex problems. The LSAT doesn't test your knowledge of law--that's what law school is for. The LSAT is designed to weed out students who don't know how to think like lawyers--that is, it filters out students who don't know how to take a problem, apply logic and reason, and find an answer. Now, that sounds like a pretty useful skill for anyone to possess.
Fail Blog posted a win yesterday that perfectly captures the elation (and frustration) of mathematics. Check it out:
You may have noticed that the link to the musical interpretation of pi I shared on Pi Day (3/14) is broken. This is because someone else had already written a composition using the first digits of pi and copyrighted that composition.
Now, if Michael Blake, who created the composition I tried to share, had used the exact same rhythm as the previous composition (which I doubt), I would understand the copyright claim. But pi is a number--how can it not be in the public domain? Well, we can still write the first digits of pi, and we can say them, but if you want to use them to create a song by numbering the notes of the scale, you can't. It's copyrighted.
Vi Hart, the recreational mathemusician, created a little video presenting her view on this copyright claim. You can watch it here. She's got a good perspective on this issue since she is a musician and a mathematician.
My main frustration with this copyright claim is that Michael Blake created something beautiful and interesting (and yes, I think he created it since I doubt he intentionally copied anything), and I couldn't share that with you.
The good news is, Blake put in a counter-claim, and it looks like his video is back up! If you didn't get to see it before, you can watch it here (and keep your fingers crossed that it stays there this time).
For some more fun, check out Vi Hart's binary hand dance.
Last time, we talked about the problem presented in the beginning of the film Good Will Hunting. I said that it was an easy problem, and now I'll show you just how easy it is!
But first, let's learn a little bit about graph theory. The graphs we use in graph theory don't use an x-axis and a y-axis. We don't have functions of x or variables that change over time.
Graph theory graphs chart relationships. In our Bridges of Konigsberg video, we looked at what would happen if we wanted to walk around a city, crossing every bridge exactly once and returning home. If I wanted to draw the city of Konigsberg as a graph, I would draw each land mass as a dot and connect the dots with lines to represent the bridges. Likewise, I could represent the people at a party by drawing a dot for each person and drawing lines to connect people who are friends with each other. These types of graphs are very useful for solving problems relating to networks, scheduling, and planning routes.
In the problem at the beginning of Good Will Hunting, we're given the graph above. Each vertex (dot) is labeled with a number. The edges (lines) connect the vertices in a pattern. The first thing we are asked to do with this graph is:
"Find the adjacency matrix A."
An adjacency matrix is simply a way of writing down which vertices are connected by edges. We'll make a little 4x4 table and write the number of edges connecting each vertex to each other vertex:
If we then wanted to know how many edges connect vertices 2 and 3, we just find the column for 2 and the row for 3 (or vice versa--it doesn't matter) and follow them until they meet--at the number 2. That means that 2 edges connect 2 and 3 (which you'll see is true if you look at the graph). If this seems confusing, just think of it like your old multiplication table. To find out what 6x7 was you had to find the 6 column and the 7 row and find where they met.
You'll notice there's some redundancy in this matrix. We really only need half of it, cut along the diagonal. But, mathematicians like things to be square, so this is what we've got!
The second task asked of us is to create "The matrix giving the number of 3 step walks." This sounds confusing, but if we think of ourselves as walking around on the graph, we just need to figure out how many ways we can get from one vertex to another by taking three steps. Each step is just a move to the next vertex--and the vertices have to be connected by edges, or else we can't walk there. It's kind of like a little board game. We'll make a 4x4 table to chart our findings, just like last time. So if I wanted to start making my table, I'd figure out how many 3-step walks I can take from vertex 1 and end up back at 1 again. If I look back at my graph, I'll see that I have two choices: I can go from 1 to 2 to 4 and back to 1, or I can go from 1 to 4 to 2 and back to 1. Any other walk would take too many steps. To get from 1 to 2, I can go from 1 to 2 to 3 and back to 2 four different ways (remember, we can walk the same edge more than once). See if you can find all four ways. Then find the three other ways you can get from 1 to 2. We build it this way, and the matrix for the number of 3-step walks will look like this:
So, as you can see, the first two parts of the problem are easy. They just involve reading the graph and counting. The second two parts, which ask for some generating functions, are a bit more advanced. But we can understand the concept behind them even if we can't do the math. If you want to see how to do the math, you can see the solution here.
We'll start by looking at the last part of the problem: "Find the generating function for walks from points 1 to 3." All this is really asking is to find a nice way to count all the walks (of any length) we could take from point 1 to 3. Of course, this will be a huge number--there are 12 walks of length 3 alone! The good news is, we don't have to give the actual number (it would be infinite anyway since we could take walks of any length we wanted). We just have to figure out how to express that number. So the generating function from walks from point 1 to 3 involves adding up all the walks of length 1, length 2, length 3, and so on.
For the third part of the problem, "Find the generating function for walks from point i to j," we are asked to find the way to write the number of walks of any length for any two vertices--i and j are just used to represent any two points on the graph.
So there you go! You're well on your way to becoming a graph theorist.
Most of you have probably seen Good Will Hunting. If you haven't seen it, go watch it. It's a really good movie.
One problem that movies like Good Will Hunting often have is that they have to either make up a new math problem, use an existing math problem, or be really vague about what the problem actually is. The movie Proof takes this last approach. Gwyneth Paltrow's character solves "a really important problem." For mathematicians like me, this approach is kind of agonizing. I want to know what the problem is! But the other approaches can be worse.
The book Uncle Petros and Goldbach's Conjecture actually names the proof. Goldbach's Conjecture is one of the big unsolved conjectures in mathematics. If anyone actually solved it, it would be a huge deal. So right from the outset you know one of two things about the outcome of the book: either he doesn't solve it (a bit of a letdown) or he solves it but for some reason never tells anyone his proof (even more of a letdown). I'll tell you this much: it ends in one of these two ways, and it is a letdown.
Good Will Hunting takes an approach somewhere between making up a math problem and being really vague. The first time I saw it, I was really impressed by the idea that Matt Damon's character could solve such a difficult problem. But, upon closer inspection, my admiration dwindled. On the very first day of graph theory class, my professor put in this movie and said, "Now pay attention."
When the big math problem is first introduced in the film, the professor says, "I also put an advanced Fourier system on the main hallway chalkboard..."
I happen to know what a Fourier system looks like (I wrote my thesis about Fourier series). Fourier series are a neat concept that only prove useful in a very specific setting. A Fourier series is a sum of sines and cosines used to approximate some function (like f(x) = x). The formula should look something like this:
At the very least, it should have sines and cosines in it. The graph for a Fourier series should look something like this:
It's got a bunch of squiggly lines that look like they're trying to be some other function. When Will looks at the hallway chalkboard (mere seconds after the professor says it's a Fourier system), this is the problem he sees written there:
Do you see any sines or cosines? Squiggly lines? Me neither. This is a graph theory problem, and this type of graph is a completely different kind from the Fourier series graph I shared above. What's more, this is an easy graph theory problem! We solved the first half of it on our first day of class, and you could solve it too. I'll walk you through the solution next time. For now, enjoy your math films (I certainly do!) but don't believe everything you hear in them.
Most of us probably think of Plato as a philosopher, but what is difficult for modern folks to understand is just how intertwined mathematics and philosophy were back in the day.
Much of Plato's philosophy was inspired by looking at the natural world and the world of mathematics (and we should probably also include a shout-out to Socrates here). Many people may know of Plato's allegory of the cave where he compares our experience of the world through our senses to people seeing shadows on the walls of a cave and thinking those shadows were real forms.
Plato's cave theory is closely linked to his idea of the Realm of the Forms. He postulated a realm of perfection where every imperfect object we experience here in the world would exist in a perfect state. Why postulate such a realm?
Let's think about shapes for a minute. Take the sphere, for example. We understand the concept of a sphere, but does it exist in it's true, perfect form here in the world? The earth is round, so is the moon, but they are not perfect spheres. They're bumpy. Have you ever seen a perfectly spherical stone? Even the marbles we manufacture today aren't perfect spheres--they have tiny imperfections. Perfect spheres don't exist in the natural world, but we understand them conceptually. Mathematical objects like spheres helped shape the idea for Plato's Theory of Forms.
Plato also has some mathematical objects named after him. The Platonic Solids are the three-dimensional shapes that can be made with sides that are all identical (and the sides have to be regular shapes like squares or equilateral triangles). The cube is the most obvious example: All its sides are squares. Four other shapes can be made from other regular polygons:
If you're still not convinced that Plato drew much of his inspiration from mathematics, consider this: over the door of his famous Academy, he inscribed the words, "Let no one ignorant of geometry enter here."
This is the second installment in my Elegant Math series. Today we'll look at the proof that there are an infinite number of primes.
We discussed prime numbers in a previous post. And we showed how they are the building blocks for all other counting numbers in Tuesday's proof. So we understand how important prime numbers are, but now the big question: How many prime numbers are there?
It seems like prime numbers go on forever. We keep finding bigger ones. But they also seem to spread out as they get higher. This makes sense because the higher the number, the more prime numbers there are below it that could divide into it. If I pick a number, and there are a hundred prime numbers smaller than it, what are the chances that none of them is a factor?
Wouldn't we eventually get to a point, if we looked at big enough numbers, that every number is composite because at least one lower number divides into it?
It turns out that this can't happen. Prime numbers do go on forever. But how do we know for sure? Here's a proof. Don't worry, it's mostly in plain English, and once again I've written my comments in italics so the math-phobic needn't be frightened.