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Last time, we talked about the problem presented in the beginning of the filmGood Will Hunting.  I said that it was an easy problem, and now I'll show you just how easy it is!


If you like easy math, click here.

 
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.


Click here to find out what approach Good Will Hunting takes!
 
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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.

 
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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?

Oh, a cliff-hanger!  Click here to find out the answer!

 
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One of the biggest stumbling blocks for many math students is the requirement that they show their work and write in proper mathematical notation.

This is simultaneously the area in which other students thrive: If you can follow directions, you will succeed in math class.

Students are told that mathematics is a language like French or Spanish.  This mostly just frightens the students, or bores them if we're lucky.

What students are not often told is why we use a special language for math and why we need to follow special rules.

Click here to find out why!

 
Check out my new video about the famous Bridges of Konigsberg problem!
 
Nate Silver, founder of fivethirtyeight.com, is my personal favorite statistician.  For one thing, I don't really enjoy doing statistics, so I appreciate that someone as smart as Nate Silver does statistics so that I don't have to.  For another thing, he uses statistics the way it was meant to be used--and his predictions have a very high level of accuracy.

He started out as a baseball statistician, predicting the careers of Major League Baseball players.  In 2007, he began running statistical analysis on the 2008 election, and he very accurately predicted the outcomes of that election year.

I bring him up today to share an article he wrote which uses statistics the way I use them in my daily life (and in the way I suspect you'd like to use them in yours).  Check out "How to Beat the Salad Bar"--his first article in a series sharing ways to use statistics to your advantage in your day-to-day life.
 
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How often did you hear this conversation as a little kid?
Kid 1: Yuh huh!
Kid 2: Nuh uh!
Kid 1: Yuh huh times infinity!
Kid 2: Nuh uh times infinity plus one!
And if you're a normal little kid, you'd think, "Wow, Kid 2 totally won!"
If you're me as a little kid, you'd remind Kid 1 and Kid 2 that you can't count to infinity, so infinity plus one doesn't make sense.

What a buzz kill.


Well, the fact that we cant count to infinity is something we have a lot of trouble wrapping our brains around.  If something is infinite, that means it goes on forever. That's not just a really long time--it's forever!  Okay, see, it's hard to imagine.

But the crazy thing about infinity is that there are different orders of infinity.  It turns out that some things are more infinite than others.  How can that be possible?  Well, I'll show you.

Let's start by talking about the counting numbers: 1, 2, 3, 4, 5, 6, 7...  They go on forever, right?  If you give me the highest number, I'll add one to it and give you a higher one, thereby proving you wrong.  This set of numbers is clearly infinite.

Now, I'm going to show you that there are more numbers between zero and one than there are counting numbers.  It sounds preposterous, I know!

Well, I want you to make a list.  All the numbers between zero and one can be expressed as decimals that go on forever.  1/2 is 0.50000000... where the zeros go on forever.  1/3 is 0.3333333333... where the threes go on forever.  Some numbers are irrational, so they just have a decimal form that goes on forever and never repeats: 0.28894503857220900...  That's not any special irrational number--I just created it by mashing on number keys.

Okay, so make your list, and make sure you include all the numbers between zero and one.  Got it?  Of course not--your list would have to be infinitely long.  But, for the sake of argument, let's say you do have a list of all the numbers between one and zero:
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To keep track of how many numbers between zero and one we've thought of, I've numbered the list (using our counting numbers).  This is only the beginning of the list--it goes on forever!

Okay, now, despite the fact that your list goes on forever, I'm going to find you a number that isn't on it.  I'm going to take the number that uses the first digit of your first number, the second digit of your second number, the third of your third and so on.  It will look like this: 0.5349918860...

But, you say, that number is on my list too!  It's way down here at spot 1,875,943!

Oh, but I'm not finished yet.

Now, I'm going to increase every digit of my number by one (and if my digit is nine, I'll make it zero).  Now my number looks like this: 0.6450019971...

This number isn't on your list.  How do I know?  This number has a different first digit from your first number, a different second digit from your second number, a different third from your third, and so on...forever.  So even if it looks like all the digits match up with your 974th number, the 974th digit won't match.

So you concede that maybe your list didn't have all the numbers on it.  You add my number to your list.  "Now it's complete," you say.  Well, I'll just do my trick again, and I'll get another number that's not on your list.  No matter how many numbers you put on your list, I'll keep being able to find a number that's not on it.

And don't forget, you have infinitely many numbers on your list--it goes on forever.  But it can never have all the numbers between zero and 1 since we can always use this technique create a number not on the list.

So when we pair the counting numbers up with the numbers between zero and one, we'll always have more numbers between zero and one than we'll have counting numbers.  The upshot of all this is that we've just seen that there are more numbers between 0 and 1 than there are counting numbers.  The numbers between 0 and 1 are more infinite than the counting numbers!

And you thought infinity was hard to wrap your brain around before...
 
One of the fun ways to do math is by playing games.  Besides the obvious math-related games like Blackjack, there are a ton of fun logic and puzzle games out there, and many are available on the internet.  Here are just a few I enjoy--please feel free to leave links to other good mathy games in the comments section!
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Entanglement (click the images for links to these games) is a great knot theory game.  You rotate tiles to form the longest path possible without hitting the edges or returning to where you started.

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Red Remover is a fun spatial reasoning and physics game where the goal is to get all the red blocks off the screen without losing any of the green blocks.  It has many levels that test your ability to reason and to predict how the objects will move.  The blocks have great facial expressions too!

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Ragdoll Cannon is a bizarre little game that is animated as though it's been drawn on graph paper.  You shoot ragdolls out of a cannon and try to hit a target.  The idea behind it is quite simple, but the levels can get difficult--they test your spatial reasoning and your timing.

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Flea Circus is a great logic game with an old-school graphic style.  You need to set up the space so the fleas can all get through the door and none of them fall in the water or get stuck.  Then release the fleas and see if your setup worked!  Sometimes you need to tweak things a bit as you go along.  I like games like this where you set everything up and then just watch it go.

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B-Cubed is a logic and spatial reasoning puzzle where you need to move a block over all of the other blocks in the puzzle before you arrive at the red block.  Gray blocks fall away when you go over them once, so you need to make sure you leave yourself a path to get to the red block in the end.  Many of the puzzles have more than one solution, so you can redo levels and see how many ways you can solve them.

Have fun playing, and if you know any other fun math games, feel free to share!
 
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Pi is the Greek letter we use to represent the number most often approximated as 3.14 (which is why March 14 is Pi Day).  Pi is one of those pesky irrational numbers.  It's an important one because it's the number that describes the relationship between a circle's diameter (the distance across the circle) and its circumference (the distance around the circle). Specifically, pi = circumference/diameter.


For a long time, we thought a circle's circumference was simply 3 times its diameter, but as we started to be able to draw more accurately, we started to notice that 3 wasn't quite right...

Mathematicians tried to figure out what the number actually was. Archimedes famously approximated pi by drawing regular 96-gons (polygons with 96 sides) inside and outside of a circle to get as close as possible to the actual measure of the circumference of the circle.  He figured out that pi is between 223/71 and 22/7.

Since pi is irrational, it can't be written as a fraction, and its decimal form goes on forever without repeating.  We're still finding digits to pi--and we always will be!

Here's a musical interpretation of pi to celebrate pi day: http://www.youtube.com/watch?v=iOjsRyxL7Rs.  Amazingly, it sounds pretty good!