Prime Divergence

Did you know that:

is equal infinity? Now, some of you might be thinking, “Oh, I already knew that.” Well, then this post is not for you, move along. Others, though, might think, “hold on, isn’t adding anything up to infinity equals infinity?” Nope because the sum (1/2)+(1/4)+(1/8)+ etc for example is equal to 1. And in fact, all geometric sum (in case you don’t know, it goes like n+n^2+n^3+… etc) with a number less than one converge into a number that is not infinite.

Some others who has learned stuff from second semester calculus class might think “Really? It looks real slow.” Well, it is really slow, but as you shall see, it nevertheless does diverge, and demonstrating this will be the topic of this post.

This post is based on a section in the book “An Imaginary Tale” by Paul Nahin. The book talks about this, and I thought the mathematics was super neat, and so I decided to put it here.

The post is in three parts:

a)I will show a proof originally done by Nicole Oresme (1320-1382) on the divergence of the harmonic series, which is 1+(1/2)+(1/3)+…

b)Then I will change the Riemann zeta function, which is f(x)=1+(1/2^x)+(1/3^x)… into a product of primes, which was done by Leonard Euler (1707-1783).

c)And finally I will use the above results to prove that the sum of all 1/prime does diverge, which was proved by Euler.

Who’s Got More Points?

So, the question is, which of the following has more points: a line segment or an infinite line? You might think the obvious answer is the infinite line. But it is not as simple as you think it is. A simple proof done by the mathematician Georg Cantor will show you the answer:

Imagine two semicircles of the same size and shape. A semicircle is pretty much half a circle. You agree, since both semicircles are the same, they both have the same number of points, right? Beneath the first semicircle, place a line segment that have the same size as the diameter of the semicircle. Beneath the other one, place a line that goes to infinity. Now, your j0b will be to match every single point on the border of the semicircle to every single point on the line segment and every single point in the second semicircle to every single point in the infinite line. The setup is done as shown in the figure below:

In the line segment, I match up every point of it to the semicircle by a straight line connection. By going from left to right (or any direction you want), you realize that you can match every single point on that semicircle to the line segment. For that reason, the line segment has as many points as the semicircle above it.

In the case of the infinite line, I have to do a cleverer maneuvering. From the center of the semicircle, I extend a line straight to the left (or right), and sweep around the semicircle counterclockwise. Doing so will match every point on the semicircle. Yes, even the one point in the infinite line that seems unimaginably distant. After all, all I would have to do is extend a line from the semicircle to that point. As you will see, no matter how far away the point is from the semicircle, it will touch a point on the semicircle. Interesting, it seems like an infinite line has the same amount of points as the semicircle. What is the implication of all of these things I have written?

Well, since the line segment has the same number of points as its corresponding semicircle, and so does the infinite line, and both semicircles have the same number of points, there is only one conclusion I can take. The infinite line has the same number of points as a line segment!

Mind Blown.

-This post has been inspired by the book To Infinite and Beyond: A Cultural History of the Infinite, so I have the book to thank-

Math Club Stuff

In college, I was in a math club for a while. And being in a math club has its own charms, including having pi ices to chill my drink:

Plus, I got to have a free mug with math equations on them!

Up there, you can see Maxwell’s equation of electromagnetism, circumference of a circle, planck’s quantum mechanic equation of energy, deBroglie’s equation of waves and momentum, Hubble’s law, and part of general relativity and Rydberg’s equation for hydrogen light emission.

Over here, you see Einstein’s energy mass equivalence, Euler’s formula, Schrodinger’s quatum mechanics wave equation, Dirac’s equation (which predicted antimatter!), Newton’s second law, Riemann’s zeta function, Fourier transform, and Boltzmann’s entropy equation. The part where you see zero, behind it is delta*S, which is path of least action.

Yeah, equation galore. Say, can you tell which one is which? I couldn’t tell what a third of them were until I looked them up. I mean, even though they did tell you the name of the equations, it is not like you know exactly what it means. And for around half of them, I wouldn’t even know how to use them. I mean really, what is that psi letter (the pitchfork looking letter, perhaps mark of the devil? ^_^) in Schrodinger’s? I know it stands for wavefunction, but that is not exactly helpful. Overall, being in the math club is a good deal. I get to have free pizzas and mug, while satisfying my math obsession.

Best-Math-Study-Ever

Title says it all: When Zombies Attack! Mathematical Modelling of an outbreak of Zombie Infection.

That is marketing genius I tell you. It might as well apply to other infections, although zombies are quiet different. I don’t understand any math in it, but the conclustion arrived is that unless aggressive actions are taken to quarantine and destroy the zombies, and done often, we are all doomed. Nice stuff to think about, eh?

hat tip neurotopia

Woo Hoo! I Did It!

I got the solution right for Monday Math Madness. Considering that it was a problem that was above my usual skill level, and that I had to do it under 6 hours because I did it 6 hours before the deadline, I did alright. Plus, I get a prize for getting it right. ^_^ Also, if you are interested, there are other solutions that he is posting, so you should look out for that. It will be interesting to see how many different kinds of solutions they come up with.

Monday Math Madness 37 Solution

For my entry on MMM 37, I wrote the following solution in my e-mail. I hope I got it right. ^_^:

Euclid’s The Elements: Preposition One-Really Fun Math Puzzle

This is the nth edition of *drum rolls* Pop Quizz! A quizz in which the answers are not obvious and if it is obvious, and you get it wrong, you are rebuked at!!!

Occasionally, I will occasionally ask a question, and the reader will answer them! For every wrong answer, I will consider one of your comments a spam. Ha! See how you like that suckers! (I am kidding, of course) Sometimes, a wrong answer won’t invite rage, some really obvious one will, and you will be called an idiot if you can’t answer the obvious one. In fact, I may have to make an extra page on my blog for the lists of idiots.

Anyways, once upon a short time ago, I was reading Euclid’s The Elements, which is basically the geometry book to rule them all. First, it defines various terms, and go over postulates. Then it goes through geometric proofs, each different one called prepostitions. Well, the first one I think is a really neat geometry problem. It goes like this:

3. On a given straight line to construct an equilateral triangle.

Meaning, considering you have a line AB, make a triangle ABC with all sides equal. Sounds hard? Well, if you hear the clue, you migh go “OOOOHHHhhhhh!” If not, well… shame on you, mate. ^_^

Calculator Contest and Monday Math Madness

Firstly, there is a contets for a calculator called Nspire. If you want to get one free, you should enter. I want one too, so I too will participate. ^_^ Secondly, the blog has Monday Math Madness, which is basically a biweekly (or something like that) feature of interesting math problems. Check out number 28!

It’s Pi Day!!!

You know, 3.14, get it? March (3) 14? Easy enough right? Just saying.

Largest Number Contest

This is the nth edition of *drum rolls* Pop Quizz! A quizz in which the answers are not obvious and if it is obvious, and you get it wrong, you are rebuked at!!!

Occasionally, I will occasionally ask a question, and the reader will answer them! For every wrong answer, I will consider one of your comments a spam. Ha! See how you like that suckers! (I am kidding, of course) Sometimes, a wrong answer won’t invite rage, some really obvious one will, and you will be called an idiot if you can’t answer the obvious one. In fact, I may have to make an extra page on my blog for the lists of idiots.

0 Today’s pop quiz is not a question. Oh nop. It is a contest between you and I! ^_^ It is to see who can write the largest numbers in 15 seconds! By the way, it is on paper, just so you know. Oh, and things like this don’t count: one more than the other guy, infinity, ubberinfinity, etc. It has to be the standard math notaion! If you don’t, not only did you lose, you have lost all dignity of yourself, and will be considered a moron by yours truly. So, get ready, get set, go! (my answer is below the fold)