## When Arithmetic Fails

Oct 082011 ## Ulam Spiral Pics

Jan 022011 via ctkrohn

My homie, Dur, pointed out this nice collection of Ulam Spiral pics. The Ulam Spiral describes the picture you get, when you make a spiral of the natural numbers circling all the primes. The basic version looks like this: via wikipedia

The ctkrohn page also blabs a little about the twin prime conjecture and some recent results of green and tao about the existence of arithmetic prime progressions of arbitrary length. The gist is that we know very little about basic patterns within the primes. My guess is that any reasonable pattern probably exists in there somewhere… probably infinitely often… just a guess tho…

## Proofs I: The Pythagorean Theorem

Dec 312010 The Pythagorean Theorem is a true classic in the history of math. Generally, peeps credit the Ionian mathematician, Pythagoras, with the first proof of this theorem back around 500 BC. The oldest proof on wikipedia is Euclid’s proof and it looks pretty complicated. This is because Euclid was working within the classical framework of Euclidean geometry. As modern mathematicians, we have more powerful and diverse tools at our disposal for this proof. However, this makes the setup for our proof somewhat more complicated. Where do we begin? How much do we assume before beginning? I guess I’ll assume a basic knowledge of modern geometry. In particular, I’ll assume a basic knowledge of angles, lengths and triangles. The theorem concerns the length of the sides of a right triangle. Recall that the two sides connected to the right angle are called legs and the third side, opposite the right angle, is called the hypotenuse. Here’s a statement of the theorem:

Theorem: The sum of the squares of the legs of a right triangle is equal to the square of the hypotenuse.

So, according to the labels on the picture to the left, this relationship can be written as [pmath]a^2 + b^2 = c^2[/pmath]1. The easiest proof involves putting four of these right triangles together to form a square. You can do this in two ways, as demonstrated at the left. If you put them together in the top square, then you get the following proof:

Top Proof: The large square has side of length c and therefore it has area [pmath]A = c^2[/pmath]. However, we can see that this square is also composed of four triangles and a tiny square inside them. The area of each triangle is [pmath]1/2 ab[/pmath] and the area of the square is [pmath](b-a)^2[/pmath]. So, the area of the big square can also be written [pmath]4( 1/2 ab ) + (b-a)^2 [/pmath]. We can use algebra to rewrite this as [pmath]2ab + b^2 – 2ab + a^2 = b^2 + a^2[/pmath]. Since we already discovered that the area of the big square was [pmath]c^2[/pmath], we deduce that [pmath]a^2 + b^2 = c^2[/pmath]. The sum of the squares of the legs is equal to the square of the hypotenuse.

The argument for the bottom construction is similar. Both of these are found on wikipedia, along with several other proofs if you’re interested. My favorite might be the following proof by animated gif: 1. I got pmath to work! And boy does it look crappy.

## Logic, Math, Computer Science and Education

Dec 312010 IMHO mathematics consists mathematical objects and their properties. We define these objects into existence and then use logic to prove theorems codifying their behavior. However, unless you major in mathematics in college, none of this is taught to us in school. Instead we’re taught to memorize theorems and use them for various computations. This was a useful skill before the invention of computers, but nowadays it’s mostly busywork. I believe that we should scrap the current high school curriculum and replace it with courses on logic and theoretical math. This would prepare students for a more sophisticated approach to the fields of computer sciencephysics, etc. I’ll be writing some basic proofs on this blog to give you a feel for what I think is important….

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