Monday, January 24, 2011

Some Math You Probably Know

I started this blog with the idea that I was going to write about music I liked and math I enjoyed. So, a few months later, and there are several music posts. Not so much on the math side though. My bad, folks. Today, I'm going to fix that once and for all. For your reading enjoyment and possibly the first time you have touched math in a long time, I present...the Pythagorean Theorem.

Originally credited to Pythagoras of Samos, one of the first "true" mathematicians. I mean, the man developed his own cult that declared their religion to based off of mathematics and how numbers had spiritual properties. Not your average middle school algebra teacher indeed. His theorem is taught to all the middle school-ers out there, the simple rule about all right triangles. Now a right triangle is any triangle with a 90 degree angle in it. Got it? Good.

Here we go, simple and nice to look at as far as math theorems go. Plus, it's very easy to plug into your calculator! But how about proving the lil' muthafucka?
Well, actually that's pretty easy too.
So easy in fact, that even people you wouldn't think were into math got into the whole mix as well - James Garfield, for instance. That's right, President James A. Garfield did some math on the side of being a politician and future president. His proof is really short, sweet, and easy to explain, hence the reason I chose it for this blog post. I found this proof (and EIGHTY-FREAKING-NINE others) at Cut-The-Knot, a really cool and thorough math site. Check it out to learn more about everything math-y. So let's get to the proof itself:

First, let's imagine a trapezoid:
This baby has two parallel lines and a right angle, like so

Then you label the sides:

Next draw the lines, making 3 triangles:

Now it's all simple area formulas from here :P

The area of a trapezoid:
Area = ((base 1 + base 2)/2)*height

Well base 1 = A, base 2 = B, and height = A+B

But this figure is also made up of 3 triangles and the triangle area formula is:
Area = .5*base*height

Area of Trapezoid = Area of Triangle 1 + Area of Triangle 2 + Area of Triangle 3

Triangle 1 = A*B*.5, Triangle 2 = A*B*.5, Triangle 3 = C*C*.5 (as you can tell from above image)

Woot, okay then

(A+B)*.5*(A+B) = .5AB + .5AB + .5C*C, thats putting area of the trapezoid equal to the 3 triangles

Let's multiply the whole line by 2:

(A+B)(A+B) = 2AB + C*C
When we do the multiplication of (A+B) squared we get,

A-squared + 2AB + B-squared = 2AB + C-squared

Subtract 2AB from both sides and you're done!

A-squared + B-squared = C-squared

As far as proofs go, this one is practically giving itself to you. Hopefully, I'll do more proofs and mathematician biographies as the year goes on. We'll see.

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