Monday, April 24, 2006

Math to Wow Your Friends - Part II

There is a building not too far from campus that looks like this

and at closer inspection like this


which raises an almost infinite number of questions.

In addition to all of the questions you have about the artist and the owner of the building and the painted guard and the little brown squirrel, you’re probably also wondering what pi and Pythagorean’s theorem have in common (besides the fact that you were made to memorize both in school.)

You may remember from Math to Wow Your Friends - Part I that most types of numbers can be written as a fraction or ratio and hence are called rational numbers.

The number pi, however, is irrational. It is a never ending, never repeating decimal that cannot be written as a fraction. Other numbers, such as the square roots of primes, are also irrational.

The first Greek mathematicians were happily devising geometry without any notion of irrational numbers until they tried to apply Pythagorean’s theorem to a very special triangle.

Pythagorean’s theorem tells us how to calculate the hypotenuse of a right triangle given the lengths of its two legs.


Let's use it to find the hypotenuse of this triangle


So, in other words


This is exactly what the Greek mathematicians did. Then they tried to write the square root of two as a fraction. And that is where they got stuck.

They could not find a way to write the square root of two as a fraction, so they were compelled to prove that it couldn’t be done.

Here is a simple, elegant proof that the square root of two is irrational:

First, we’ll assume that it CAN be written as a fraction, and see if this assumption leads to a contraction. If it does, we will have proven our point.

Assuming that the square root of two can be written as a fraction, we can write


where a and b are whole numbers (and b isn’t zero, since it’s impossible to divide by zero.)

We can re-write this as


and by squaring both sides we can write
Now comes a bit of logic. Think about perfect squares like 4 and 9 and 16 and 25 and 36 and so on and the number of prime factors each one has. Here are several perfect squares broken down into their prime factors.

4 = 2 * 2

9 = 3 * 3

16 = 2 * 2 * 2* 2

25 = 5 * 5

36 = 2 * 3 * 2 * 3

and so on

Do you notice that each perfect square has an EVEN number of prime factors? Try some more on your own if you’re not convinced.

Back to our equation

b-squared must have an even number of prime factors and a-squared must have an even number of prime factors.

But the 2 on the left of the equal sign means that the left side of the equation has an ODD number of prime factors while the right side of the equation has an EVEN number of prime factors.

If two numbers are equal they will have the exact same prime factorization. Since the left side of the equation and the right side of the equation don’t even have the same NUMBER of prime factors the two sides of the equation cannot be equal.

This is a contraction. And that means we have proven that the square root of two cannot be written as a fraction and is therefore irrational.

Oh, and by the way, the squirrel says:

“I love you Mary!”

3 comments:

jen said...

I really love the Pythagorean's Theorem. I know you think I'm luing or teasing, but I'm not. I think it's almost as sexy as Picasso.

jen said...

make that "lying."

KK said...

Jen - we all want to know when your blog is coming out. We think you NEED to explore this Picasso fascination and cerebral love affair. Really. You must.