So I'm going to show you one of the most elegant proofs from mathematics. Now, don't get scared! You don't need to know much math to understand the beauty of this proof, and I'll walk you through the basic ideas you need to know.
A^2 + B^2 = C^2 or
1^2 + 1^2 = 2 (I'm using little carrots to show "squared" because this program doesn't have many fancy fonts. 1^2 is "one squared.")
To find the length of the third side, we just need to take the square root of two. Easy! It's uh.... what number do we square to get two? 1^2 is 1, 2^2 is 4. What squares to make two? It must be between 1 and 2. Well, Pythagoras was stumped, because he couldn't find a fraction between 1 and 2 that squared to make 2.
Before we move on, I want to quickly talk about factors here. The factors of a number are numbers that divide into that number. For example, 2 is a factor of 6 because 6/2 = 3 (and 3 is a whole number). In fact, 2 is a factor of every even number but not a factor of any odd numbers. That's one way to define even and odd numbers.
We say that two numbers have no common factors if, well, they have no factors in common! When we talk about fractions, we have a reduced fraction when the numerator and the denominator have no common fractions.
6/2 is not reduced because 6 and 2 have a common factor: 2. We could write 6/2 as (3*2)/(1*2) or (3/1)*(2/2). 2/2 = 1, so we can just get rid of it, and we see that 6/2 can be written more simply as 3/1 or 3.
If all those numbers freaked you out, don't worry about it. Just read on. The proof is better:
So let's say, for the sake of argument, that Pythagoras was wrong, and there is a way to represent root2 as a fraction. Then:
- root2 = a/b where a and b have no common factors
- 2 = (a/b)^2 I just squared both sides because I don't like having square roots in my equations.
- 2 = (a^2)/(b^2) You'll have to trust me on this one (or look it up); we're not going to prove this step here.
- 2*(b^2) = a^2 Just some quick algebra: I multiplied both sides by b^2.
- 2*b*b = a*a That looks a little nicer, doesn't it? It's an unnecessary step in the proof, but I want to make this easy to read!
- This means that a is a multiple of two (if a^2 is a multiple of two, a must be also--another thing I'll let you figure out on your own), so we could say a=2*m. That means a*a = 2*m*2*m = 4*m*m, so:
- 2*b*b = 4*m*m Let's simplify by dividing both sides by two...
- b*b = 2*m*m
- So b is also a multiple of two!
- But that's a contradiction: a and b have no common factors. Our proof started by setting a and b as numbers without common factors and ended by saying they were both even (and therefore had 2 as a common factor).
- This means that there are no numbers a and b for which root2 = a/b. Since two numbers can't both be even and have no common factors!
- Therefore, root2 cannot be expressed as a fraction. In other words, it is irrational.