The year is 500 BC. You are an ancient Greek mathematician. A very popular one, I might add. You have many students who love you and all your Greek peers agree with you. In fact there is a measure of perfection in your world. You love numbers, and you love ratios. You have some favourite ratios, like ½ and 3/5, because they just look nice, or sound good in music. You construct shapes and lines and everything is wonderful in your world.
Then along comes this rogue mathematician. He doesn’t have the same affinity for fractions that you do. In fact, he proposes that there are numbers in our world that are not fractions. This concept has never before been put forward in all of your years of study.
“Preposterous!” you say to him, “everything in nature is ratios and fractions and whole numbers and perfection. There is no way you can create some abomination that does not submit to my fraction powers.”
So he just smiles at you, and draws the following shape.
“So what?” you say. “It’s just a simple square. What could possibly be so wrong with that?”
Then he does something dastardly; he adds the following lengths:
You still do not see a problem with this shape. It is at this moment, that the rogue mathematician shatters your understanding of reality. He draws the diagonal.
You gasp as the pieces fall into place.
Pythagoras’s theorem tells you that
The length of this seemingly innocent diagonal is !
But wait, what’s so wrong with ? Who’s to say that we couldn’t find a fraction to represent this quantity? Why does this number have to break our perfectly constructed laws of mathematics? Below is an adaptation of that proof. The proof that cannot be expressed as a fraction. The proof that changed the world!
Before we start the proof we need a few useful facts. An even number is a number that can be represented as 2k where k is an integer (whole number). For example, 16 is even because 16 = 2*8.
If a number is even, then the square of that number is even. For example, 10 is even. The square of 10, 100, is also even.
This result works backwards as well. If a number is even, then the square root of that number is even. For example, 16 is even. The square root of 16, 4, is also even. Ok, on with the proof. We will use proof by contradiction.
Suppose that can be represented as a fraction. We don’t yet know what this fraction is, so we must represent it arbitrarily as: .
We will assume this fraction has been reduced to lowest terms. For example, if the fraction is , we reduce it to .
Square both sides of this equation.
Multiply both sides by q2:
That 2 in front of the q2 tells us that the p2 is even. And we know that if p2 is even then p must also be even. Ok, p is even, what`s the big deal? Well if p is even, then we can rewrite p as 2k, where k is an integer (we don’t yet know which integer). And then:
Combining this with our equation from before:
Divide both sides by 2:
Like before, that 2 in front of the k2 tells us that q2 is even. And we know that if q2 is even then q must also be even. Ok, q is even, what`s the big deal?
So what have we established here? We have shown that q is even and p is even. Can anyone see why this would be a problem?
If both q and p are even, then both of these numbers are divisible by 2 and the fraction is not in lowest terms. But we assumed that the fraction was in lowest terms. This is a contradiction!
This may seem like a small matter. We assumed the fraction was expressed in lowest terms and we proved that it was not. Big deal. Well, actually, it is a big deal. Mathematics is so precise and so structured that the ability to prove a contradiction, no matter how small the contradiction may seem, has huge ramifications.
In short, we assumed that could be represented as a fraction and then went on to produce a contradiction. We are therefore left with only one, terrible, inescapable, conclusion. The can never, no matter how hard you try, be represented as a fraction.
Back to ancient Greece. You ponder this proof. You think and think, trying to find a way out. You try to find some esoteric fraction that could represent . But it is impossible. There is no fraction for this evil number. There is no perfect ratio. There is no hope.
The length of the diagonal is just as real as the length of the sides of the square. You could measure it. You could walk down it. You can see it and touch it and it is real. But, this length cannot be represented as fraction. In your utter disgust for this number, you call it “irrational,” meaning “cannot be represented as fraction.”
You gain the ability to represent and believe in . But you lose the world of perfect fractions and ratios. It is a bittersweet day for mathematics. And this insight was accomplished by using a powerful tool, proof by contradiction.