Numbers & Functions · 2 The Hole in the Number Line

2.2 The proof (and why it's gorgeous)

You already have the fact you need: √ 2 is irrational, not a fraction. What follows is the proof of why — one of the most beautiful arguments ever made, and completely within your reach. If it gets heavy, you can skim it, trust the result, and come back later; the rest of the chapter won’t depend on the details. But if you want to feel what a real mathematical proof is, slow down here. ♦Why it’s true — skip on a first read, but the real brain-training The trick is sneaky. We’re going to assume the opposite of what we want to show and watch it destroy itself. This is called proof by contradiction, and it’s a bit like a detective assuming a suspect is innocent and then showing that assumption leads to something impossible.

So: suppose √ 2 IS a fraction. Let’s see where that leads.

If it’s a fraction, we can write it as a b , where a and b are whole numbers. And we’ll insist on one reasonable thing: that we’ve already reduced the fraction to lowest terms. (You can always do this. 6 4 becomes 3 2.) In particular, a and b are not both even — because if they were, we could cancel a 2 and reduce further. Remember that. It’s the trap that’s going to spring. √ 2 = a b =⇒ 2 = a2 b2 =⇒ 2b2 = a2.

Look at that last equation. The left side is 2 times something, so it is even. That means a2 is even. And here’s a small fact you can check yourself: if a number’s square is even, the number itself must be even (an odd number squared is always odd — try a few). So a is even.

If a is even, we can write it as a = 2k for some whole number k. Substitute that back in: 2b2 = (2k)2 = 4k2 =⇒ b2 = 2k2.

But now look — the right side is 2 times something, so b2 is even, which means b is even too.

And there’s the contradiction. We just proved a is even and b is even. But we started by insisting they weren’t both even! Our assumption has eaten itself. The only thing we assumed was that √ 2 could be written as a fraction. That assumption led to an impossibility. So the assumption must be false.

√ 2 is not a fraction. It is irrational.

There’s a legend that the man who first discovered this — a follower of Pythagoras named Hippasus — revealed the secret while at sea, and was thrown overboard for it. The Pythagoreans believed that everything in the universe was built from whole-number ratios. A number that wasn’t a ratio wasn’t just inconvenient. It was heresy. It meant their whole picture of reality had a hole in it.

They were right to be shaken. Because it wasn’t one hole.

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