Numbers & Functions

2 The Hole in the Number Line

CHAPTER 2 The Hole in the Number Line At the end of the last chapter I made you a promise. I said the rational numbers — all those fractions packed infinitely tightly along the line — don’t actually fill it up. I said there are holes.

Let me show you one.

2.1 A perfectly innocent question

Draw a square. Make each side exactly 1 unit long. Now draw the diagonal — the line from one corner to the opposite corner.

1 1 1 1 d Figure 2.1. The unit square and its diagonal d. The diagonal cuts the square into two right triangles, each with short sides 1 — the perfect setup for Pythagoras: 12 + 12 = d2. The diagonal is a real, drawable length, so it must be a number.

How long is that diagonal?

This isn’t a trick question. It’s about the most ordinary shape there is. A square with sides of length 1. Surely its diagonal has some length. And surely that length is a number — a number we can write down as a fraction, like everything else we’ve met so far.

There’s a famous fact — you may have already bumped into it — that tells us exactly how long the diagonal is. It’s called the Pythagorean theorem, and for a right triangle it says: the two short sides, squared and added together, equal the long side squared. Our diagonal cuts the square into two right triangles, each with short sides of length 1. So if we call the diagonal’s length d: 12 + 12 = d2 =⇒ 2 = d2.

So d is the number whose square is 2. We write it √ 2, the square root of 2.

Now here’s the question that’s going to crack the world open. We know this length exists — you just drew it. So it must be a number. And every number we’ve met so far is a fraction. So: What fraction is √ 2?

Take a moment and try to find it. Hunt for two whole numbers, one over the other, that give you 2 when you square them.

You might try 3 2 = 1.5. Square it: 2.25. Too big. Try 7 5 = 1.4. Square it: 1.96.

Too small, but close! Try 17 12 ≈1.4167. Square it: about 2.0069. So close!

p 2 = 1.41421 never exactly a fraction 1.400 1.405 1.410 1.415 1.420 Fractions creep toward p 2 but never land on it 7 5 41 29 17 12 fraction (square) 7/5 1.96 17/12 2.0069 41/29 1.99881 99/70 2.00020 239/169 1.999986 approaching 2, never reaching Figure 2.2. The hunt for √ 2 as a fraction. The convergents creep closer and closer from both sides, and their squares crowd in on 2 — but never land on it exactly. No fraction ever will.

You can keep going. You can get astonishingly close. But you will never, ever land exactly on 2. And the reason is not that you’re not trying hard enough. The reason is that no such fraction exists.

√ 2 is not a ratio of two whole numbers.

It is not rational.

That word — not rational — gives these numbers their name. They are the irrational numbers. And the wild thing is we can prove, with total certainty, that √ 2 is one of them. Not “we haven’t found the fraction yet.” We can prove the fraction cannot exist.

The Hole in the Number Line

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