Numbers & Functions · 2 The Hole in the Number Line

2.8 Where we are now

We started with counting. We were forced to invent negatives, then fractions, each time to repair a broken operation. And now we’ve been forced one more time — to fill the holes the fractions couldn’t reach — and we’ve arrived at the real numbers: the complete, unbroken number line.

But notice what we’ve been doing. We keep talking about numbers, and then about operations on numbers. An operation takes a number in and gives a number out. That idea — a machine you feed a number and it hands you back another number — turns out to be even more important than the numbers themselves. It’s called a function, and it’s where we’re going next.

Things worth thinking about 1. The proof that √ 2 is irrational hinged on one small claim: if a number’s square is even, the number itself is even. Convince yourself by checking what happens when you square an odd number. (Try 3, 5, 7. Can you see why odd times odd is always odd?)

2. We proved √ 2 is irrational. Would the same proof work for √ 4? Try running the argument with 4 in place of 2 and watch where it breaks — that break is telling you something true, because √ 4 really is a nice number. What is it?

3. A fraction’s decimal always stops or repeats; an irrational’s never does. Here’s a puzzle: I’m thinking of the decimal 0.1010010001000010 . . ., where the gaps between the 1’s keep growing by one each time. It never stops. Does it repeat? Is this number rational or irrational?

4. Now you’ve learned there are vastly more irrationals than fractions. So here’s the unsettling question to carry forward: are these all the same size of infinity, or are some infinities bigger than others? (They are. We’ll get there.) 5. We write 13 as 1101 in binary. Work out what the binary number 10000 is in our ordinary system. (Hint: the places double — 1, 2, 4, 8, 16.) Then write the number 10 in binary.

6. A computer says 0.1 + 0.2 is not quite 0.3. A friend says “then computers are bad at math.” Are they right? Explain what’s actually going on — and whether the mistake is the computer’s fault or just the price of having only finitely many boxes.

Problems to work These first three build on each other on purpose. Do them in order. Problem 1 — Turn the crank.

We proved √ 2 is irrational using a fact about even and odd. Now prove that √ 3 is irrational. You don’t need a new idea — you need the same proof with one word swapped. Everywhere we used “even” (divisible by 2), use “divisible by 3” instead. Start: suppose √ 3 = a b in lowest terms — and follow the trail until it contradicts itself. The point isn’t the answer; it’s the discovery that the method transfers.

⋆Teach it — explain it to someone Problem 2 — Teach it. Find a person who has never seen why √ 2 is irrational. Without reading from this book, explain it to them out loud, start to finish, until they nod. The moment you have to say it to someone else, the gaps in your understanding light up like warning lamps. Three load-bearing steps your explanation must include: • You assumed √ 2 was a fraction in lowest terms — so a and b are not both even.

• You showed a must be even, then used that to show b must be even too. • You pointed out that “both even” contradicts “lowest terms” — and that’s what kills the assumption.

Problem 3 — Stretch: adapt the idea.

Now show that 3√ 2 — the cube root of 2 — is irrational. Suppose 3√ 2 = a b in lowest terms, cube both sides, and hunt for a contradiction. You may use this fact for free: if 2 divides a3, then 2 divides a. Push the argument through and make a and b both turn out even, contradicting “lowest terms.” You took a method built for square roots and bent it to fit cube roots — that is most of what doing real mathematics actually is.

Going further The Hole in the Number Line ⋆Teach it — explain it to someone Problem 4 — Teach it (the computer’s sketch). Sit a family member down at any computer or phone with a Python prompt, and have them type 0.1 + 0.2. When the surprising 0.30000000000000004 appears, explain why — using the “finite grid of dots” picture. Get them to see the computer isn’t broken; it rounds every number to the nearest dot it can store, and two roundings didn’t quite cancel. If they walk away saying “oh — it’s a sketch, not the real number,” you’ve taught the whole chapter in one sentence. Problem 5 — Build a repeating decimal back into a fraction.

A repeating decimal is secretly a fraction. Take x = 0.27 = 0.272727 . . . Multiply both sides by 100 to get 100x = 27.2727 . . ., then subtract the first equation from the second so the infinite tail cancels. Solve for x and write it as a fraction in lowest terms. (This proves the converse of the chapter’s claim: every repeating decimal really is rational.)

Problem 6 — [challenge] Rational, or not?

Decide whether each number is rational or irrational, and justify it in one line: (a) 0.45454545 . . .

(b) √ 9 (c) √ 2 + √ 2 (d) π −π.

(Two of these look irrational but aren’t. The lesson: “has a square root in it” does not mean “irrational.”) Problem 7 — [challenge] A sum that hides.

We know √ 2 is irrational. Is it possible for an irrational number plus another irrational number to come out rational? Either argue it can never happen, or find two irrational numbers whose sum is a plain whole number. (Hint: think about √ 2 and what you’d add to it to get back to 0 . . . or to 4.)

Problem 8 — [challenge] Irrational to an irrational power.

A famous, almost magical question: can an irrational number raised to an irrational power ever be rational? Consider N = √ 2 √ 2. You don’t know whether N is rational — and you don’t need to: • If N is rational, you already have your example.

• If N is irrational, compute N √ 2 using (ab)c = abc — and watch it collapse to a whole number.

Either way, such a pair exists. (This is a real theorem, and this two-case trick is the actual proof. You proved something exists without ever finding out which case is true — a strange and powerful kind of argument.)

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