Numbers & Functions · 1 Counting Isn't Enough

1.4 A picture: the number line

There’s a beautiful way to see all of this at once. Draw a straight line. Pick a spot, call it 0. Pick another spot to its right, call it 1. Now you’ve got a ruler for the whole universe of numbers.

−3 −2 −1 0 1 2 3 1 2 7 3 −4 3 Figure 1.2. The integers are evenly spaced fence posts; the fractions live between them. Here 1 2 sits exactly halfway between 0 and 1, 7 3 a little past 2, and −4 3 between −2 and −1.

The integers are evenly spaced fence posts marching off in both directions. The fractions live between the fence posts: 1 2 sits exactly halfway between 0 and 1, and 7 3 sits a bit past 2.

And here’s something that feels almost magical about the rationals: between any two of them, no matter how close, there’s always another one. Pick 1 2 and 1 3. Right between them sits their average, 5 12. And between those two there’s another. And another. Forever. The rationals are packed in infinitely tightly. There are no gaps — or so it seems.

Counting Isn’t Enough 1 1 2 zoom 1 3 1 2 5 12 5 12 1 2 another one . . . forever Figure 1.3. Density of the rationals. Between any two fractions there is always another — and between those, another still. Zoom in as far as you like; the packing never runs out.

Sit with that for a second. Between any two fractions, however close together, you can always squeeze another fraction. It feels like the rationals must fill up the entire line, leaving no room for anything else.

That feeling is wrong.

And the fact that it’s wrong — the discovery that the number line has holes in it even after you’ve packed in every possible fraction — is one of the most shocking things in all of mathematics. It broke the people who first found it. There’s a legend that the man who discovered it was thrown off a boat for telling the secret.

That’s the next chapter.

Things worth thinking about 1. We invented negatives to make subtraction always work, and fractions to make division always work. Addition and multiplication never needed fixing. Why do you think those two operations were already fine, but their “opposites” caused trouble?

2. We said division always works for rationals — almost. The one thing you still can’t do is divide by zero. Try it: what number could 1 0 possibly be? See if you can talk yourself into an answer, and then see why it falls apart. 3. Between any two fractions there’s always another fraction. Does that mean there are “more” fractions than integers? It seems obvious that there must be. Hold onto your answer — it turns out to be one of the great surprises of mathematics, and it’s not what you’d guess.

4. Every time an operation broke, we invented new numbers to fix it. Can you imagine an operation that even the fractions can’t handle — something that would force us to invent yet another kind of number? (You already know one, secretly. Think about what number, times itself, gives 2.)

Problems to work ⋆Teach it — explain it to someone Problem 1 — Teach it (the invention pattern). Find someone at home — a brother or sister, a parent — and tell them the cookie story. Start with the plate that has 3 cookies and someone asking you to take 5 away. Walk them all the way to “so we invented a number, −2, to be the answer.” Then do it again with sharing 12 cookies among 5 people, arriving at fractions.

The thing you are really teaching is the pattern: an operation broke, so we built the number that fixes it. If your listener can repeat that one sentence back to you, you’ve got it — and you’ll never again think of negative numbers or fractions as things that were just “always there.” You’ll know they were built.

Problem 2 — Sort the zoo.

Put each number into the smallest family it belongs to: counting number (N), integer (Z), or rational (Q). Remember the families are nested, so “smallest” means the innermost ring it fits in.

7 −4 3 8 0 −10 2 9 3 (Two of these are sneakier than they look — simplify before you decide.) Problem 3 — Squeeze one in.

For each pair of fractions, find a fraction that lies strictly between them. (The average — add them and halve — always works, but see if you can sometimes spot an easier one.)

(a) 1 4 and 1 2 (b) 2 5 and 3 5 (c) 3 7 and 4 7 Problem 4 — The divide-by-zero trap.

Suppose 1 0 were some number — call it n. Then, just like 1 2 = n would mean 1 = 2 × n, having 1 0 = n would mean 1 = 0 × n. What does 0 × n equal, for any number n? Use that to explain, in one sentence, why no value of n can possibly Counting Isn’t Enough work — and why dividing by zero is banned, not by decree, but because it’s genuinely impossible.

Problem 5 — [challenge] Always room for one more.

You can keep squeezing fractions between fractions forever, and Problem 3 hinted at why. Prove it in general: take any two different fractions a b and c d, and show that their average 1 2 a b + c d
is (i) again a fraction, and (ii) strictly between the two you started with. Once you’ve shown the average is always a brand-new fraction in the gap, explain why that means the squeezing can never stop. (This is your first real proof of something true for infinitely many cases at once — a taste of what’s coming.)

Problem 6 — [challenge] Is there a smallest positive fraction? A friend claims that 1 1,000,000,000 is “basically the smallest fraction there is.”

Show they’re wrong by naming a positive fraction that is even smaller — and then explain why their search is hopeless: why there is no smallest positive fraction at all. (Hint: whatever tiny fraction someone hands you, what could you do to its bottom number?)

Problem 7 — [challenge] The next wall.

Every family we built repaired a broken operation. Here is a brand-new operation: “take the square root.” Experiment with a calculator: √ 9 = 3 (a tidy rational), q 1 4 = 1 2 (also tidy). Now try √ 2. Write down the first eight digits it shows you, then square that number by hand and see whether you land exactly on 2. You won’t — and chasing why you can’t is precisely where Chapter 2 begins.

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