Numbers & Functions · 1 Counting Isn't Enough

1.3 When division breaks

Now let’s do the same thing again, one level up.

Counting Isn’t Enough Multiplication of integers? No problem. 4 × 3 = 12. Multiply integers all you like, you stay among the integers.

Division?

12 ÷ 3 = 4.

Fine.

12 ÷ 5 = ?

Hmm. If all you have is integers, this question has no answer. You can’t split 12 into 5 equal whole pieces. With actual cookies, you’d be stuck — unless you’re willing to break the cookies.

And of course you are. Of course you’d break the cookies. Each person gets two cookies and then a little bit more — two and two-fifths, in fact. So once again, we have an operation — division — that refuses to always work if we only allow integers. And once again, instead of accepting a broken operation, we invent the numbers that fix it.

These are the fractions: numbers like 12 5 , 1 2, 7 3, −4 9. A fraction is just one integer divided by another (with the bottom one not zero — we’ll get to why later). Mathematicians call these the rational numbers, and the word rational has nothing to do with being sensible. It comes from ratio — a fraction is a ratio of two integers, one number “on top of” another.

The rational numbers are a huge upgrade. Now both subtraction and division always work. Almost any arithmetic you can think of stays safely inside the rationals.

Do you see the pattern now? Let me say it plainly, because it’s the most important idea in this chapter.

Every time we extended our numbers, we did it for the same reason: an operation broke, and we built the numbers that would let it work again.

• We added negatives so that subtraction always works.

• We added fractions so that division always works.

Each new family of numbers contains all the old ones and then some. The counting numbers live inside the integers, which live inside the rationals. We never threw anything away — we only ever added what was missing. Q rationals Z integers N counting 1 2 3 −2 −7 7 3 −4 9 1 2 Figure 1.1. The nested number systems: N ⊂Z ⊂Q. Each larger system contains all the smaller ones — we never threw anything away, we only ever added what was missing.

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