Chapter 10: The Other Side of Zero

For most of your life, "number" has meant counting things: $0, 1, 2, 3, \dots$. You cannot have $-3$ apples, after all. But mathematics needs numbers that go below zero , to talk about temperatures below freezing, depths below sea level, debts, falls in price, or stairs going down from the ground floor.

These numbers , $-1, -2, -3, -4, \dots$ , are called negative numbers, and together with the whole numbers ($0, 1, 2, 3, \dots$) they form the integers: $\dots, -3, -2, -1, 0, 1, 2, 3, \dots$. Mathematics has been using them for over $2{,}000$ years, but the rules for adding and subtracting them with confidence were first written down by the great Indian mathematician Brahmagupta in the year $628$ CE.

In this chapter we will visit "Bela's Building of Fun" , a building with floors above and below ground. Zero is the ground floor. Positive numbers are the floors above. Negative numbers are the floors below. As we move the lift up and down, addition and subtraction of integers become wonderfully clear.

By the end you will have a feel for the number line that extends in both directions, know how to add and subtract any integers, and understand Brahmagupta's elegant rules for combining positives and negatives.

What's inside

1. Below zero , meeting negative numbers.
2. The number line extends both ways.
3. Adding integers.
4. Subtracting integers.
5. Real-world integers , temperature, money, sea level, and stairs.

Key results / Formula card

Order: For integers on a number line, the number further to the right is bigger. So $5 > 0 > -3$, and $-1 > -7$.

How to read this chapter

Have a number line drawn out in front of you, with marks from $-10$ to $+10$. Use it like a road. Every addition is a step right; every subtraction is a step left. Visual movement turns abstract rules into common sense.