What integers are

In win­ter the news may say that the tem­per­a­ture in Leh fell to $-12$ degrees. A lift in a mall may show floor $-1$ for the car park. A bank pass­book may show money going out. All of these use num­bers that sit below zero. Together with the count­ing num­bers and zero, they make a fam­ily called the inte­gers.

Some num­bers are smaller than noth­ing at all. A very cold day is one. A floor below the ground is another. Num­bers like these carry a minus sign. This page shows you what they are called. It also shows you how to put them in order.

Why num­bers go below zero

Count­ing goes up from $1$. Zero is the point you start from. But some things go past zero. Water freezes at $0$ degrees. A day can be colder than that. Count­ing num­bers can­not say how cold. So maths keeps num­bers on the other side of zero. Each one is writ­ten with a minus sign in front.

The set of inte­gers

All those num­bers make one fam­ily. In maths a fam­ily of num­bers like this is called a set. The set of them is called the inte­gers. It is writ­ten with the let­ter $Z$, like this.

Remem­ber. The set of inte­gers is $Z = \{ \dots -3, -2, -1, 0, 1, 2, 3, 4 \dots \}$.

Read that line like this. $Z$ is the name of the set. The curly brack­ets hold the num­bers in it. Zero sits in the mid­dle of them. The num­bers below zero are on its left. The num­bers above zero are on its right.

There are dots at both ends of the list. They are not a mis­take. They mean the list car­ries on. Go right and $5$, $6$, $7$ come next. Go left and $-4$, $-5$, $-6$ come next. Nei­ther end ever stops. So there is no biggest inte­ger. There is no small­est one either.

Pos­i­tive, neg­a­tive and zero

A num­ber below zero is called a neg­a­tive num­ber. It is writ­ten with a minus sign in front. A num­ber above zero is called a pos­i­tive num­ber. So the minus sign tells you which side a num­ber is on.

Zero is not above zero. It is not below zero either. So it is nei­ther pos­i­tive nor neg­a­tive. It is the point the two sides start from. Think of it as the ground floor.

The minus sign in front is part of the num­ber. $-3$ is one num­ber on its own. It is not a sum wait­ing to be done. Here the minus sign does not mean take away. It means the num­ber is below zero. You can read $-3$ as minus three. You can also read it as three below zero.

The num­ber line

A num­ber line is a straight line with the inte­gers marked on it. Zero is marked in the mid­dle. The pos­i­tive inte­gers go to the right of zero. The neg­a­tive inte­gers go to the left. The marks are the same dis­tance apart. Every gap is the same size, so every step is worth $1$. That is true wher­ever you are on the line.

So the marks read $-3$, $-2$, $-1$, $0$, $1$, $2$ from left to right. Move to the right and the num­ber goes up. Move to the left and it goes down.

The pic­ture below shows a num­ber line from $-6$ to $6$. The arrow­heads at both ends remind you that the line goes on for ever in both direc­tions, just like the dots in the set $Z$.

Put your fin­ger on a mark and try it. Here are two start­ing points.

Remem­ber. The neg­a­tive inte­gers are on the left. Zero is in the mid­dle. The pos­i­tive inte­gers are on the right.

Which inte­ger is smaller

The num­ber line puts every inte­ger in order. The order runs from left to right. The num­ber fur­ther left is the smaller one. The num­ber fur­ther right is the big­ger one. That rule works for every pair.

It helps to count the steps from zero. To do that, for­get the minus sign. Then count the marks back to zero. $-5$ is five steps. $-2$ is two steps.

This next one looks wrong at first. $-5$ is smaller than $-2$. After all, $5$ is big­ger than $2$. So look at the line again. $-5$ is five steps left of zero. $-2$ is only two steps left. $-5$ is fur­ther left. That makes $-5$ the smaller num­ber.

The big­ger the num­ber after the minus sign, the fur­ther left it sits. Fur­ther left means smaller. So a big-look­ing neg­a­tive num­ber is a small num­ber.

Remem­ber. Fur­ther left on the line means smaller. So $-5$ is smaller than $-2$.

A ther­mome­ter works the same way

A ther­mome­ter is a num­ber line stood on end. Zero is the freez­ing point. Warm days are above zero. Cold days are below it. A day at $-9$ degrees is colder than a day at $-4$ degrees. Colder means lower down the ther­mome­ter. Lower means smaller. So $-9$ is smaller than $-4$.

Floors below the ground

Some build­ings have floors below the ground. The ground floor is floor $0$. One floor down is floor $-1$. Three floors down is floor $-3$. Floor $-3$ is lower than floor $-1$. So $-3$ is smaller than $-1$.

Com­par­ing any two inte­gers

Each rule below is just the line read from left to right.

1. A pos­i­tive inte­ger is always big­ger than a neg­a­tive one. It sits on the other side of zero.
2. Zero is big­ger than every neg­a­tive inte­ger. Every neg­a­tive inte­ger sits to its left.
3. Zero is smaller than every pos­i­tive inte­ger. Every pos­i­tive inte­ger sits to its right.
4. Take two neg­a­tive inte­gers. The one fur­ther from zero is the smaller one.
5. Take two pos­i­tive inte­gers. The one fur­ther from zero is the big­ger one.

Worked exam­ples

Put $3$, $-4$, $0$, $-1$ in order, small­est first.

First pick out the neg­a­tive num­bers: $-4$ and $-1$. They come before zero. $-4$ is fur­ther from zero, so it comes first. Then comes zero, and then the pos­i­tive num­ber $3$.

$$-4 \lt -1 \lt 0 \lt 3$$

Start at $2$ and move five steps to the left. Where do you land?

Count the steps one at a time: $1$, $0$, $-1$, $-2$, $-3$. You land on $-3$. Notice that you crossed zero on the way. Two of the steps took you down to zero, and the other three took you below it.

A diver is $6$ metres below the sea. A bird is $4$ metres above it. Write both as inte­gers.

Take sea level as zero. Below the sea is neg­a­tive, so the diver is at $-6$. Above the sea is pos­i­tive, so the bird is at $4$. The diver is lower, so $-6 \lt 4$.

Your turn

Which of the two is smaller

Copy each pair into your book. Write down the smaller num­ber of the two. Pic­ture the line if a pair is hard.

Answer these in words

1. Put these in order, small­est first: $4$, $-2$, $0$, $-7$, $1$.
2. Put these in order, small­est first: $-1$, $-10$, $-5$.
3. Which inte­ger is one step to the left of $-6$?
4. Which inte­ger is one step to the right of $-1$?
5. Which is colder, $-15$ degrees or $-6$ degrees?
6. A lift is on floor $-2$. It goes down two floors. Which floor is it on now?
7. Write down the three inte­gers that come just before $0$ on the line.
8. Which is the small­est num­ber here: $-3$, $2$, $-9$, $0$?
9. Which is big­ger, $-100$ or $-1$?

Com­mon mis­takes

• Think­ing $-9$ is big­ger than $-4$ because $9$ is big­ger than $4$. With neg­a­tive num­bers, the big­ger the num­ber after the sign, the smaller the inte­ger.
• Call­ing zero pos­i­tive or neg­a­tive. Zero is nei­ther.
• Read­ing the minus sign in $-3$ as "take away". Here it tells you the num­ber is below zero.
• Count­ing the start­ing mark as a step when mov­ing along the line.
• Leav­ing out zero when putting inte­gers in order.
• Draw­ing the marks on a num­ber line at uneven gaps. Every step must be the same size.

Key terms

Inte­ger

Any of the num­bers $\dots, -3, -2, -1, 0, 1, 2, 3, \dots$

Set

A fam­ily of num­bers taken together; the set of inte­gers is called $Z$.

Pos­i­tive inte­ger

An inte­ger above zero, to the right of zero on the line.

Neg­a­tive inte­ger

An inte­ger below zero, writ­ten with a minus sign and placed to the left of zero.

Zero

The mid­dle of the num­ber line; nei­ther pos­i­tive nor neg­a­tive.

Num­ber line

A straight line with the inte­gers marked at equal steps.

Smaller and big­ger

On the num­ber line, fur­ther left is smaller and fur­ther right is big­ger.

Answers

Answers to the pairs

Answers to the ques­tions in words

1. $-7, -2, 0, 1, 4$
2. $-10, -5, -1$
3. $-7$
4. $0$
5. $-15$ degrees is colder.
6. Floor $-4$.
7. $-3, -2, -1$
8. $-9$
9. $-1$ is big­ger.