Dividing integers

Divi­sion shares a quan­tity into equal parts. With inte­gers the quan­tity can be below zero: a loss of ₹15 shared equally over 5 days, or a tem­per­a­ture that falls $15$ degrees in $5$ equal steps. Let us see how to divide such num­bers, and how to be sure of the sign of the answer.

You already count with $1$, $2$, $3$ and zero. Now think about the num­bers below zero. You write a minus sign in front of them, like $-1$ and $-2$. Those are called neg­a­tive num­bers. Num­bers above zero are called pos­i­tive num­bers. Put them all together and you have the inte­gers.

The inte­gers carry on both ways: $\dots, -3, -2, -1, 0, 1, 2, 3, 4, \dots$

Now we divide them. Since you already know how to mul­ti­ply them, there is hon­estly very lit­tle new to learn here.

Divid­ing undoes mul­ti­ply­ing#

Every divi­sion that works out exactly comes from a mul­ti­pli­ca­tion. Take $(-3) \times 5 = -15$.

Now turn it round. You started at $-3$ and mul­ti­plied by $5$. To get back to $-3$, you divide $-15$ by $5$. So $(-15) \div 5 = -3$.

You can go back the other way too. Divide $-15$ by $-3$ and you land on $5$. So $(-15) \div (-3) = 5$.

One mul­ti­pli­ca­tion gives two divi­sions

$$\begin{aligned}(-3) \times 5 &= -15 \\ (-15) \div 5 &= -3 \\ (-15) \div (-3) &= 5\end{aligned}$$

This is why divid­ing needs no new sign rules. The signs work the same way as they do in mul­ti­ply­ing.

A num­ber line makes this easy to see. Divid­ing $-15$ by $5$ asks: if you make five equal jumps from $0$ and land on $-15$, how big is each jump? Each jump is $3$ steps to the left, which is $-3$.

The rule for signs#

The lit­tle mark in front of a num­ber is called its sign. A minus sign looks like this: $-$. A num­ber with no mark in front of it has a plus sign. You just do not write that one down.

Mul­ti­ply­ing can go two ways. Two signs the same give a pos­i­tive answer. Two signs dif­fer­ent give a neg­a­tive answer.

So $3 \times 5 = 15$, and $(-3) \times (-5) = 15$ as well. But $(-3) \times 5 = -15$, and $3 \times (-5) = -15$.

Divid­ing works in exactly the same way. This is true when one num­ber goes into the other exactly.

Remem­ber. Two signs the same give a pos­i­tive answer. Two signs dif­fer­ent give a neg­a­tive answer. This works for divid­ing too, just as it does for mul­ti­ply­ing.

Why two signs the same give a pos­i­tive answer#

Take $(-15) \div (-3)$. Ask what num­ber times $-3$ gives $-15$. Try $5$. You get $5 \times (-3) = -15$. That is right. So the answer is $5$, and it is pos­i­tive. Try it with any other pair of num­bers and the same thing hap­pens. The ques­tion you ask always has the same shape.

Why two signs dif­fer­ent give a neg­a­tive answer#

Take $(-15) \div 3$. Ask what num­ber times $3$ gives $-15$. Try $-5$. You get $(-5) \times 3 = -15$. That is right. So the answer is $-5$, and it is neg­a­tive. Try it with any other pair of num­bers and the same thing hap­pens again. The ques­tion you ask always has the same shape.

Each check is a mul­ti­pli­ca­tion you already know. That is why the two rules match.

All four together#

The first two have signs the same, so the answers are pos­i­tive. The last two have dif­fer­ent signs, so the answers are neg­a­tive.

How to do one#

1. Look at the two signs.
2. Work out the sign of the answer.
3. Cover up the minus signs. Divide the two num­bers as you would in any other divi­sion.
4. Write the sign in front of the answer.

The sign and the num­bers are two sep­a­rate jobs. You can do them one at a time.

Worked exam­ples#

Divide $56$ by $8$. Both signs are the same, so the answer is pos­i­tive.

Both num­bers pos­i­tive

$$\begin{aligned}56 \div 8 &= 7 \\ 7 \times 8 &= 56\end{aligned}$$

Divide $-56$ by $-8$. Both signs are the same again, so the answer is pos­i­tive.

Both num­bers neg­a­tive

$$\begin{aligned}(-56) \div (-8) &= 7 \\ 7 \times (-8) &= -56\end{aligned}$$

Divide $-56$ by $8$. The signs are dif­fer­ent, so the answer is neg­a­tive.

A neg­a­tive divided by a pos­i­tive

$$\begin{aligned}(-56) \div 8 &= -7 \\ (-7) \times 8 &= -56\end{aligned}$$

Divide $56$ by $-8$. The signs are dif­fer­ent again, so the answer is neg­a­tive.

A pos­i­tive divided by a neg­a­tive

$$\begin{aligned}56 \div (-8) &= -7 \\ (-7) \times (-8) &= 56\end{aligned}$$

The last step in each one is the check. It is the mul­ti­pli­ca­tion you started from. Mul­ti­ply your answer back and see where you land. If you land on the num­ber you started with, your answer is right. Divid­ing is just mul­ti­ply­ing undone.

Harder exam­ples#

A big­ger divi­sion

Divide $-144$ by $12$. The signs are dif­fer­ent, so the answer is neg­a­tive. Now cover the signs: $144 \div 12 = 12$.

$$\begin{aligned}(-144) \div 12 &= -12 \\ (-12) \times 12 &= -144\end{aligned}$$

Two divi­sions in a row

Work out $(-120) \div (-4) \div (-5)$. Work from left to right, one divi­sion at a time.

$$\begin{aligned}(-120) \div (-4) &= 30 \\ 30 \div (-5) &= -6\end{aligned}$$

The first pair had signs the same, so it gave a pos­i­tive $30$. The sec­ond pair had signs dif­fer­ent, so the final answer is neg­a­tive.

A prob­lem in words

The tem­per­a­ture in a hill town fell by $18$ degrees in $6$ hours, by the same amount each hour. A fall is a change of $-18$. So the change each hour is

$$(-18) \div 6 = -3$$

The tem­per­a­ture dropped $3$ degrees every hour.

Divid­ing by zero#

There is one divi­sion you can never do. You can­not divide by $0$.

Here is why. Divid­ing asks a ques­tion. $\displaystyle \frac{12}{4}$ asks what num­ber times $4$ gives $12$. The answer is $3$. Now try $\displaystyle \frac{12}{0}$. It asks what num­ber times $0$ gives $12$. Noth­ing does. Every num­ber times $0$ gives $0$.

So there is no answer to find. We say it is not defined.

Do not mix this up with $0 \div 5$. That one is fine. It asks what num­ber times $5$ gives $0$, and the answer is $0$. Shar­ing noth­ing between five peo­ple gives each of them noth­ing.

Remem­ber. Zero shared out is fine: $0 \div 5 = 0$. Zero as the num­ber you share BETWEEN is not allowed. You can never divide by $0$.

Your turn#

Work these out. Sort out the sign first. Then divide the num­bers.

Remem­ber. If you can mul­ti­ply inte­gers, you can divide them. The sign rule is the same one.

Com­mon mis­takes#

• Think­ing two neg­a­tives give a neg­a­tive. In divi­sion, as in mul­ti­pli­ca­tion, $(-24) \div (-6) = 4$, a pos­i­tive num­ber.
• Divid­ing the num­bers cor­rectly but for­get­ting to write the minus sign in front of the answer.
• Swap­ping the order. $(-24) \div 6$ is not the same as $6 \div (-24)$.
• Writ­ing an answer for a divi­sion by $0$. It is not defined.
• Say­ing $0 \div 9$ has no answer. Zero divided by any non-zero num­ber is $0$.

Key terms#

Inte­ger

A whole num­ber that is pos­i­tive, neg­a­tive or zero.

Sign

The mark in front of a num­ber that shows whether it is above or below zero.

Div­i­dend

The num­ber being divided, such as $-15$ in $(-15) \div 5$.

Divi­sor

The num­ber you divide by, such as $5$ in $(-15) \div 5$.

Quo­tient

The answer to a divi­sion.

Not defined

Hav­ing no answer at all, as with any divi­sion by $0$.

Answers#

Each answer was checked by mul­ti­ply­ing back.

1. $24 \div 6 = 4$
2. $(-24) \div 6 = -4$
3. $24 \div (-6) = -4$
4. $(-24) \div (-6) = 4$
5. $(-45) \div 9 = -5$
6. $45 \div (-9) = -5$
7. $(-72) \div (-8) = 9$
8. $72 \div 8 = 9$
9. $(-100) \div 10 = -10$
10. $(-48) \div (-6) = 8$
11. $81 \div (-9) = -9$
12. $(-56) \div (-7) = 8$
13. $0 \div 9 = 0$

The same answers as a table: