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 18 degrees in 6 equal steps. In this les­son you learn the rule for the sign when you divide, and see that it is the same rule you used for mul­ti­ply­ing.

The rule for the sign#

Remem­ber. When you divide one inte­ger by another:

Same signs give a pos­i­tive answer. Dif­fer­ent signs give a neg­a­tive answer. This is exactly the rule for mul­ti­ply­ing.

Why the rule is the same#

Every exact divi­sion comes from a mul­ti­pli­ca­tion. Take $(-3) \times 5 = -15$. Turn­ing it round gives two divi­sions:

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

On the num­ber line, $(-15) \div 5$ asks: if five equal jumps from 0 land on $-15$, how big is each jump? Each jump is 3 steps to the left, which is $-3$.

To check any divi­sion, mul­ti­ply back. For $(-15) \div (-3) = 5$, check that $5 \times (-3) = -15$. It does, so the answer is right.

How to do one#

1. Look at the two signs and decide the sign of the answer.
2. Cover up the minus signs and divide the num­bers as usual.
3. Write the sign in front of the answer.

A divi­sion can be writ­ten with a line, as a frac­tion: $\displaystyle \frac{2478}{2}$ means $2478 \div 2$.

Exam­ple 1#

Work out $\displaystyle \frac{2478}{2}$.

Both num­bers are pos­i­tive, so the answer is pos­i­tive. By long divi­sion, $2478 \div 2 = 1239$.

$$\displaystyle \frac{2478}{2} = 1239$$

Check: $1239 \times 2 = 2478$.

Exam­ple 2#

Work out $\displaystyle \frac{-2133}{-3}$.

The signs are the same, so the answer is pos­i­tive. Divide the num­bers: $2133 \div 3 = 711$.

$$\displaystyle \frac{-2133}{-3} = +711$$

Check: $711 \times (-3) = -2133$.

Exam­ple 3#

Work out $\displaystyle \frac{101010}{-2}$.

The signs are dif­fer­ent, so the answer is neg­a­tive. $101010 \div 2 = 50505$.

$$\displaystyle \frac{101010}{-2} = -50505$$

Exam­ple 4#

Work out $\displaystyle \frac{-5328}{4}$.

The signs are dif­fer­ent, so the answer is neg­a­tive. $5328 \div 4 = 1332$.

$$\displaystyle \frac{-5328}{4} = -1332$$

Check: $(-1332) \times 4 = -5328$.

Exam­ple 5#

Work out $\displaystyle \frac{37105}{-5}$.

The signs are dif­fer­ent, so the answer is neg­a­tive. $37105 \div 5 = 7421$.

$$\displaystyle \frac{37105}{-5} = -7421$$

Exam­ple 6#

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 the same signs, so it gave $+30$. The sec­ond pair had dif­fer­ent signs, so the final answer is $-6$.

Exam­ple 7#

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

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

The tem­per­a­ture dropped 3 degrees every hour.

Divid­ing by zero#

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

Divid­ing asks a ques­tion. $12 \div 4$ asks what num­ber times 4 gives 12, and the answer is 3. But $12 \div 0$ asks what num­ber times 0 gives 12. Noth­ing does, because every num­ber times 0 gives 0. So there is no answer. 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.

Remem­ber. $0 \div 5 = 0$, but divid­ing by 0 is not allowed.

Com­mon mis­takes#

• Think­ing two neg­a­tives give a neg­a­tive. $(-24) \div (-6) = 4$, a pos­i­tive num­ber.
• For­get­ting to write the minus sign in front of the answer.
• Swap­ping the order. $(-24) \div 6$ is not the same as $6 \div (-24)$.
• Giv­ing an answer for a divi­sion by 0. It is not defined.

Key terms#

• 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.

Prac­tice#

1. $24 \div (-6)$
2. $(-24) \div (-6)$
3. $(-45) \div 9$
4. $(-72) \div (-8)$
5. $81 \div (-9)$
6. $\displaystyle \frac{2478}{2}$
7. $\displaystyle \frac{-30531}{-3}$
8. $\displaystyle \frac{-46028}{4}$
9. $\displaystyle \frac{36105}{-5}$
10. $0 \div (-9)$
11. A shop made a loss of ₹420 over 7 days, the same each day. Write the loss for one day as an inte­ger.
12. What num­ber mul­ti­plied by $-8$ gives $56$?

Answers#