Multiplying integers

Inte­gers turn up when­ever some­thing can go below zero: a tem­per­a­ture of $-4$ degrees in Leh, a bank bal­ance that has gone over­drawn, or a lift that goes down three floors. Mul­ti­ply­ing inte­gers answers ques­tions like "the tem­per­a­ture fell $2$ degrees every hour for $5$ hours; how much did it change?" The size of the answer comes from your times tables. The only new thing is the sign, and this les­son shows where the sign rule comes from.

Inte­gers are the whole num­bers. They are the ones above zero, the ones below zero, and zero itself: $\ldots -3, -2, -1, 0, 1, 2, 3 \ldots$ There are no halves or bits among them.

A num­ber with a minus in front, like $-3$, is called a neg­a­tive num­ber. It sits below $0$. A num­ber with no minus, like $3$, is called a pos­i­tive num­ber. It sits above $0$.

You have already added inte­gers, and you know that $-3 + 5 = 2$. Mul­ti­ply­ing comes next. The rule for the sign is short. This page gives you the rule and shows you why it is true.

Mul­ti­ply­ing is adding again and again

$3 \times 5$ means $3$ added five times. That is $3 + 3 + 3 + 3 + 3 = 15$. You can add a neg­a­tive num­ber again and again too. $(-3) \times 5$ means $-3$ added five times.

$(-3) \times 5$

$$\begin{aligned}&(-3) + (-3) + (-3) + (-3) + (-3) \\ &= -15\end{aligned}$$

Think of a num­ber line. It is the num­bers writ­ten out in a row. Zero sits in the mid­dle. The pos­i­tive num­bers go to the right of it. The neg­a­tive num­bers go to the left.

Start at $0$ on that line. Each step takes you down $3$. After five steps you are at $-15$. So a neg­a­tive num­ber times a pos­i­tive one gives a neg­a­tive answer.

$5 \times (-3)$ gives $-15$ too. Pic­ture three rows of five coun­ters. Now pic­ture five rows of three coun­ters. Both hold the same num­ber of coun­ters. So turn­ing a mul­ti­pli­ca­tion round does not change the answer.

Why two neg­a­tive num­bers give a pos­i­tive answer

Adding again and again can­not help you here. You can­not add a num­ber $-5$ times. So look at a pat­tern instead.

Keep the first num­ber as $-3$. Let the sec­ond num­ber drop by $1$ each line.

The sec­ond num­ber drops by $1$ each line

$$\begin{aligned}(-3) \times 3 &= -9 \\ (-3) \times 2 &= -6 \\ (-3) \times 1 &= -3 \\ (-3) \times 0 &= 0 \\ (-3) \times (-1) &= 3 \\ (-3) \times (-2) &= 6\end{aligned}$$

Read the answers down the list. They go up by $3$ each line. Here is why. Each new line takes away one lot of $-3$. Tak­ing away a lot of $-3$ makes the answer go up by $3$. That is true on every line. The num­bers do not change their habits when you pass $0$. So the climb of $3$ car­ries on below zero.

So the line after $0$ has to be $3$. That is how you get $(-3) \times (-1) = 3$. Any other answer breaks the climb of $3$. This is why two neg­a­tive num­bers give a pos­i­tive answer. You do not have to take the rule on trust. The pat­tern makes it the only choice.

Zero is the odd one out. It is nei­ther pos­i­tive nor neg­a­tive. Any num­ber mul­ti­plied by zero gives zero. The rule below is for the num­bers on either side of zero.

The sign of a num­ber is the minus in front of it, or no minus at all. So $-3$ has a minus sign, and $3$ has none.

Remem­ber. When you mul­ti­ply two inte­gers, look at the two signs. When the two signs are the same, the answer is pos­i­tive. When the two signs are dif­fer­ent, the answer is neg­a­tive.

The four cases

Look down the exam­ple col­umn. Ignore the minus signs for a moment. Every answer is $15$. Only the sign changes from row to row.

How to work one out

The numer­i­cal value of a num­ber is how big it is. You ignore the sign in front of it. So the numer­i­cal value of $-7$ is $7$. The numer­i­cal value of $7$ is $7$ as well.

The table shows the same $15$ in every row. Only the sign changes. So you can do the two jobs one at a time. First mul­ti­ply the numer­i­cal val­ues. Then look at the two signs. Take the sign of the answer from the rule.

$(-7) \times 4$

$$\begin{aligned}7 \times 4 &= 28 \\ (-7) \times 4 &= -28\end{aligned}$$

The two signs are dif­fer­ent here. So the answer is neg­a­tive.

$(-6) \times (-8)$

$$\begin{aligned}6 \times 8 &= 48 \\ (-6) \times (-8) &= 48\end{aligned}$$

The two signs are the same here. So the answer is pos­i­tive.

More worked exam­ples

$(-12) \times 5$

$$\begin{aligned}12 \times 5 &= 60 \\ (-12) \times 5 &= -60\end{aligned}$$

The signs are dif­fer­ent, so the answer is neg­a­tive.

$(-8) \times (-9)$

$$\begin{aligned}8 \times 9 &= 72 \\ (-8) \times (-9) &= 72\end{aligned}$$

The signs are the same, so the answer is pos­i­tive.

Three num­bers: $(-2) \times (-3) \times (-4)$

Mul­ti­ply two at a time. Take the first pair, then bring in the third num­ber.

$$\begin{aligned}(-2) \times (-3) \times (-4) &= 6 \times (-4) \\ &= -24\end{aligned}$$

The first pair has the same signs, so it gives $+6$. Then $6$ and $-4$ have dif­fer­ent signs, so the final answer is neg­a­tive.

A prob­lem in words

The tem­per­a­ture in a hill town falls by $2$ degrees every hour for $5$ hours. A fall of $2$ is writ­ten $-2$. The total change is

$$(-2) \times 5 = -10$$

so the tem­per­a­ture ends $10$ degrees lower than it started.

Prac­tice

Work these out. Find the sign first if that helps you.

Answers

Com­mon mis­takes

• Using the adding rule by mis­take. $(-4) + (-7) = -11$, but $(-4) \times (-7) = 28$.
• Think­ing two neg­a­tives always give a pos­i­tive, even when adding. The "same signs give pos­i­tive" rule is for mul­ti­ply­ing only.
• Drop­ping the brack­ets. Write $(-4) \times (-7)$, not $-4 \times -7$, so the signs are easy to read.
• Get­ting the size wrong while wor­ry­ing about the sign. Do the two jobs one at a time: size first, then sign.
• Giv­ing zero a sign. $(-3) \times 0 = 0$, which is nei­ther pos­i­tive nor neg­a­tive.

Key terms

Inte­ger

A whole num­ber that may be pos­i­tive, neg­a­tive or zero.

Pos­i­tive num­ber

A num­ber above zero, such as $3$.

Neg­a­tive num­ber

A num­ber below zero, writ­ten with a minus sign, such as $-3$.

Num­ber line

The num­bers set out in a row, with zero in the mid­dle, pos­i­tives to the right and neg­a­tives to the left.

Sign

The minus in front of a num­ber, or no minus at all.

Numer­i­cal value

How big a num­ber is when you ignore its sign; for $-7$ it is $7$.

Answers

1. $4 \times 7 = 28$ (same signs, pos­i­tive)
2. $(-4) \times 7 = -28$ (dif­fer­ent signs, neg­a­tive)
3. $4 \times (-7) = -28$ (dif­fer­ent signs, neg­a­tive)
4. $(-4) \times (-7) = 28$ (same signs, pos­i­tive)
5. $(-9) \times 3 = -27$ (dif­fer­ent signs, neg­a­tive)
6. $(-6) \times (-5) = 30$ (same signs, pos­i­tive)
7. $8 \times (-5) = -40$ (dif­fer­ent signs, neg­a­tive)
8. $(-11) \times (-2) = 22$ (same signs, pos­i­tive)
9. $(-1) \times 15 = -15$ (dif­fer­ent signs, neg­a­tive)