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 you are going to see exactly 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, and it is not magic. Here is the rule, and here is 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#