Multiplying negative numbers
Negative numbers can feel a little unintuitive, especially when it comes to multiplying them.
Picturing negative numbers
In everyday life we do not really come across negative numbers. Hardly anyone says “Look, Gran – there are minus eight cows up on that hill!” That makes multiplying negative numbers even harder to explain.
There is at least one case, though, where negative numbers are clear to everyone: a household budget, with its income and spending. If you look at your bank account, for example, you will see entries like these:
- +£500: wages
- −£60: supermarket shop
- −£30: a year’s supply of chewing gum
- +£80: government “good mood” bonus
- −£300: rent
If you receive some money, the amount has a plus sign. If you spend some money, it has a minus sign, because you are taking that spent money away from your balance.
So we can think of negative numbers as a kind of spending. If we gain a thousand pounds, we write it as +1000, and if we spent five thousand, we write it as −5000. It does not have to be only money: if we take six packs of chewing gum out of the basket, we can write that as −6.
Multiplying a negative number by a positive one
So what will the product 7 · (−2) be? Really, the question is which sign the result will have. It is fairly clear that the result will be either 14 or −14. How can we justify the right choice?
Imagine that one pack of chewing gum costs £2 and we buy 7 of them. Now we ask how much money we spent in total on these seven packs. That is easy: we spent 7 · 2 = 14 pounds.
But because this is money we really spent and did not gain, we put a minus sign on it. If we left the result as 14, it would mean this is our income, but it is our spending: that is why 7 · (−2) = −14.
We can turn this into addition again: suppose we bought just two packs. Each pack cost £2, so in our accounts we would have
- −£2: chewing gum
- −£2: chewing gum
We can rewrite this as 2 · (−2), or we can add up the spending: −2 + (−2) = −2 − 2 = −4. Then we can simply write:
- −£4: two packs of chewing gum
When we multiply a positive number by a negative one (or a negative by a positive, it makes no difference), we can think of the negative part as an item of spending and the positive number as how many of those items there are. And since we are adding up spending, the result must be negative too.
An algebraic explanation
The fact that multiplying a positive and a negative number gives a negative number can also be justified using algebra. Let’s try working out 5 · (4 − 4). The result is clearly zero, because 4 − 4 = 0 and 5 · 0 = 0. However, if we know about taking out common factors and expanding brackets, we can also rewrite the previous calculation like this:
We have rewritten the expression as a different but equivalent one. The left-hand side must still equal zero, because the original expression 5 · (4 − 4) was also equal to zero. Now we work out 5 · 4 = 20:
All that is left is the expression 5 · (−4). But for the equation to stay true, what must this expression equal? What number do we have to add to 20 to get zero? It is −20. So the only possible value of the expression 5 · (−4) is −20. Once again we see that the product of a positive and a negative number is negative.
The product of two negative numbers
The product of two negative numbers is a positive number. At first sight this may seem strange, but it is quite understandable. It works like this in speech too, if you use a double negative: in the announcement “we will not be rude” there are two negatives, yet the sentence tells us the same as “we will be polite”. (Here we assume that we can only be polite or rude. In theory we might be neither.)
You will find a simple explanation in the next section, about the movement of a man with a briefcase. For now, here is a handy table of how positive and negative numbers behave in a product:
Explaining it with movement
A product with negative factors can be explained nicely using movement. Imagine we are standing at the origin of a number line. Something like this:
Now take a product of two numbers, say 2 · 3. We will show this product on the number line as the movement of the man with the briefcase. The first number will be the length of a step and the second the number of steps the man takes, so we can see it as the product step length · number of steps.
But we will take negative numbers into account too. If the step length is positive, the man walks forwards as normal, but if the step length is negative, he walks backwards. If the number of steps is positive, the man faces right (as in the picture above), and if it is negative, he faces left.
The number the man with the briefcase arrives at is the result of the product of the given numbers. Let’s see how it turns out for every combination of positive and negative numbers.
For the product 2 · 3 the man faces right and walks forwards. So he takes three steps forwards, each two squares long.
The man with the briefcase arrives at the number 6, so the result is 2 · 3 = 6.
Now we try the product −2 · 3. The first number is negative, which means the man will now walk backwards. But the second number is positive, so he will be looking right. So the man looks right and takes three steps backwards:
We can see that the man with the briefcase arrived at the number −6, which agrees with −2 · 3 = −6.
Next we try 2 · (−3). The first number is positive, so the man will walk forwards. But the second number is negative, so the man will face left. He looks left and takes three steps forwards:
The man again arrived at the number −6, which agrees with 2 · (−3) = −6. You can see that it makes no difference whether the man faces left and walks forwards, or faces right and walks backwards: either way he ends up among the negative numbers.
Finally, we send the man off to work out −2 · (−3), so we turn him to face left and let him walk backwards three steps:
The man, perhaps a little surprisingly, arrived at the number 6. So we can say that −2 · (−3) = 6: the product of two negative numbers is a positive number.
Chapters: Multiplication, Long multiplication, Multiplying negative numbers
