Multiplying negative numbers

Negative numbers can be a bit counterintuitive, especially when it comes to their product.

How to visualize negative numbers

In everyday life, we don't come across negative numbers all that often. I don't think many people will say "hey, Grandma, look - there's minus eight cows on the hill over there!" Multiplying negative numbers is even harder to explain.

But there is at least one case where negative numbers are clear to all: the household budget, its income and expenditure. If you look at your bank account, for example, you'll see items like:

• +15 000 Kč: salary payment
• −1 753 CZK: purchase at the hypermarket
• −980 Kč: buying chewing gum for the whole year
• +2500 CZK: state allowance for good mood
• −8000 Kč: rent

If you get some money, there is a plus sign next to the amount, if you spend some money, there is a minus sign - because you subtract this spent money from your balance.

So we can think of negative numbers as some spending. If we have added a thousand crowns, we write it as +1000, if we have spent five thousand, we write it as −5000. It doesn't have to be just money, if we take six pieces of gum from the basket, we can mark it as −6.

Multiplying a negative number by a positive number

So what will be the result of the product 7 · (−20)? Basically, it's about what sign the result will have. It is probably obvious that the result will be either 140, or −140. How do you justify the correct procedure?

Suppose one pack of gum costs 20 and we buy 7. Now we ask how much money we spent in total on these seven chewing gums. Clearly, we have spent 7 · 20 = 140 Kč.

But since that's money we actually spent, we didn't earn it, so we add a minus sign. If we left the result 140, it would mean that it is our income, but it is our spending: hence the result 7 · (−20) = −140.

Again, we can convert this to addition: suppose we only bought two pieces of chewing gum. They both cost twenty crowns, so in our books it would be

• −20 CZK: chewing gum
• −20 CZK: chewing gum

We can rewrite this at 2 · (−20), or we can add up the spending: −20 + (−20) = −20 − 20 = −40. Then we can write simply:

• −40 Kč: two chewing gum

The moment we multiply a positive number by a negative (or a negative by a positive, it doesn't matter), we can think of it as the negative part representing some spending and the positive number representing the number of those spending. And since we're adding up the spending, the result must also be negative.

Algebraic justification

That multiplying a positive and a negative number gives a negative number can be justified algebraically. Let's try to calculate this example: 5 · (4 − 4). Obviously the result is zero, because 4 − 4 = 0 and 5 · 0 = 0. However, if we know how to factor and multiply parentheses, we can rewrite the previous example like this:

$$ 5 \cdot 4 + 5 \cdot (-4) = 0 $$

We rewrote the expression to a different but equivalent one. The left-hand side of the equation must still equal zero, because the original expression 5 · (4 − 4) was also equal to zero. Now we multiply 5 · 4 = 20:

$$ 20 + 5 \cdot (-4) = 0 $$

Now all we have left is the expression 5 · (−4). But in order for the equation to remain valid, what must this expression equal? What number must we add to 20 to get zero? It's the number −20. So the only possible result of the expression 5 · (−4) is the number −20. So again we see that the product of a positive and a negative number gives a negative number.

The product of two negative numbers

The product of two negative numbers gives us a positive number. At first sight this may seem strange, but it is quite understandable. It works the same way in speech if you use a double negative: in the announcement "we will not be rude" we have two negatives, yet the sentence tells us the same thing as "we will be polite". Theoretically, we could be neither.)

A slight justification is found in the next section with the movement of the gentleman with the briefcase. Now follows an overview table with the relationships of positive and negative numbers with respect to the product:

$$\begin{eqnarray} \mbox{ Positively } \cdot \mbox{ Positively } &=& \mbox{ Positively }\\ \mbox{ Positively } \cdot \mbox{ Forward } &=& \mbox{ Forward }\\ \mbox{ Forward } \cdot \mbox{ Positively } &=& \mbox{ Forward }\\ \mbox{ Forward } \cdot \mbox{ Forward } &=& \mbox{ Positively }\\ \end{eqnarray}$$

Justification by movement

The product with negative factors can be nicely explained on motion. Imagine that we are standing at the origin of the number axis. It goes something like this:

Next, let's have the product of two numbers, say 2 · 3. We'll show this product on the number line as the motion of the gentleman with the briefcase. The first number will represent the length of the step and the second number the number of steps the gentleman takes, so we can see it as the product delka kroku · počet kroků.

But in doing so, we will also take into account negative numbers. If the step length is positive, the master normally goes forward, but if the step length is negative, he goes backwards. If the number of steps is positive, the master is facing right (as in the picture above), if it is negative, it is facing left.

Whatever number the gentleman with the briefcase reaches, that is the result of the product of the given numbers. Let's see how it works out for all combinations of positive and negative numbers.

For the product of 2 · 3, the gentleman is facing right and walking forward. So he takes three steps of two squares forward.

The gentleman with the briefcase reaches the number 6, so the result is 2 · 3 = 6.

Now we try the product −2 · 3. The first number is negative, which means that the gentleman will now go backwards. But the second number is positive, so he will look to the right. So the gentleman looks to the right and takes three steps backwards:

We see that the gentleman with the briefcase has reached the number −6, which is consistent with the fact that −2 · 3 = −6.

Next we try 2 · (−3). The first number is positive, so the gentleman will go forward. But the second number is negative, so the gentleman will be facing left. The gentleman will look to the left and take three steps forward:

The gentleman has again arrived at the number −6, which agrees with 2 · (−3) = −6. You can see that it doesn't matter if the gentleman is facing left and going forward, or if he is facing right and going backwards - in both cases he will arrive at negative numbers.

Last we send the master to compute −2 · (−3), so we turn to the left and let him go backwards three steps:

The gentleman, perhaps a little surprisingly, arrives at the number 6. So we can say that −2 · (−3) = 6, the product of two negative numbers, is equal to a positive number.