第1章基礎

We will continue to use counters to model the subtraction. Remember, the blue counters represent positive numbers and the red counters represent negative numbers.

Perhaps when you were younger, you read $\text{“}5-3\text{”}$ as $\text{“}5$ take away $3.\text{”}$ When you use counters, you can think of subtraction the same way!

We will model the four subtraction facts using the numbers $5$ and $3.$

To subtract $5-3,$ we restate the problem as $\text{“}5$ take away $3.\text{”}$

Now we will subtract $\mathrm{-5}-\left(\mathrm{-3}\right).$ Watch for similarities to the last example $5-3=2.$

To subtract $\mathrm{-5}-\left(\mathrm{-3}\right),$ we restate this as $\text{“}\mathrm{–5}$ take away $\mathrm{–3}\text{”}$

Notice that these two examples are much alike: The first example, we subtract 3 positives from 5 positives and end up with 2 positives.

In the second example, we subtract 3 negatives from 5 negatives and end up with 2 negatives.

Each example used counters of only one color, and the “take away” model of subtraction was easy to apply.

What happens when we have to subtract one positive and one negative number? We’ll need to use both blue and red counters as well as some neutral pairs. Adding a neutral pair does not change the value. It is like changing quarters to nickels—the value is the same, but it looks different.

We start with 5 negatives. We need to take away 3 positives, but we do not have any positives to take away.

Remember, a neutral pair has value zero. If we add 0 to 5 its value is still 5. We add neutral pairs to the 5 negatives until we get 3 positives to take away.

And now, the fourth case, $5-\left(\mathrm{-3}\right).$ We start with 5 positives. We need to take away 3 negatives, but there are no negatives to take away. So we add neutral pairs until we have 3 negatives to take away.

Have you noticed that subtraction of signed numbers can be done by adding the opposite? In Example 1.42, $\mathrm{-3}-1$ is the same as $\mathrm{-3}+\left(\mathrm{-1}\right)$ and $3-\left(\mathrm{-1}\right)$ is the same as $3+1.$ You will often see this idea, the subtraction property, written as follows:

Look at these two examples.

Of course, when you have a subtraction problem that has only positive numbers, like $6-4,$ you just do the subtraction. You already knew how to subtract $6-4$ long ago. But knowing that $6-4$ gives the same answer as $6+\left(\mathrm{-4}\right)$ helps when you are subtracting negative numbers. Make sure that you understand how $6-4$ and $6+\left(\mathrm{-4}\right)$ give the same results!

Look at what happens when we subtract a negative.

You will often see this written as $a-(\text{−}b)=a+b.$

Does that work for other numbers, too? Let’s do the following example and see.

Let’s look again at the results of subtracting the different combinations of $5,\mathrm{-5}$ and $3,\mathrm{-3}.$

What happens when there are more than three integers? We just use the order of operations as usual.