Operations on Integers
There are three appealing ways to understand how to add integers. We can use movement, temperature and money. Lastly, we will take a look at the rules for addition.
Instead of coming up with a new method for explaining how to subtract integers, let us borrow from the explanation above under the addition of integers. We will learn how to transform subtraction problems into addition problems.
The technique for changing subtraction problems into addition problems is extremely mechanical. There are two steps:
Let's take a look at example 1:
According to step #1, we have to change the subtraction sign to an addition sign. According to step #2, we have to take the opposite of 4, which is -4. Therefore the problem becomes:
Using the rules for addition, the answer is -1.
Here are a few other examples:
Example 2: -2 - 8 = -2 + (-8) = -10
Example 3: 6 - (-20) = 6 + 20 = 26
Example 4: -7 - (-1) = -7 + 1 = -6
uizmaster: Subtracting Integers
The best place to start with multiplication, is with the rules:
Now we have to understand the rules. The first rule is the easiest to remember because we learned it so long ago. Working with positive numbers under multiplication always yeilds positive answers. However, the last three rules are a bit more challenging to understand.
The second and third steps can be explained simultaneously. This is because numbers can be multiplied in any order. -3 x 7 has the same answer as 7 x -3, which is always true for all integers. [This property has a special name in mathematics. It is called the commutative property.] For us, this means the second and third rules are equivalent.
One reason why mathematics has so much value is because its usefulness is derived from its consistency. It behaves with strict regularity. This is no accident, mind you. This is quite purposeful.
Keeping this in mind, let's take a look at Figure 2a below. There is a definite pattern to the problems in the table. The values in the first column remain constant but the values in the second column are decreasing by one, each step down the table. Consequently, the answer column is changing. The answers to each problem have a definite pattern as we go down the table too. It should be relatively easy to determine the two missing answers.
If you understand the pattern, you will see that the values in the answer column are decreasing. Each value in the answer column is two less than the value above it. See Figure 2b to see the solutions to those last two problems.
This should provide some meaning why a positive number times a negative number is always a negative number. Since we can multiply numbers in any order, it also explains why a negative number times a positive number is always a negative number.
Now, let us turn our attention to Figure 3a below. This table has a pattern similar to the one in Figure 2a. However, Figure 3a begins with a negative number in the first column. As we scan the list of answers in the last column, we can see that the last two problems remain unanswered.
With a little concentration, we can see that the two unanswered questions must have positive answers to maintain mathematical consistency. Why? The pattern within the last column shows that the values are increasing by 8 as we travel down the column. Figure 3b shows the table completely filled in.
This should help us understand why a negative number times a negative number is always a postive number.
Example 1: 4 x -8 = -32
Example 2: -6 x 8 = -48
Example 3: -20 x -3 = 60
The rules for division are exactly the same as those for multiplication. If we were to take the rules for multiplication and change the multiplication signs to division signs, we would have an accurate set of rules for division.
Here are three examples:
Example 1: -9 ÷ 3 = -3
Example 2: 20 ÷ (-4) = -5
Example 3: -18 ÷ (-3) = 6
uizmaster: Dividing Integers
Use these videos to solidify your knowledge of the skills and concepts presented within this section of MATHguide:
Try this related lesson.