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:
Here is another problem: -2 - 8. Switching the problem to an addition problem, it becomes -2 + (-8), which is equal to -10.
6 - (-20) is equal to 6 + 20, which is 26.
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 2 below. There is a definite pattern to the problems in the table. The first number in each row remains constant but the second number is decreasing by one, each step down the table. Consequently, the answer is changing. The answers 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 first unanswered problem is -2 and the second unanswered problem is -4. This should provide some meaning why a negative number is always the result when multiplying two numbers of opposite sign.
Likewise, lets turn our attention to Figure 3 below. This table has a pattern similar to the one in Figure 1. However, this table begins with a negative number. As we scan the list of answers, 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. This should help us understand why a positive number is always the result of multiplying two numbers of the same sign.
Here are some examples:
4 x -8 = -32,
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.
For example, -9 ÷ 3 = -3.