In Hours 1 and 2, your math world expanded beyond the whole numbers to include positive and negative integers, real numbers, inequalities, and absolute value. Now, to use this knowledge, you learn the rules for adding, subtracting, multiplying, and dividing real numbers. These four operations with real numbers form an important foundation that allows you to set up and solve equations and advance to higher algebra skills.

Adding two whole numbers is probably one of the first math problems you learned to solve, right after you learned to count from 1 to 20. Because algebraic expressions include all the real numbers, not just whole numbers, you also need to be able to add real numbers.

Even though your understanding of numbers has grown quite a bit in the past two hours of study, the basics remain the same. It’s still true that 2 + 3 = 5 and that 10 + 6 = 16. As long as an algebra problem has only whole numbers, you will be fine using your basic math facts. Many algebra problems do include negative numbers, and fortunately, some basic rules will guide you. But before we get to the rules, it might be helpful to know when you will use them.

You’ll see negative signs on some of the terms in the next examples. As you reach the end of this section, you will be able to simplify each of these problems:

These examples are addition problems involving some negative and some positive numbers.

Look closely at the first example: -1 + -10 = .

The signs can be confusing, but in words, you would say “negative 1 plus negative 10.” This helps you realize that you are trying to add a negative number to another negative number.

A real-life example of this situation is when you owe someone $1 and then borrow another $10. Now you owe this person a total of $11. But because it’s a debt, it’s really -11. You can see clearly that adding -1 and -10 will give a sum of -11.

Let’s get back to the algebra. The rule is that, when adding negative numbers, you remove their negative signs, add them, and put the negative sign on the final answer. Mathematically, adding two or more negative numbers gives a negative number.

The next four problems show the different ways you will see parentheses in algebra problems.

You probably noticed that each of these problems is a different version of the same problem, which, at its simplest, is -4 plus -2. The parentheses do not show multiplication, which, you will recall from Hour 1, can sometimes be the case. They are also not grouping symbols. They are simply a form of emphasis and may or may not appear in a given problem.

Being aware that the parentheses have no power when they surround a single term frees you to focus on the operation itself, which, in this case, is just adding two negative numbers. Because you will see parentheses in some problems and not in others, be prepared to determine whether they are showing multiplication, as they did in Hour 1, or are just separating the terms, as in these examples. You will see some examples of parentheses used for multiplying real numbers later this hour, on page 29. Remember that if the numbers being added are positive, the sign of the answer will be positive. If the signs of all the numbers being added are negative, the answer will be negative.

Now try these six problems and check your answers on page 30.

The problems that follow have more than two negative numbers, but the same rule applies: remove their negative signs, add them, and put the negative sign on the final answer.

Now consider the following problem:

This is an addition problem; in words, you would say “9 plus negative 7.” The rule you just learned will work perfectly if all the numbers are positive or all are negative. But one of these numbers is positive and the other is negative, so the rule doesn’t apply here.

The problem is actually 9 plus the opposite of 7. In effect, then, this is a subtraction problem. The key concept here is that adding a negative number is the same as adding the opposite of a positive number.

The formal rule for adding two numbers with different signs is to subtract their absolute values and then use the sign of the larger absolute value for the sign in the answer. More simply, you remove the signs from the numbers, subtract them, and use the larger number’s sign for the answer.

Look again at the example in this section: 9 + -7 = ___.

Using this rule, you think 9 - 7 = 2, and because 9 is larger and positive, the answer is +2, which you usually write as 2.

Look at some more examples and solutions of adding numbers with different signs:

Now try the following problems, and check your answers on pages 30-31. (Watch the signs carefully to know which rule you will use.)

To review what you’ve learned so far about adding real numbers:

You are probably wondering what to do if there is a mixture of positive and negative numbers within an addition problem. Once you’ve mastered the previous rules, having several numbers to add, no matter what their signs, is not difficult.

Consider the following problem:

The 3 and 14 are both positive, and the 9 is negative. You have two choices here. The first is to work the problem from left to right, as you read it. What is 3 + -9? Since the signs are different, remove the signs, subtract the numbers, and use the sign of the larger number. So 3 + -9 = -6. To continue the problem, you need to ask, what is -6 + 14? Remove the signs, subtract the numbers, and use the sign of the larger number for the answer. Since -6 + 14 = 8, this is the final answer. (See the FYI note for another way to do this problem.)

The following examples include both positive and negative terms:

Now try these problems and check your answers on page 31.

Now that you have learned how to use the rules for adding real numbers, you are ready to shift gears and try the following basic subtraction problems:

These simple problems are easily solved. Because they are all examples of subtracting a smaller whole number from a larger one, they don’t require any higher skills than basic arithmetic. (Did you get 3, 5, 5, and 2 for the answers?)

Now consider these more complex subtraction problems:

These subtraction problems include positive and negative integers, so they are not as easy as the first four. To learn how to solve them, you need to first see the close link between the operations of addition and subtraction.

You will see this link most clearly if you begin by looking back at a problem that involves adding positive and negative integers, such as this one:

Because you are adding and the signs are different, you know the rule: remove the signs, subtract the numbers, and use the sign of the larger number for the sign of the answer.

So you know that 10 + (-6) = 4.

Now look at the simple subtraction problem 10 - 6 = 4. The answer to both of these problems is clearly 4. Is it a coincidence that you got the same answer? No, because adding -6 to 10 is exactly the same as subtracting 6 from 10, so, of course, the answers are the same.

The close link between subtraction and addition is most easily stated this way: subtracting a number is the same as adding a number with the opposite sign. To see this for yourself, consider these subtraction-to-addition examples:

(Did you get 2, 3, and -5 for your answers?)

In the previous three problems, the first term in each problem is a positive number, but the same rule applies if the first term is negative:

(Did you get -8 and -6?)

In fact, all subtraction problems can be changed to addition, as long as you change the sign of the second term. The beauty of this process is that you can always get the answer to any addition problem because you know all the addition rules. You don’t need any subtraction rules! Look at some samples:

To review, when you have a subtraction problem, change it to addition and change the sign on the second number.

Now try these problems and check your answers on page 31.

If a problem has many positive and negative numbers to be added and subtracted, you should change the subtraction to addition first and then follow the rules for adding positive and negative numbers. Here are some examples, including each step for finding the final solution:

In the previous example, all the subtraction became addition of positive terms, so it is actually an easy problem to solve.

In the previous example, the steps of the solution show the addition done from left to right. You could also group the positives together and get a total, group the negatives together and get a total, and then combine the positive and negative totals. The final answer will still be 7, using either method.

Now try these problems and check your answers on page 31.

Now that you have mastered adding and subtracting real numbers, the most difficult part of this hour is behind you. The rules for multiplying (and dividing) are simpler and very easy to use. In fact, let’s go straight to the rules.

Multiplying positive numbers always gives a positive answer. This is true no matter how many positive numbers are in the problem. Examples:

Multiplying two negative numbers always gives a positive answer. Yes, this seems strange, but it’s true. Just be careful not to confuse the multiplying rule with the adding rule, since adding two or more negative terms gives a negative answer.

Examples:

Multiplying a positive number times a negative number always gives a negative answer. It doesn’t matter which is larger or smaller; multiplying numbers with two different signs always gives a negative product.

Examples:

When there are more than two numbers in the problem, apply the three multiplication rules as needed. The following examples show the solution steps:

Now try these problems and check your answers on page 31.

Knowing the rules for multiplying positive and negative numbers helps you know how to divide positive and negative numbers. In fact, the rules are exactly the same for both operations. Here are the division rules:

Examples follow:

Now try these problems and check your answers on page 31.

Based on what you’ve learned in Hour 3, solve the following about adding, subtracting, multiplying, and dividing real numbers. Check your answers with the solution key in Appendix B.