## Easy Algebra Step-by-Step: Master High-Frequency Concepts and Skills for Algebra Proficiency—FAST! (2012)

### Chapter 2. Computation with Real Numbers

This chapter presents the rules for computing with real numbers—often called signed numbers. Before proceeding with addition, subtraction, multiplication, and division of signed numbers, the discussion begins with comparing numbers and finding the absolute value of a number.

**Comparing Numbers and Absolute Value**

Comparing numbers uses the inequality symbols shown in __Table 2.1__.

**Table 2.1 Inequality Symbols**

Graphing the numbers on a number line is helpful when you compare two numbers. The number that is farther to the right is the greater number. If the numbers coincide, they are equal; otherwise, they are unequal.

**Problem** Which is greater –7 or –2?

**Solution**

*Step 1*. Graph –7 and –2 on a number line.

*Step 2*. Identify the number that is farther to the right as the greater number.

–2 is to the right of –7, so

The concept of absolute value plays an important role in computations with signed numbers. The *absolute value* of a real number is its distance from 0 on the number line. For example, as shown in __Figure 2.1__, the absolute value of –8 is 8 because –8 is 8 units from 0.

**Figure 2.1** The absolute value of –8

Absolute value is a distance, so it is *never* negative.

You indicate the absolute value of a number by placing the number between a pair of vertical bars like this: |–8| (read as “the absolute value of negative eight”). Thus,

**Problem** Find the indicated absolute value.

**Solution**

*Step 1*. Recalling that the absolute value of a real number is its distance from 0 on the number line, determine the absolute value.

because –30 is 30 units from 0 on the number line.

*Step 1*. Recalling that the absolute value of a real number is its distance from 0 on the number line, determine the absolute value.

because 0.4 is 0.4 units from 0 on the number line.

**c**.

As you likely noticed, the absolute value of a number is the value of the number with no sign attached. This strategy works for a number whose value you know, but do not use it when you don’t know the value of the number.

*Step 1*. Recalling that the absolute value of a real number is its distance from 0 on the number line, determine the absolute value.

because units from 0 on the number line.

**Problem** Which number has the greater absolute value?

**Solution**

*Step 1*. Determine the absolute values.

*Step 2*. Compare the absolute values.

60 has the greater absolute value because 60 > 35.

*Step 1*. Determine the absolute values.

*Step 2*. Compare the absolute values.

–60 has the greater absolute value because 60 > 35.

**c**.

*Step 1*. Determine the absolute values.

*Step 2*. Compare the absolute values.

has the greater absolute value because .

**d**.

*Step 1*. Determine the absolute values.

*Step 2*. Compare the absolute values.

has the greater absolute value because .

Don’t make the mistake of trying to compare the numbers without first finding the absolute values.

**Addition and Subtraction of Signed Numbers**

Real numbers are called signed numbers because these numbers may be positive, negative, or 0. From your knowledge of arithmetic, you already know how to do addition, subtraction, multiplication, and division with positive numbers and 0. To do these operations with all signed numbers, you simply use the absolute values of the numbers and follow these eight rules.

**Addition of Signed Numbers**

**Rule 1**. To add two numbers that have the same sign, add their absolute values and give the sum their common sign.

**Rule 2**. To add two numbers that have opposite signs, subtract the lesser absolute value from the greater absolute value and give the sum the sign of the number with the greater absolute value; if the two numbers have the same absolute value, their sum is 0.

**Rule 3**. The sum of 0 and any number is the number.

These rules might sound complicated, but practice will make them your own. One helpful hint is that when you need the absolute value of a number, just use the value of the number with no sign attached.

**Problem** Find the sum.

**Solution**

*Step 1*. Determine which addition rule applies.

The signs are the same (both negative), so use Rule 1.

*Step 2*. Add the absolute values, 35 and 60.

*Step 3*. Give the sum a negative sign (the common sign).

*Step 1*. Determine which addition rule applies.

The signs are opposites (one positive and one negative), so use Rule 2.

*Step 2*. Subtract 35 from 60 because

*Step 3*. Make the sum negative because –60 has the greater absolute value.

*Step 1*. Determine which addition rule applies.

The signs are opposites (one negative and one positive), so use Rule 2.

*Step 2*. Subtract 35 from 60 because

*Step 3*. Keep the sum positive because 60 has the greater absolute value.

**d**.

*Step 1*. Determine which addition rule applies.

The signs are opposites (one positive and one negative), so use Rule 2.

*Step 2*. Subtract from because .

*Step 3*. Keep the sum positive because has the greater absolute value.

**e**.

*Step 1*. Determine which addition rule applies.

The signs are opposites (one positive and one negative), so use Rule 2.

*Step 2*. Subtract from because .

*Step 3*. Make the sum negative because has the greater absolute value.

*Step 1*. Determine which addition rule applies.

The signs are the same (both negative), so use Rule 1.

*Step 2*. Add the absolute values 9.75 and 8.12.

*Step 3*. Give the sum a negative sign (the common sign).

*Step 1*. Determine which addition rule applies.

0 is added to a number, so the sum is the number (Rule 3).

You subtract signed numbers by changing the subtraction problem to an addition problem in a special way, so that you can apply the rules for addition of signed numbers. Here is the rule.

**Subtraction of Signed Numbers**

**Rule 4**. To subtract two numbers, keep the first number and add the opposite of the second number.

To apply this rule, think of the minus sign, –, as “add the opposite of.” In other words, “subtracting a number” and “adding the opposite of the number” give the same answer.

**Problem** Change the subtraction problem to an addition problem.

**Solution**

*Step 1*. Keep –35.

–35

*Step 2*. Add the opposite of 60.

*Step 1*. Keep 35.

35

*Step 2*. Add the opposite of 60.

*Step 1*. Keep 60.

60

*Step 2*. Add the opposite of 35.

*Step 1*. Keep –35.

–35

*Step 2*. Add the opposite of –60.

*Step 1*. Keep 0.

0

*Step 2*. Add the opposite of 60.

*Step 1*. Keep –60.

–60

*Step 2*. Add the opposite of 0.

Remember 0 is its own opposite.

**Problem** Find the difference.

A helpful mnemonic to remember how to subtract signed numbers is “Keep, change, change.” You *keep* the first number, you *change* minus to plus, and you *change* the second number to its opposite.

**Solution**

*Step 1*. Keep –35 and add the opposite of 60.

*Step 2*. The signs are the same (both negative), so use Rule 1 for addition.

= –95

*Step 3*. Review the main results.

Cultivate the habit of reviewing your main results. Doing so will help you catch careless mistakes.

*Step 1*. Keep 35 and add the opposite of 60.

*Step 2*. The signs are opposites (one positive and one negative), so use Rule 2 for addition.

= –25

*Step 3*. Review the main results.

*Step 1*. Keep 60 and add the opposite of 35.

*Step 2*. The signs are opposites (one positive and one negative), so use Rule 2 for addition.

= 25

*Step 3*. Review the main results.

*Step 1*. Keep –35 and add the opposite of –60.

*Step 2*. The signs are opposites (one positive and one negative), so use Rule 2 for addition.

= 25

*Step 3*. Review the main results.

*Step 1*. Keep 0 and add the opposite of –60.

*Step 2*. 0 is added to a number, so the sum is the number (Rule 3 for addition).

= 60

*Step 3*. Review the main results.

*Step 1*. Keep –60 and add the opposite of 0.

*Step 2*. 0 is added to a number, so the sum is the number (Rule 3 for addition).

= –60

*Step 3*. Review the main results.

Notice that subtraction is *not* commutative. That is, in general, for real numbers *a* and

Before going on, it is important that you distinguish the various uses of the short horizontal – symbol. Thus far, this symbol has three uses: (1) as part of a number to show that the number is negative, (2) as an indicator to find the opposite of the number that follows, and (3) as the minus symbol indicating subtraction.

**Problem** Given the statement

**a**. Describe the use of the – symbols at (1), (2), (3), and (4).

**b**. Express the statement in words.

**Solution**

**a**. Describe the use of the – symbols at (1), (2), (3), and (4).

*Step 1*. Interpret each – symbol.

The – symbol at (1) is an indicator to find the opposite of –35.

Don’t make the error of referring to negative numbers as “minus numbers.”

The – symbol at (2) is part of the number –35 that shows –35 is negative.

The – symbol at (3) is the minus symbol indicating subtraction.

The minus symbol always has a number to its immediate left.

The – symbol at (4) is part of the number –60 that shows –60 is negative.

There is never a number to the immediate left of a negative sign.

**b**. Express the statement in words.

*Step 1*. Translate the statement into words.

is read “the opposite of negative thirty-five minus sixty is thirty-five plus negative sixty.”

**Multiplication and Division of Signed Numbers**

For multiplication of signed numbers, use the following three rules:

**Multiplication of Signed Numbers**

**Rule 5**. To multiply two numbers that have the same sign, multiply their absolute values and keep the product positive.

**Rule 6**. To multiply two numbers that have opposite signs, multiply their absolute values and make the product negative.

**Rule 7**. The product of 0 and any number is 0.

When you multiply two positive or two negative numbers, the product is *always* positive no matter what. Similarly, when you multiply two numbers that have opposite signs, the product is *always* negative—it doesn’t matter which number has the greater absolute value.

**Problem** Find the product.

**Solution**

*Step 1*. Determine which multiplication rule applies.

The signs are the same (both negative), so use Rule 5.

*Step 2*. Multiply the absolute values, 3 and 40.

*Step 3*. Keep the product positive.

**b**. (3)(40)

*Step 1*. Determine which multiplication rule applies.

(3)(40)

The signs are the same (both positive), so use Rule 5.

*Step 2*. Multiply the absolute values, 3 and 40.

(3)(40) = 120

*Step 3*. Keep the product positive.

*Step 1*. Determine which multiplication rule applies.

(–3)(40)

The signs are opposites (one negative and one positive), so use Rule 6.

*Step 2*. Multiply the absolute values, 3 and 40.

(3)(40) = 120

*Step 3*. Make the product negative.

(–3)(40) = –120

**d**. (3)(–40)

*Step 1*. Determine which multiplication rule applies.

(3) (–40)

The signs are opposites (one positive and one negative), so use Rule 6.

*Step 2*. Multiply the absolute values, 3 and 40.

(3)(40) = 120

*Step 3*. Make the product negative.

(3)(–40) = –120

**e**. (358)(0)

*Step 1*. Determine which multiplication rule applies.

(358)(0)

0 is one of the factors, so use Rule 7.

*Step 2*. Find the product.

(358)(0) = 0

Rules 5, 6, and 7 tell you how to multiply two numbers, but often you will want to find the product of more than two numbers. To do this, multiply in pairs. You can keep track of the sign as you go along, or you simply can use the following guideline:

When 0 is one of the factors, the product is *always* 0; otherwise, products that have an even number of *negative* factors are positive, whereas those that have an odd number of *negative* factors are negative.

Notice that if there is no zero factor, then the sign of the product is determined by how many *negative* factors you have.

**Problem** Find the product.

**Solution**

*Step 1*. 0 is one of the factors, so the product is 0.

*Step 1*. Find the product ignoring the signs.

*Step 2*. You have five negative factors, so make the product negative.

*Step 1*. Find the product ignoring the signs.

*Step 2*. You have four negative factors, so leave the product positive.

**Division of Signed Numbers**

**Rule 8**. To divide two numbers, divide their absolute values (being careful to make sure you don’t divide by 0) and then follow the rules for multiplication of signed numbers.

If 0 is the dividend, the quotient is 0. For instance, . But if 0 is the divisor, the quotient is undefined. Thus, and has no answer because division by 0 is undefined!

**Problem** Find the quotient.

**a**.

**b**.

**c**.

**d**.

**e**.

In algebra, division is commonly indicated by the fraction bar.

**Solution**

**a**.

*Step 1*. Divide 120 by 3.

*Step 2*. The signs are the same (both negative), so keep the quotient positive.

**b**.

*Step 1*. Divide 120 by 3.

*Step 2*. The signs are opposites (one negative and one positive), so make the quotient negative.

**c**.

*Step 1*. Divide 120 by 3.

*Step 2*. The signs are opposites (one positive and one negative), so make the quotient negative.

**d**.

*Step 1*. The divisor (denominator) is 0, so the quotient is undefined.

**e**.

*Step 1*. The dividend (numerator) is 0, so the quotient is 0.

To be successful in algebra, you must memorize the rules for adding, subtracting, multiplying, and dividing signed numbers. Of course, when you do a computation, you don’t have to write out all the steps. For instance, you can mentally ignore the signs to obtain the absolute values, do the necessary computation or computations, and then make sure your result has the correct sign.

**Exercise 2**

*For 1–3, simplify*.

*For 4 and 5, state in words*.

*For 6–20, compute as indicated*.

__12__.

__14__.

__15__.

__17__.

__18__.

__19__.