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 −2 > −7.

Absolute value is a distance, so it is always positive or zero. It is never negative.

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

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, |−8| = 8.

Problem   Find the indicated absolute value.

a. |−30|

b. |0.4|

c.

Solution

a. |−30|

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

|−30| = 30 because −30 is 30 units from 0 on the number line.

b. |0.4|

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

|0.4| = 0.4 because 0.4 is 0.4 unit from 0 on the number line.

c.

As you likely noticed, the absolute value of a specific number is the value of the number with no sign attached. This strategy works for a number whose value you know (that is, a number that you can locate on a number line), 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.

units from 0 on the number line.

Problem   Which number, if any, has the greater absolute value?

a. −35, 60

b. 35, −60

c. −25, 25

d.

Solution

a. −35, 60

Step 1.   Determine the absolute values.

|−35| = 35, |60| = 60

Step 2.   Compare the absolute values.

60 has the greater absolute value because 60 > 35.

b. 35, −60

Step 1.   Determine the absolute values.

|35| = 35, |−60| = 60

Step 2.   Compare the absolute values.

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

c. −25, 25

A number and its opposite have the same absolute value.

Step 1.   Determine the absolute values.

|−25| = 25, |25| = 25

Step 2.   Compare the absolute values.

Neither number has the greater absolute value because 25 = 25.

d.

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

Step 1.   Determine the absolute values.

Step 2.   Compare the absolute values.

has the greater absolute value because .

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 the eight rules in this chapter.

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 specific number, just use the value of the number with no sign attached.

Problem   Find the sum.

a. −35 + −60

b. 35 + −60

c. −35 + 60

d.

e.

f. −9.75 + −8.12

g. −990.36 + 0

Solution

a. −35 + −60

Step 1.   Determine which addition rule applies.

−35 + −60

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

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

35 + 60 = 95

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

−35 + −60 = −95

b. 35 + −60

Step 1.   Determine which addition rule applies.

35 + −60

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

Step 2.   Subtract 35 from 60 because |−60| > |35|.

60 − 35 = 25

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

35 + −60 = −25

c. −35 + 60

Step 1.   Determine which addition rule applies.

−35 + 60

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

Step 2.   Subtract 35 from 60 because |60| > |−35|.

60 − 35 = 25

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

−35 + 60 = 25

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.

f. −9.75 + −8.12

Step 1.   Determine which addition rule applies.

−9.75 + −8.12

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

Step 2.   Add the absolute values 9.75 and 8.12.

9.75 + 8.12 = 17.87

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

−9.75 + −8.12 = −17.87

g. −990.36 + 0

Step 1.   Determine which addition rule applies.

−990.36 + 0

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

−990.36 + 0 = −990.36

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.

a. −35 − 60

b. 35 − 60

c. 60 − 35

d. −35 − (−60)

e. 0 − 60

f. −60 − 0

Solution

a. −35 − 60

Step 1.   Keep −35.

−35

Step 2.   Add the opposite of 60.

= −35 + −60

b. 35 − 60

Step 1.   Keep 35.

35

Step 2.   Add the opposite of 60.

= 35 + −60

c. 60 − 35

Step 1.   Keep 60.

60

Step 2.   Add the opposite of 35.

= 60 + −35

d. −35 − (−60)

Step 1.   Keep −35.

−35

Step 2.   Add the opposite of −60.

= −35 + 60

e. 0 − 60

Step 1.   Keep 0.

0

Step 2.   Add the opposite of 60.

= 0 + −60

f. −60 − 0

Step 1.   Keep −60.

−60

Step 2.   Add the opposite of 0.

= −60 + 0

Remember 0 is its own opposite.

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

Problem   Find the difference.

a. −35 − 60

b. 35 − 60

c. 60 − 35

d. −35 − (−60)

e. 0 − (−60)

f. −60 − 0

Solution

a. −35 − 60

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

−35 − 60

= −35 + −60

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

= −95

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

Step 3.   Review the main results.

−35 − 60 = −35 + −60 = −95

a. 35 − 60

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

35 − 60

= 35 + −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.

35 − 60 = 35 + −60 = −25

c. 60 − 35

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

60 − 35

= 60 + −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.

60 − 35 = 60 + −35 = 25

d. −35 − (−60)

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

−35 − (−60)

= −35 + 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.

−35 − (−60) = −35 + 60 = 25

e. 0 − (−60)

Step 1.   Keep 0 and add the opposite of −60.

0 − (−60)

= 0 + 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.

0 − (−60) = 0 + 60 = 60

f. −60 − 0

Step 1.   Keep −60 and add the opposite of 0.

−60 − 0

= −60 + 0

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

= −60

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

Step 3.   Review the main results.

−60 − 0 = −60 + 0 = −60

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 −(−35) − 60 = 35 + −60 in words.

Solution

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

Step 1.   Interpret each − symbol.

b. Express the statement −(−35) − 60 = 35 + −60 in words.

Step 1.   Translate the statement into words.

−(−35) − 60 = 35 + −60 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.

a. (−3)(−40)

b. (3)(40)

c. (−3)(40)

d. (3)(−40)

e. (358)(0)

Solution

a. (−3)(−40)

Step 1.   Determine which multiplication rule applies.

(−3)(−40)

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

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

(3)(40) = 120

Step 3.   Keep the product positive.

(−3)(−40) = 120

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.

(3)(40) = 120

c. (−3)(40)

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:

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

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.

Problem   Find the product.

a. (600)(−40)(−1,000)(0)(−30)

b. (−3)(−10)(−5)(25)(−1)(−2)

c. (−2)(−4)(−10)(1)(−20)

Solution

a. (600)(−40)(−1,000)(0)(−30)

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

(600)(−40)(−1,000)(0)(−30) = 0

b. (−3)(−10)(−5)(25)(−1)(−2)

Step 1.   Find the product, ignoring the signs.

(3)(10)(5)(25)(1)(2) = 7,500

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

(−3)(−10)(−5)(25)(−1)(−2) = −7,500

c. (−2)(−4)(−10)(1)(−20)

Step 1.   Find the product, ignoring the signs.

(2)(4)(10)(1)(20) = 1,600

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

(−2)(−4)(−10)(1)(−20) = 1,600

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 the divisor is nonzero and 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!

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

Problem   Find the quotient.

a.

b.

c.

d.

e.

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 divisor is nonzero and 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.

1. |−45|

2. |5.8|

3.

For 4 and 5, state in words.

4. −9 + −(−4) = −9 + 4

5. −9 − (−4) = −9 + 4

For 6–20, compute as indicated.

6. −80 + −40

7. 0.7 + −1.4

8.

9.

10. (−100)(−8)

11.

12.

13. −450.95 − (−65.83)

14.

15.

16. −458 + 0

17.

18.

19.

20. (−3)(1)(−1)(−5)(−2)(2)(−10)