Chapter 10. Signed Numbers
10.1. Objectives*
After completing this chapter, you should
Variables, Constants, and Real Numbers (Section 10.2)
be able to distinguish between variables and constants
be able to recognize a real number and particular subsets of the real numbers
understand the ordering of the real numbers
Signed Numbers (Section 10.3)
be able to distinguish between positive and negative real numbers
be able to read signed numbers
understand the origin and use of the double-negative product property
Addition of Signed Numbers (Section 10.5)
be able to add numbers with like signs and with unlike signs
be able to use the calculator for addition of signed numbers
Subtraction of Signed Numbers (Section 10.6)
understand the definition of subtraction
be able to subtract signed numbers
be able to use a calculator to subtract signed numbers
Multiplication and Division of Signed Numbers (Section 10.7)
be able to multiply and divide signed numbers
be able to multiply and divide signed numbers using a calculator
10.2. Variables, Constants, and Real Numbers *
Section Overview
Variables and Constants
Real Numbers
Subsets of Real Numbers
Ordering Real Numbers
Variables and Constants
A basic distinction between algebra and arithmetic is the use of symbols (usually letters) in algebra to represent numbers. So, algebra is a generalization of arithmetic. Let us look at two examples of situations in which letters are substituted for numbers:
Suppose that a student is taking four college classes, and each class can have at most 1 exam per week. In any 1-week period, the student may have 0, 1, 2, 3, or 4 exams. In algebra, we can let the letter x represent the number of exams this student may have in a 1-week period. The letter x may assume any of the various values 0, 1, 2, 3, 4.
Suppose that in writing a term paper for a biology class a student needs to specify the average lifetime, in days, of a male housefly. If she does not know this number off the top of her head, she might represent it (at least temporarily) on her paper with the letter t (which reminds her of time). Later, she could look up the average time in a reference book and find it to be 17 days. The letter t can assume only the one value, 17, and no other values. The value t is constant.
Variable, Constant
A letter or symbol that represents any member of a collection of two or more numbers is called a variable.
A letter or symbol that represents one specific number, known or unknown, is called a constant.
In example 1, the letter x is a variable since it can represent any of the numbers 0, 1, 2, 3, 4. The letter t example 2 is a constant since it can only have the value 17.
Real Numbers
The study of mathematics requires the use of several collections of numbers. The real number line allows us to visually display (graph) the numbers in which we are interested.
A line is composed of infinitely many points. To each point we can associate a unique number, and with each number, we can associate a particular point.
The number associated with a point on the number line is called the coordinate of the point.
The point on a number line that is associated with a particular number is called the graph of that number.
We construct a real number line as follows:
Draw a horizontal line.
- Origin
Choose any point on the line and label it 0. This point is called the origin.
Choose a convenient length. Starting at 0, mark this length off in both directions, being careful to have the lengths look like they are about the same.
We now define a real number.
A real number is any number that is the coordinate of a point on the real number line.
Real numbers whose graphs are to the right of 0 are called positive real numbers, or more simply, positive numbers. Real numbers whose graphs appear to the left of 0 are called negative real numbers, or more simply, negative numbers.
The number 0 is neither positive nor negative.
Subsets of Real Numbers
The set of real numbers has many subsets. Some of the subsets that are of interest in the study of algebra are listed below along with their notations and graphs.
The natural or counting numbers ( N ): 1, 2, 3, 4, . . . Read “and so on.”
The whole numbers ( W ): 0, 1, 2, 3, 4, . . .
Notice that every natural number is a whole number.
The integers ( Z ): . . . -3, -2, -1, 0, 1, 2, 3, . . .
Notice that every whole number is an integer.
The rational numbers ( Q ): Rational numbers are sometimes called fractions. They are numbers that can be written as the quotient of two integers. They have decimal representations that either terminate or do not terminate but contain a repeating block of digits. Some examples are
Some rational numbers are graphed below.
Notice that every integer is a rational number.
Notice that there are still a great many points on the number line that have not yet been assigned a type of number. We will not examine these other types of numbers in this text. They are examined in detail in algebra. An example of these numbers is the number π , whose decimal representation does not terminate nor contain a repeating block of digits. An approximation for π is 3.14.
Sample Set A
Example 10.1.
Is every whole number a natural number?
No. The number 0 is a whole number but it is not a natural number.
Example 10.2.
Is there an integer that is not a natural number?
Yes. Some examples are 0, -1, -2, -3, and -4.
Example 10.3.
Is there an integer that is a whole number?
Yes. In fact, every whole number is an integer.
Practice Set A
Ordering Real Numbers
A real number b is said to be greater than a real number a , denoted b > a , if b is to the right of a on the number line. Thus, as we would expect, 5 > 2 since 5 is to the right of 2 on the number line. Also, – 2 > – 5 since -2 is to the right of -5 on the number line.
If we let a and b represent two numbers, then a and b are related in exactly one of three ways: Either
Sample Set B
Example 10.4.
What integers can replace x so that the following statement is true?
– 3 ≤ x < 2
The integers are -3, -2, -1, 0, 1.
Example 10.5.
Draw a number line that extends from -3 to 5. Place points at all whole numbers between and including -1 and 3.
-1 is not a whole number
Practice Set B
Exercise 10.2.6. (Go to Solution)
What integers can replace x so that the following statement is true? – 5 ≤ x < 2
Exercise 10.2.7. (Go to Solution)
Draw a number line that extends from -4 to 3. Place points at all natural numbers between, but not including, -2 to 2.
Exercises
For the following 8problems, next to each real number, note all collections to which it belongs by writing N for natural number, W for whole number, or Z for integer. Some numbers may belong to more than one collection.
Exercise 10.2.9.
12
Exercise 10.2.11.
1
Exercise 10.2.13.
-7
Exercise 10.2.15.
-900
Exercise 10.2.16. (Go to Solution)
Is the number 0 a positive number, a negative number, neither, or both?
Exercise 10.2.17.
An integer is an even integer if it is evenly divisible by 2. Draw a number line that extends from -5 to 5 and place points at all negative even integers and all positive odd integers.
Exercise 10.2.18. (Go to Solution)
Draw a number line that extends from -5 to 5. Place points at all integers that satisfy – 3 ≤ x < 4.
Exercise 10.2.19.
Is there a largest two digit number? If so, what is it?
For the pairs of real numbers in the following 5 problems, write the appropriate symbol (<, >, =) in place of the □.
Exercise 10.2.21.
-7 □ -2
Exercise 10.2.23.
-1 □ 4
Exercise 10.2.25.
10 □ 10
For the following 5 problems, what numbers can replace m so that the following statements are true?
Exercise 10.2.27.
– 7 < m < – 1, m an integer.
Exercise 10.2.29.
– 15 < m ≤ – 1, m a natural number.
For the following 10 problems, on the number line, how many units are there between the given pair of numbers?
Exercise 10.2.31.
0 and 3
Exercise 10.2.33.
-1 and 6
Exercise 10.2.35.
-3 and 3
Exercise 10.2.37.
Are all positive numbers greater than all negative numbers?
Exercise 10.2.39.
Is there a largest natural number?
Exercises for Review
Solutions to Exercises
10.3. Signed Numbers *
Section Overview
Positive and Negative Numbers
Reading Signed Numbers
Opposites
The Double-Negative Property
Positive and Negative Numbers
Each real number other than zero has a sign associated with it. A real number is said to be a positive number if it is to the right of 0 on the number line and negative if it is to the left of 0 on the number line.
THE NOTATION OF SIGNED NUMBERS
A number is denoted as positive if it is directly preceded by a plus sign or no sign at all. A number is denoted as negative if it is directly preceded by a minus sign.
Reading Signed Numbers
The plus and minus signs now have two meanings:
The plus sign can denote the operation of addition or a positive number.
The minus sign can denote the operation of subtraction or a negative number.
To avoid any confusion between "sign" and "operation," it is preferable to read the sign of a number as "positive" or "negative." When "+" is used as an operation sign, it is read as "plus." When " – " is used as an operation sign, it is read as "minus."
Sample Set A
Read each expression so as to avoid confusion between "operation" and "sign."
Example 10.6.
– 8 should be read as "negative eight" rather than "minus eight."
Example 10.7.
4 + ( – 2) should be read as "four plus negative two" rather than "four plus minus two."
Example 10.8.
– 6 + ( – 3)should be read as "negative six plus negative three" rather than "minus six plus minus three."
Example 10.9.
– 15 – ( – 6)should be read as "negative fifteen minus negative six" rather than "minus fifteen minus minus six."
Example 10.10.
– 5 + 7 should be read as "negative five plus seven" rather than "minus five plus seven."
Example 10.11.
0 – 2 should be read as "zero minus two."
Practice Set A
Write each expression in words.
Opposites
On the number line, each real number, other than zero, has an image on the opposite side of 0. For this reason, we say that each real number has an opposite. Opposites are the same distance from zero but have opposite signs.
The opposite of a real number is denoted by placing a negative sign directly in front of the number. Thus, if a is any real number, then − a is its opposite.
Note
The letter " a " is a variable. Thus, " a " need not be positive, and " – a " need not be negative.
If a is any real number, − a is opposite a on the number line.
The Double-Negative Property
The number a is opposite − a on the number line. Therefore, − ( − a) is opposite − a on the number line. This means that
− ( − a ) = a
From this property of opposites, we can suggest the double-negative property for real numbers.
If a is a real number, then − ( − a ) = a
Sample Set B
Find the opposite of each number.
Example 10.12.
If a = 2, then – a = – 2. Also, − ( − a) = − ( − 2) = 2.
Example 10.13.
If a = – 4, then – a = – ( – 4) = 4. Also, – ( – a) = a = – 4.
Practice Set B
Find the opposite of each number.
Exercise 10.3.15. (Go to Solution)
Suppose we do not know the sign of the number k . Is − k positive, negative, or do we not know?
Exercises
Exercise 10.3.16. (Go to Solution)
A number is denoted as positive if it is directly preceded by ____________________.
Exercise 10.3.17.
A number is denoted as negative if it is directly preceded by ____________________.
How should the number in the following 6 problems be read? (Write in words.)
Exercise 10.3.19.
− 5
Exercise 10.3.21.
11
Exercise 10.3.23.
− ( − 5)
For the following 6 problems, write each expression in words.
Exercise 10.3.25.
3 + 8
Exercise 10.3.27.
1 + ( − 9)
Exercise 10.3.29.
0 − ( − 12)
For the following 6 problems, rewrite each number in simpler form.
Exercise 10.3.31.
− ( − 16)
Exercise 10.3.33.
− [ − ( − 20)]
Exercise 10.3.35.
6 − ( − 4)
Exercises for Review
Exercise 10.3.38. (Go to Solution)
(Section 8.2) Use the method of rounding to estimate the sum: 5829 + 8767
Solutions to Exercises
Solution to Exercise 10.3.38. (Return to Exercise)
6,000 + 9,000 = 15,000 (5,829 + 8,767 = 14,596) or 5,800 + 8,800 = 14,600
10.4. Absolute Value*
Section Overview
Geometric Definition of Absolute Value
Algebraic Definition of Absolute Value
Geometric Definition of Absolute Value
Geometric definition of absolute value: The absolute value of a number a , denoted ∣ a ∣ , is the distance from a to 0 on the number line.
Absolute value answers the question of "how far," and not "which way." The phrase "how far" implies "length" and length is always a nonnegative quantity. Thus, the absolute value of a number is a nonnegative number.
Sample Set A
Determine each value.
Example 10.14.
∣ 4 ∣ = 4
Example 10.15.
∣ − 4 ∣ = 4
Example 10.16.
∣ 0 ∣ = 0
Example 10.17.
− ∣ 5 ∣ = − 5. The quantity on the left side of the equal sign is read as "negative the absolute value of 5." The absolute value of 5 is 5. Hence, negative the absolute value of 5 is -5.
Example 10.18.
− ∣ − 3 ∣ = − 3. The quantity on the left side of the equal sign is read as "negative the absolute value of -3." The absolute value of -3 is 3. Hence, negative the absolute value of -3 is – ( 3 ) = – 3 .
Practice Set A
By reasoning geometrically, determine each absolute value.
Algebraic Definition of Absolute Value
From the problems in the section called “Sample Set A”, we can suggest the following algebraic definition of absolute value. Note that the definition has two parts.
Algebraic definition of absolute value
The absolute value of a number a is
The algebraic definition takes into account the fact that the number a could be either positive or zero (a ≥ 0) or negative (a < 0).
If the number a is positive or zero (a ≥ 0), the upper part of the definition applies. The upper part of the definition tells us that if the number enclosed in the absolute value bars is a nonnegative number, the absolute value of the number is the number itself.
The lower part of the definition tells us that if the number enclosed within the absolute value bars is a negative number, the absolute value of the number is the opposite of the number. The opposite of a negative number is a positive number.
Note
The definition says that the vertical absolute value lines may be eliminated only if we know whether the number inside is positive or negative.
Sample Set B
Use the algebraic definition of absolute value to find the following values.
Example 10.19.
∣ 8 ∣ . The number enclosed within the absolute value bars is a nonnegative number, so the upper part of the definition applies. This part says that the absolute value of 8 is 8 itself.
∣ 8 ∣ = 8
Example 10.20.
∣ − 3 ∣ . The number enclosed within absolute value bars is a negative number, so the lower part of the definition applies. This part says that the absolute value of -3 is the opposite of -3, which is − ( − 3). By the definition of absolute value and the double-negative property,
∣ − 3 ∣ = − ( − 3 ) = 3
Practice Set B
Use the algebraic definition of absolute value to find the following values.
Exercises
Determine each of the values.
Exercise 10.4.16.
∣ 3 ∣
Exercise 10.4.18.
∣ − 9 ∣
Exercise 10.4.20.
∣ − 4 ∣
Exercise 10.4.22.
− ∣ 7 ∣
Exercise 10.4.24.
∣ 0 ∣
Exercise 10.4.26.
− ∣ − 26 ∣
Exercise 10.4.28.
− ( − ∣ 2 ∣ )
Exercise 10.4.30.
− ( − ∣ − 42 ∣ )
Exercise 10.4.32.
∣ − 2 ∣ 3
Exercise 10.4.34.
∣ − 2 ∣ − ∣ − 9 ∣
Exercise 10.4.36.
( ∣ − 1 ∣ − ∣ 1 ∣ )3
Exercise 10.4.38.
– [ | – 10 | – 6 ] 2
Exercise 10.4.40.
A Mission Control Officer at Cape Canaveral makes the statement “lift-off, T minus 50 seconds.” How long is it before lift-off?
Exercise 10.4.41. (Go to Solution)
Due to a slowdown in the industry, a Silicon Valley computer company finds itself in debt $2,400,000. Use absolute value notation to describe this company’s debt.
Exercise 10.4.42.
A particular machine is set correctly if upon action its meter reads 0. One particular machine has a meter reading of – 1.6 upon action. How far is this machine off its correct setting?
Exercises for Review
Exercise 10.4.46.
(Section 7.3) The ratio of acid to water in a solution is
. How many mL of acid are there in a solution that contain 112 mL of water?
Solutions to Exercises
10.5. Addition of Signed Numbers*
Section Overview
Addition of Numbers with Like Signs
Addition with Zero
Addition of Numbers with Unlike Signs
Calculators
Addition of Numbers with Like Signs
The addition of the two positive numbers 2 and 3 is performed on the number line as follows.
Begin at 0, the origin.
Since 2 is positive, move 2 units to the right.
Since 3 is positive, move 3 more units to the right.
We are now located at 5.
Thus, 2 + 3 = 5.
Summarizing, we have
( 2 positive units ) + ( 3 positive units ) = ( 5 positive units )
The addition of the two negative numbers -2 and -3 is performed on the number line as follows.
Begin at 0, the origin.
Since -2 is negative, move 2 units to the left.
Since -3 is negative, move 3 more units to the left.
We are now located at -5.
Thus, ( − 2) + ( − 3) = − 5.
Summarizing, we have
( 2 negative units ) + ( 3 negative units ) = ( 5 negative units )
Observing these two examples, we can suggest these relationships:
( positive number ) + ( positive number ) = ( positive number )
( negative number ) + ( negative number ) = ( negative number )
Addition of numbers with like sign: To add two real numbers that have the same sign, add the absolute values of the numbers and associate with the sum the common sign.
Sample Set A
Find the sums.
Example 10.21.
3 + 7
Add these absolute values.
3 + 7 = 10
The common sign is “+.”
Thus, 3 + 7 = + 10, or 3 + 7 = 10.
Example 10.22.
( − 4) + ( − 9)
Add these absolute values.
4 + 9 = 13
The common sign is “ – .“
Thus, ( − 4) + ( − 9) = − 13.
Practice Set A
Find the sums.
Addition With Zero
Notice that (0) + (a positive number) = (that same positive number). (0) + (a negative number) = (that same negative number).
Since adding zero to a real number leaves that number unchanged, zero is called the additive identity.
Addition of Numbers with Unlike Signs
The addition 2 + ( – 6), two numbers with unlike signs, can also be illustrated using the number line.
Begin at 0, the origin.
Since 2 is positive, move 2 units to the right.
Since -6 is negative, move, from 2, 6 units to the left.
We are now located at -4.
We can suggest a rule for adding two numbers that have unlike signs by noting that if the signs are disregarded, 4 can be obtained by subtracting 2 from 6. But 2 and 6 are precisely the absolute values of 2 and -6. Also, notice that the sign of the number with the larger absolute value is negative and that the sign of the resulting sum is negative.
Addition of numbers with unlike signs: To add two real numbers that have unlike signs, subtract the smaller absolute value from the larger absolute value and associate with this difference the sign of the number with the larger absolute value.
Sample Set B
Find the following sums.
Example 10.23.
7 + ( – 2)
Subtract absolute values: 7 – 2 = 5.
Attach the proper sign: "+."
Thus, 7 + ( – 2) = + 5 or 7 + ( – 2) = 5.
Example 10.24.
3 + ( – 11 )
Subtract absolute values: 11 – 3 = 8.
Attach the proper sign: " – ."
Thus, 3 + ( – 11) = – 8.
Example 10.25.
The morning temperature on a winter's day in Lake Tahoe was -12 degrees. The afternoon temperature was 25 degrees warmer. What was the afternoon temperature?
We need to find – 12 + 25.
Subtract absolute values: 25 – 12 = 16.
Attach the proper sign: "+."
Thus, – 12 + 25 = 13.
Practice Set B
Find the sums.
Calculators
Calculators having the 
key can be used for finding sums of signed numbers.
Sample Set C
Use a calculator to find the sum of -147 and 84.
| Display Reads | |||
| Type | 147 | 147 | |
| Press |
 | -147 | This key changes the sign of a number. It is different than – . |
| Press | + | -147 | |
| Type | 84 | 84 | |
| Press | = | -63 |
Practice Set C
Use a calculator to find each sum.
Exercises
Find the sums in the following 27 problems. If possible, use a calculator to check each result.
Exercise 10.5.24.
8 + 6
Exercise 10.5.26.
( − 6) + ( − 20)
Exercise 10.5.28.
8 + ( − 15)
Exercise 10.5.30.
− 22 + ( − 1)
Exercise 10.5.32.
0 + ( − 4)
Exercise 10.5.34.
− 6 + 1 + ( − 7)
Exercise 10.5.36.
− 5 + 5
Exercise 10.5.38.
− 14 + 14
Exercise 10.5.40.
9 + ( − 9)
Exercise 10.5.42.
13 + ( − 56)
Exercise 10.5.44.
636 + ( − 989)
Exercise 10.5.46.
− 373 + ( − 14)
Exercise 10.5.48.
− 47.03 + ( − 22.71)
Exercise 10.5.50.
In order for a small business to break even on a project, it must have sales of $21,000. If the amount of sales was $15,000, by how much money did this company fall short?
Exercise 10.5.51. (Go to Solution)
Suppose a person has $56 in his checking account. He deposits $100 into his checking account by using the automatic teller machine. He then writes a check for $84.50. If an error causes the deposit not to be listed into this person’s account, what is this person’s checking balance?
Exercise 10.5.52.
A person borrows $7 on Monday and then $12 on Tuesday. How much has this person borrowed?
Exercise 10.5.53. (Go to Solution)
A person borrows $11 on Monday and then pays back $8 on Tuesday. How much does this person owe?
Exercises for Review
Solutions to Exercises
10.6. Subtraction of Signed Numbers*
Section Overview
Definition of Subtraction
The Process of Subtraction
Calculators
Definition of Subtraction
We know from experience with arithmetic that the subtraction 5 – 2 produces 3, that is 5 – 2 = 3. We can suggest a rule for subtracting signed numbers by illustrating this process on the number line.
Begin at 0, the origin.
Since 5 is positive, move 5 units to the right.
Then, move 2 units to the left to get to 6. (This reminds us of addition with a negative number.)
From this illustration we can see that 5 – 2 is the same as 5 + ( – 2). This leads us directly to the definition of subtraction.
If a and b are real numbers, a – b is the same as a + ( – b), where – b is the opposite of b .
The Process of Subtraction
From this definition, we suggest the following rule for subtracting signed numbers.
To perform the subtraction a – b , add the opposite of b to a , that is, change the sign of b and add.
Sample Set A
Perform the indicated subtractions.
Example 10.26.
5 − 3 = 5 + ( − 3) = 2
Example 10.27.
4 – 9 = 4 + ( – 9) = – 5
Example 10.28.
− 4 − 6 = − 4 + ( − 6) = − 10
Example 10.29.
− 3 − ( − 12) = − 3 + 12 = 9
Example 10.30.
0 – ( – 15) = 0 + 15 = 15
Example 10.31.
The high temperature today in Lake Tahoe was 26°F. The low temperature tonight is expected to be -7°F. How many degrees is the temperature expected to drop?
We need to find the difference between 26 and -7.
26 – ( – 7 ) = 26 + 7 = 33
Thus, the expected temperature drop is 33°F.
Example 10.32.
Practice Set A
Perform the indicated subtractions.
Calculators
Calculators can be used for subtraction of signed numbers. The most efficient calculators are those with a 
key.
Sample Set B
Use a calculator to find each difference.
Example 10.33.
3,187 − 8,719
| Display Reads | ||
| Type | 3187 | 3187 |
| Press | - | 3187 |
| Type | 8719 | 8719 |
| Press | = | -5532 |
Thus, 3,187 − 8,719 = − 5,532.
Example 10.34.
− 156 − ( − 211)
Method A:
| Display Reads | ||
| Type | 156 | 156 |
| Press |
 | -156 |
| Type | - | -156 |
| Press | 211 | 211 |
| Type |
 | -211 |
| Press | = | 55 |
Thus, − 156 − ( − 211) = 55.
Method B:
We manually change the subtraction to an addition and change the sign of the number to be subtracted.
− 156 − ( − 211) becomes − 156 + 211
| Display Reads | ||
| Type | 156 | 156 |
| Press |
 | -156 |
| Press | + | -156 |
| Type | 211 | 211 |
| Press | = | 55 |
Practice Set B
Use a calculator to find each difference.
Exercises
For the following 18 problems, perform each subtraction. Use a calculator to check each result.
Exercise 10.6.21.
12 − 7
Exercise 10.6.23.
14 − 30
Exercise 10.6.25.
− 1 − 12
Exercise 10.6.27.
− 11 − ( − 8)
Exercise 10.6.29.
0 − 15
Exercise 10.6.31.
0 − ( − 10)
Exercise 10.6.33.
142 − 85
Exercise 10.6.35.
105 − 421
Exercise 10.6.37.
− 15.016 − (4.001)
For the following 4 problems, perform the indicated operations.
Exercise 10.6.39.
− 15 − 21 − ( − 2)
Exercise 10.6.41.
− 0.012 − ( − 0.111) − (0.035)
Exercise 10.6.42. (Go to Solution)
When a particular machine is operating properly, its meter will read 34. If a broken bearing in the machine causes the meter reading to drop by 45 units, what is the meter reading?
Exercise 10.6.43.
The low temperature today in Denver was − 4° F and the high was − 42° F. What is the temperature difference?
Exercises for Review
Solutions to Exercises
10.7. Multiplication and Division of Signed Numbers*
Section Overview
Multiplication of Signed Numbers
Division of Signed Numbers
Calculators
Multiplication of Signed Numbers
Let us consider first, the product of two positive numbers. Multiply: 3 ⋅ 5.
3 ⋅ 5 means 5 + 5 + 5 = 15
This suggests[1] that
( positive number ) ⋅ ( positive number ) = ( positive number )
More briefly,
( + ) ( + ) = ( + )
Now consider the product of a positive number and a negative number. Multiply: (3)( − 5).
(3)( − 5) means ( − 5) + ( − 5) + ( − 5) = − 15
This suggests that
( positive number ) ⋅ ( negative number ) = ( negative number )
More briefly,
( + ) ( – ) = ( – )
By the commutative property of multiplication, we get
( negative number ) ⋅ ( positive number ) = ( negative number )
More briefly,
( − ) ( + ) = ( − )
The sign of the product of two negative numbers can be suggested after observing the following illustration.
Multiply -2 by, respectively, 4, 3, 2, 1, 0, -1, -2, -3, -4.
We have the following rules for multiplying signed numbers.
Multiplying signed numbers:
To multiply two real numbers that have the same sign, multiply their absolute values. The product is positive. ( + ) ( + ) = ( + ) ( − ) ( − ) = ( + )
To multiply two real numbers that have opposite signs, multiply their absolute values. The product is negative. ( + ) ( − ) = ( − ) ( − ) ( + ) = ( − )
Sample Set A
Find the following products.
Example 10.35.
8 ⋅ 6
Multiply these absolute values.
8 ⋅ 6 = 48
Since the numbers have the same sign, the product is positive.
Thus, 8⋅6=+48, or 8⋅6 = 48.
Example 10.36.
( − 8)( − 6)
Multiply these absolute values.
8 ⋅ 6 = 48
Since the numbers have the same sign, the product is positive.
Thus, ( − 8)( − 6)=+48, or ( − 8)( − 6) = 48.
Example 10.37.
( − 4)(7)
Multiply these absolute values.
4 ⋅ 7 = 28
Since the numbers have opposite signs, the product is negative.
Thus, ( − 4)(7) = − 28.
Example 10.38.
6( − 3)
Multiply these absolute values.
6 ⋅ 3 = 18
Since the numbers have opposite signs, the product is negative.
Thus, 6( − 3) = − 18.
Practice Set A
Find the following products.
Division of Signed Numbers
To determine the signs in a division problem, recall that
since
12 = 3⋅4
This suggests that
since
( + ) = ( + )( + )
What is
?
− 12 = ( − 3)( − 4) suggests that
. That is,
( + ) = ( − )( − ) suggests that
What is
?
− 12 = (3)( − 4) suggests that
. That is,
( − ) = ( + )( − ) suggests that
What is
?
− 12 = ( − 3)(4) suggests that
. That is,
( − ) = ( − )( + ) suggests that
We have the following rules for dividing signed numbers.
Dividing signed numbers:
To divide two real numbers that have the same sign, divide their absolute values. The quotient is positive.
To divide two real numbers that have opposite signs, divide their absolute values. The quotient is negative.
Sample Set B
Find the following quotients.
Example 10.39.
Divide these absolute values.
Since the numbers have opposite signs, the quotient is negative.
Thus
.
Example 10.40.
Divide these absolute values.
Since the numbers have the same signs, the quotient is positive.
Thus,
.
Example 10.41.
Divide these absolute values.
Since the numbers have opposite signs, the quotient is negative.
Thus,
.
Practice Set B
Find the following quotients.
Sample Set C
Example 10.42.
Find the value of
.
Using the order of operations and what we know about signed numbers, we get,
Calculators
Calculators with the 
key can be used for multiplying and dividing signed numbers.
Sample Set D
Use a calculator to find each quotient or product.
Example 10.43.
( − 186)⋅( − 43)
Since this product involves a (negative)⋅(negative), we know the result should be a positive number. We'll illustrate this on the calculator.
| Display Reads | ||
| Type | 186 | 186 |
| Press |
 | -186 |
| Press | × | -186 |
| Type | 43 | 43 |
| Press |
 | -43 |
| Press | = | 7998 |
Thus, ( − 186)⋅( − 43) = 7,998.
Example 10.44.
. Round to one decimal place.
| Display Reads | ||
| Type | 158.64 | 158.64 |
| Press | ÷ | 158.64 |
| Type | 54.3 | 54.3 |
| Press |
 | -54.3 |
| Press | = | -2.921546961 |
Rounding to one decimal place we get -2.9.
Practice Set D
Use a calculator to find each value.
Exercises
Find the value of each of the following. Use a calculator to check each result.
Exercise 10.7.16.
( − 3)( − 9)
Exercise 10.7.18.
( − 5)( − 2)
Exercise 10.7.20.
(4)( − 18)
Exercise 10.7.22.
( − 6)(4)
Exercise 10.7.24.
( − 8)(7)
Exercise 10.7.26.
Exercise 10.7.28.
Exercise 10.7.30.
Exercise 10.7.32.
Exercise 10.7.34.
14 − ( − 20)
Exercise 10.7.36.
− 4 − ( − 1)
Exercise 10.7.38.
0 − ( − 1)
Exercise 10.7.40.
15 − 12 − 20
Exercise 10.7.42.
2 + 7 − 10 + 2
Exercise 10.7.44.
8(5 − 12)
Exercise 10.7.46.
− 8(4 − 12) + 2
Exercise 10.7.48.
− 9(0 − 2) + 4(8 − 9) + 0( − 3)
Exercise 10.7.50.
Exercise 10.7.52.
Exercise 10.7.54.
− 1(4 + 2)
Exercise 10.7.56.
− (8 + 21)
Exercises for Review
Exercise 10.7.60.
(Section 6.2) Write this number in decimal form using digits: “fifty-two three-thousandths”
Exercise 10.7.61. (Go to Solution)
(Section 7.4) The ratio of chlorine to water in a solution is 2 to 7. How many mL of water are in a solution that contains 15 mL of chlorine?
Solutions to Exercises
10.8. Summary of Key Concepts*
Summary of Key Concepts
A variable is a letter or symbol that represents any member of a set of two or more numbers. A constant is a letter or symbol that represents a specific number. For example, the Greek letter π (pi) represents the constant 3.14159 . . . .
The real number line allows us to visually display some of the numbers in which we are interested.
The number associated with a point on the number line is called the coordinate of the point. The point associated with a number is called the graph of the number.
A real number is any number that is the coordinate of a point on the real number line.
The set of real numbers has many subsets. The ones of most interest to us are: The natural numbers: {1, 2, 3, 4, . . .} The whole numbers: {0, 1, 2, 3, 4, . . .} The integers: {. . . ,-3,-2,-1,0, 1, 2, 3, . . .} The rational numbers: {All numbers that can be expressed as the quotient of two integers.}
A number is denoted as positive if it is directly preceded by a plus sign (+) or no sign at all. A number is denoted as negative if it is directly preceded by a minus sign (–).
Opposites are numbers that are the same distance from zero on the number line but have opposite signs. The numbers a and − a are opposites.
− ( − a) = a
The absolute value of a number a , denoted ∣ a ∣ , is the distance from a to 0 on the number line.
To add two numbers with
like signs, add the absolute values of the numbers and associate with the sum the common sign.
unlike signs, subtract the smaller absolute value from the larger absolute value and associate with the difference the sign of the larger absolute value.
0 + (any number) = that particular number.
Since adding 0 to any real number leaves that number unchanged, 0 is called the additive identity.
a − b = a + ( − b)
To perform the subtraction a − b , add the opposite of b to a , that is, change the sign of b and follow the addition rules (Section 10.5).
(
+
)
(
+
)
=
(
+
)
(
−
)
(
−
)
=
(
+
)
(
+
)
(
−
)
=
(
−
)
(
−
)
(
+
)
=
(
−
)
10.9. Exercise Supplement *
Exercise Supplement
Variables, Constants, and Real Numbers (Section 10.2)
For problems 1-5, next to each real number, note all subsets of the real numbers to which it belongs by writing N for natural numbers, W for whole numbers, or Z for integers. Some numbers may belong to more than one subset.
Exercise 10.9.2.
Exercise 10.9.4.
1
Exercise 10.9.6.
Write all the integers that are between and including
and
For each pair of numbers in problems 7-10, write the appropriate symbol (<, >, =) in place of the □.
Exercise 10.9.8.
0 □ 2
Exercise 10.9.10.
-1 □ 0
For problems 11-15, what numbers can replace x so that each statement is true?
Exercise 10.9.12.
, x is a whole number.
Exercise 10.9.14.
, x is a natural number
For problems 16-20, how many units are there between the given pair of numbers?
Exercise 10.9.16.
0 and 4
Exercise 10.9.18.
and
Exercise 10.9.20.
Exercise 10.9.21. (Go to Solution)
A number is positive if it is directly preceded by a _______________ sign or no sign at all.
Exercise 10.9.22.
A number is negative if it is directly preceded by a _______________ sign.
Signed Numbers (Section 10.3)
For problems 23-26, how should each number be read?
Exercise 10.9.24.
Exercise 10.9.26.
For problems 27-31, write each expression in words.
Exercise 10.9.28.
Exercise 10.9.30.
For problems 32-36, rewrite each expression in simpler form.
Exercise 10.9.32.
Exercise 10.9.34.
Exercise 10.9.36.
Absolute Value (Section 10.4)
For problems 37-52, determine each value.
Exercise 10.9.38.
Exercise 10.9.40.
Exercise 10.9.42.
Exercise 10.9.44.
Exercise 10.9.46.
Exercise 10.9.48.
Exercise 10.9.50.
Exercise 10.9.52.
Addition, Subtraction, Multiplication and Division of Signed Numbers (Section 10.5,Section 10.6,Section 10.7)
For problems 53-71, perform each operation.
Exercise 10.9.54.
Exercise 10.9.56.
Exercise 10.9.58.
Exercise 10.9.60.
Exercise 10.9.62.
Exercise 10.9.64.
Exercise 10.9.66.
Exercise 10.9.68.
Exercise 10.9.70.
Solutions to Exercises
10.10. Proficiency Exam*
Proficiency Exam
Exercise 10.10.1. (Go to Solution)
(Section 10.2) Write all integers that are strictly between –8 and –3.
Exercise 10.10.2. (Go to Solution)
(Section 10.2) Write all integers that are between and including –2 and 1.
For problems 3-5, write the appropriate symbol (<, >, =) in place of the □ for each pair of numbers.
For problems 6 and 7, what numbers can replace
so that the statement is true?
For problems 9-20, find each value.
Solutions to Exercises
[1] In later mathematics courses, the word "suggests" turns into the word "proof." One example does not prove a claim. Mathematical proofs are constructed to validate a claim for all possible cases.