What You Should Learn
1 䉴Multiply integers with like signs and with unlike signs.
2䉴Divide integers with like signs and with unlike signs.
3䉴Find factors and prime factors of an integer.
4䉴Represent the definitions and rules of arithmetic symbolically.
Why You Should Learn It You can multiply and divide integers to solve real-life problems. For instance, in Exercise 109 on page 30, you will divide integers to find the average speed of a space shuttle.
1 䉴Multiply integers with like signs and with unlike signs.
NASA
Multiplying Integers
Multiplicationof two integers can be described as repeated addition or subtraction.
The result of multiplying one number by another is called a product.Here are three examples.
Multiplication Repeated Addition or Subtraction
Add 5 three times.
Add four times.
Subtract three times.
Multiplication is denoted in a variety of ways. For instance, and
all denote the product of “7 times 3,” which is 21.
To find the product of more than two numbers, first find the product of their absolute values. If the number of negative factors is even, then the product is positive. If the number of negative factors is odd, then the product is negative. For instance, the expression
Expression has 2 negative factors.
is positive because it has an even number of negative factors.
5共3兲共4兲共7兲
共7兲共3兲 共7兲3,
7共3兲, 73,
73,
4
共4兲共4兲共4兲12
共3兲共4兲12
2
共2兲共2兲共2兲共2兲 8
4共2兲 8
55515 3515
Rules for Multiplying Integers 1. The product of an integer and zero is 0.
2. The product of two integers with likesigns is positive.
3. The product of two integers with unlikesigns is negative.
CHECKPOINT CHECKPOINT
EXAMPLE 1 Multiplying Integers
a.
b.
c.
d.
e.
f. Odd number of negative factors
Answer is negative.
Now try Exercise 11.
Be careful to distinguish properly between expressions such as and or and The first of each pair is a multiplicationproblem, whereas the second is a subtractionproblem.
Multiplication Subtraction
To multiply two integers having two or more digits, we suggest the vertical multiplication algorithmdemonstrated in Figure 1.27. The sign of the product is determined by the usual multiplication rule.
EXAMPLE 2 Geometry: Volume of a Box
Find the volume of the rectangular box shown in Figure 1.28.
Figure 1.28 Solution
To find the volume, multiply the length, width, and height of the box.
So, the box has a volume of 900 cubic inches.
Now try Exercise 113.
900 cubic inches
共15 inches兲共12 inches兲共5 inches兲
Volume共Length兲共Width兲共Height兲
5 in.
12 in.
15 in.
35 8 3共5兲15
35 2
3共5兲 15
35.
3共5兲 35
3共5兲 48
2共8兲共3兲共1兲 共2831兲
共Negative兲共zero兲zero 1200
共Positive兲共negative兲negative
3共12兲 36
共Negative兲共negative兲positive 5共7兲35
共Negative兲共positive兲negative 69 54
共Positive兲共positive兲positive 4共10兲40
Figure 1.27 Vertical Multiplication Algorithm
47 23 141 94 1081
Multiply 3 times 47.
Multiply 2 times 47.
Add columns.
⇐
⇐
⇐
CHECKPOINT
Dividing Integers
Just as subtraction can be expressed in terms of addition, you can express division in terms of multiplication. Here are some examples.
Division Related Multiplication because
because because because
The result of dividing one integer by another is called the quotient of the integers. Division is denoted by the symbol or by or by a horizontal line.
For instance,
and
all denote the quotient of 30 and 6, which is 5. Using the form 30 is called the dividend and 6 is the divisor. In the forms and 30 is the numeratorand 6 is the denominator.
It is important to know how to use 0 in a division problem. Zero divided by a nonzero integer is always 0. For instance,
because
On the other hand, division by zero, such as is undefined.
Because division can be described in terms of multiplication, the rules for dividing two integers with like or unlike signs are the same as those for multiplying such integers.
EXAMPLE 3 Dividing Integers
a. because
b. because
c. because
d. because
e. is undefined.
Now try Exercise 43.
970
共15兲共7兲 105.
1057 15
共0兲共13兲0.
0共13兲0
共4兲共9兲36.
36共9兲 4
427共6兲. 42
6 7
130, 0013.
0 130
30 6, 30兾6
306, 30
30兾6, 6 306,
兾, ,
155共3兲
15共3兲5
15共5兲共3兲
15共3兲5
1553
1535
1553
1535
2 䉴Divide integers with like signs and with unlike signs.
Technology: Discovery Does Does Write each division above in terms of multiplication. What does this tell you about division by zero? What does your calculator display when you perform the division?
2 00?
1 00?
Rules for Dividing Integers
1. Zero divided by a nonzero integer is 0, whereas a nonzero integer divided by zero is undefined.
2. The quotient of two nonzero integers with likesigns is positive.
3. The quotient of two nonzero integers with unlikesigns is negative.
CHECKPOINT
When dividing large numbers, the long division algorithm can be used. For instance, the long division algorithm in Figure 1.29 shows that
Remember that division can be checked by multiplying the answer by the divisor.
So, it is true that
because
All four operations on integers (addition, subtraction, multiplication, and division) are used in the following real-life example.
EXAMPLE 4 Stock Purchase
On Monday you bought $500 worth of stock in a company. During the rest of the week, you recorded the gains and losses in your stock’s value, as shown in the table.
a. What was the value of the stock at the close of Wednesday?
b. What was the value of the stock at the end of the week?
c. What would the total loss have been if Thursday’s loss had occurred on each of the four days?
d.What was the average daily gain (or loss) for the four days recorded?
Solution
a.The value at the close of Wednesday was
b.The value of the stock at the end of the week was
c. The loss on Thursday was $23. If this loss had occurred each day, the total loss would have been
d.To find the average daily gain (or loss), add the gains and losses of the four days and divide by 4.
This means that during the four days, the stock had an average loss of $4 per day.
Now try Exercise 103.
Average 15共18兲共23兲10
4 16
4 4
4共23兲$92.
50015182310$484.
5001518$497.
Tuesday Wednesday Thursday Friday Gained $15 Lost $18 Lost $23 Gained $10
27共13兲351.
3511327 3511327.
To find the averageof numbers, add the numbers and divide the result by n.
n Study Tip
Figure 1.29 Long Division Algorithm 9 1
9 1 2 6 13) 3 5 1 2 7
Factors and Prime Numbers
The set of positive integers
is one subset of the real numbers that has intrigued mathematicians for many centuries.
Historically, an important number concept has been factors of positive integers. In a multiplication problem such as the numbers 3 and 7 are called factorsof 21.
Factors Product
It is also correct to call the numbers 3 and 7 divisorsof 21, because 3 and 7 each divide evenly into 21.
The concept of factors allows you to classify positive integers into three groups:
primenumbers,compositenumbers, and the number 1.
The numbers 2, 3, 5, 7, and 11 are primes because they have only themselves and 1 as factors. The numbers 4, 6, 8, 9, and 10 are composites because each has more than two factors. The number 1 is neither prime nor composite because 1 is its only factor.
Every composite number can be expressed as a unique product of prime factors. Here are some examples.
According to the definition of a prime number, is it possible for any negative number to be prime? Consider the number Is it prime? Are its only factors 1 and itself? No, because
or 2共1兲共1兲共2兲.
2共1兲共2兲, 21共2兲,
2.
1242231
42237,
18233,
1535,
623,
3721
3721,
再1, 2, 3, . . .冎
3 䉴Find factors and prime factors of an integer.
Definition of Factor (or Divisor)
If and are positive integers, then is a factor(or divisor) of if and only if there is a positive integer such that ac cb.
b a
b a
Definitions of Prime and Composite Numbers 1. A positive integer greater than 1 with no factors other than itself and 1 is
called a prime number,or simply a prime.
2. A positive integer greater than 1 with more than two factors is called a composite number,or simply a composite.
CHECKPOINT
One strategy for factoring a composite number into prime factors is to begin by finding the smallest prime number that is a factor of the composite number.
Dividing this factor into the number yields a companionfactor. For instance, 3 is the smallest prime number that is a factor of 45, and its companion factor is 15 because Continue identifying factors and companion factors until each factor is prime. As shown in Figure 1.30, a tree diagramis a nice way to record your work. From the tree diagram, you can see that the prime factorization of 45 is
EXAMPLE 5 Prime Factorization
Write the prime factorization of each number.
a. 84 b. 78 c. 133 d. 43 Solution
a. 2 is a recognized divisor of 84. So,
b. 2 is a recognized divisor of 78. So,
c. If you do not recognize a divisor of 133, you can start by dividing any of the prime numbers 2, 3, 5, 7, 11, 13, etc., into 133. You will find 7 to be the first prime to divide 133. So,
d.In this case, none of the primes less than 43 divides 43. So, 43 is prime.
Now try Exercise 85.
To say that a number is divisibleby nmeans that ndivides into the number without leaving a remainder. Other aids to finding prime factors of a number include the following divisibility tests.
133719.
782392313.
8424222212237.
45335.
15453.
45
3
3 5
15 3
Figure 1.30 Tree Diagram
Divisibility Tests
Test Example
1. A number is divisible by 2 if 364 is divisible by 2 because
it is even. it is even.
2. A number is divisible by 3 if the 261 is divisible by 3 because sum of its digits is divisible by 3.
3. A number is divisible by 9 if the 738 is divisible by 9 because sum of its digits is divisible by 9.
4. A number is divisible by 5 if its 325 is divisible by 5 because units digit is 0 or 5. its units digit is 5.
5. A number is divisible by 10 if its 120 is divisible by 10 because units digit is 0. its units digit is 0.
73818.
2619.
CHECKPOINT
Summary of Definitions and Rules
So far in this chapter, rules and procedures have been described more with words than with symbols. For instance, subtraction is verbally defined as “adding the opposite of the number being subtracted.” As you move to higher and higher levels of mathematics, it becomes more and more convenient to use symbols to describe rules and procedures. For instance, subtraction is symbolically defined as
At its simplest level, algebra is a symbolic form of arithmetic. This arithmetic–
algebra connection can be illustrated in the following way.
Arithmetic Algebra
An illustration of this connection is shown in Example 6.
EXAMPLE 6 Writing a Rule of Arithmetic in Symbolic Form
Write an example and an algebraic description of the arithmetic rule:
The product of two integers with unlike signs is negative.
Solution
Example
For the integers and 7,
Also, for the integers 3 and
Algebraic Description If and are positive integers, then
Unlike Negative
signs product
and
Unlike Negative
signs product
Now try Exercise 95.
The list on the following page summarizes the algebraic versions of important definitions and rules of arithmetic. In each case, a specific example is included for clarification.
a共b兲 共ab兲.
共a兲b 共ab兲
b a
3共7兲 21.
7, 37 21.
3 aba共b兲.
4 䉴Represent the definitions and rules of arithmetic symbolically.
Symbolic rules and definitions Verbal rules and definitions
Specific examples of rules and definitions
Arithmetic Summary Definitions:Let and be integers.
Definition Example
1. Subtraction:
2. Multiplication: ( is a positive integer)
aterms
3. Division:
if and only if because
4. Less than:
if there is a positive real number such that because 5. Absolute value:
6. Divisor:
is a divisor of if and only if there is an integer such that 7 is a divisor of 21 because
Rules: Let and be integers.
Rule Example
1. Addition:
(a) To add two integers with likesigns, add their absolute values and attach the common sign to the result.
(b) To add two integers with unlikesigns, subtract the smaller absolute value from the larger absolute value and attach the sign of the integer with the larger absolute value.
2. Multiplication:
(a)
(b) Like signs:
(c) Unlike signs:
3. Division:
(a)
(b) is undefined. is undefined.
(c) Like signs:
(d) Unlike signs: 5
7 5 7 a
b < 0
2 32
3 a
b > 0
6 0 a
0
0 40 0
a0
共2兲共5兲 10
ab < 0
共2兲共5兲10 ab > 0
30003
a000a
37ⱍ3ⱍⱍ7ⱍ10
b a
7321.
acb. b c
a
ⱍ3ⱍ 共3兲3
ⱍaⱍ冦a,a, if a if a < 00
231.
2 < 1 acb.
c a < b
1234.
1243 acb.
abc 共b0兲
35555
abbb. . . b
a
575共7兲 aba共b兲
c a, b,
3 85 58ⱍ8ⱍⱍ5ⱍ
CHECKPOINT CHECKPOINT
EXAMPLE 7 Using Definitions and Rules
a. Use the definition of subtraction to complete the statement.
b. Use the definition of multiplication to complete the statement.
c. Use the definition of absolute value to complete the statement.
d.Use the rule for adding integers with unlike signs to complete the statement.
e. Use the rule for multiplying integers with unlike signs to complete the statement.
Solution a.
b.
c.
d.
e.
Now try Exercise 97.
EXAMPLE 8 Finding a Pattern
Complete each pattern. Decide which rules the patterns demonstrate.
a. b.
Solution
a. b.
The product of integers with unlike signs is negative, and the product of integers with like signs is positive.
Now try Exercise 101.