Download SECTION 1-1 Algebra and Real Numbers

2
1 Basic Algebraic Operations
Algebra is often referred to as “generalized arithmetic.” In arithmetic we
deal with the basic arithmetic operations of addition, subtraction, multiplication, and division performed on specific numbers. In algebra we continue to use all that we know in arithmetic, but, in addition, we reason
and work with symbols that represent one or more numbers. In this chapter we review some important basic algebraic operations usually studied
in earlier courses. The material may be studied systematically before commencing with the rest of the book or reviewed as needed.
SECTION
1-1
Algebra and Real Numbers
•
•
•
•
•
•
Sets
The Set of Real Numbers
The Real Number Line
Basic Real Number Properties
Further Properties
Fraction Properties
The rules for manipulating and reasoning with symbols in algebra depend, in large
measure, on properties of the real numbers. In this section we look at some of the
important properties of this number system. To make our discussions here and elsewhere in the text clearer and more precise, we first introduce a few useful notions
about sets.
• Sets
Georg Cantor (1845–1918) developed a theory of sets as an outgrowth of his studies
on infinity. His work has become a milestone in the development of mathematics.
Our use of the word “set” will not differ appreciably from the way it is used in
everyday language. Words such as “set,” “collection,” “bunch,” and “flock” all convey the same idea. Thus, we think of a set as a collection of objects with the important property that we can tell whether any given object is or is not in the set.
Each object in a set is called an element, or member, of the set. Symbolically,
aA
means
“a is an element of set A”
3 {1, 3, 5}
aA
means
“a is not an element of set A”
2 {1, 3, 5}
Capital letters are often used to represent sets and lowercase letters to represent elements of a set.
A set is finite if the number of elements in the set can be counted and infinite
if there is no end in counting its elements. A set is empty if it contains no elements.
The empty set is also called the null set and is denoted by . It is important to observe
that the empty set is not written as {}.
A set is usually described in one of two ways—by listing the elements between
braces, { }, or by enclosing within braces a rule that determines its elements. For
example, if D is the set of all numbers x such that x2 4, then using the listing
method we write
D {2, 2}
Listing method
1-1
Algebra and Real Numbers
3
or, using the rule method we write
D {x
x
2
4}
Rule method
Note that in the rule method, the vertical bar represents “such that,” and the entire
symbolic form {x x2 4} is read, “The set of all x such that x2 4.”
The letter x introduced in the rule method is a variable. In general, a variable is
a symbol that is used as a placeholder for the elements of a set with two or more elements. This set is called the replacement set for the variable. A constant, on the
other hand, is a symbol that names exactly one object. The symbol “8” is a constant,
since it always names the number eight.
If each element of set A is also an element of set B, we say that A is a subset
of set B, and we write
AB
{1, 5} {1, 3, 5}
Note that the definition of a subset allows a set to be a subset of itself.
Since the empty set has no elements, every element of is also an element
of any given set. Thus, the empty set is a subset of every set. For example,
{1, 3, 5}
and
{2, 4, 6}
If two sets A and B have exactly the same elements, the sets are said to be equal,
and we write
AB
{4, 2, 6} {6, 4, 2}
Notice that the order of listing elements in a set does not matter.
We can now begin our discussion of the real number system. Additional set concepts will be introduced as needed.
• The Set of
Real Numbers
The real number system is the number system you have used most of your life. Informally, a real number is any number that has a decimal representation. Table 1 on the
next page describes the set of real numbers and some of its important subsets. Figure 1 illustrates how these sets of numbers are related to each other.
Figure 1 Real numbers and important subsets.
Real numbers (R )
NZQR
Rational numbers (Q ) Irrational numbers (I )
Integers (Z )
Natural numbers (N )
Zero
Noninteger ratios of integers
Negatives of natural numbers
4
1 Basic Algebraic Operations
TABLE 1
Symbol
The Set of Real Numbers
Name
Description
Examples
N
Natural numbers
Counting numbers (also called positive
integers)
1, 2, 3, . . .
Z
Integers
Natural numbers, their negatives, and 0
. . . , 2, 1, 0, 1, 2, . . .
Q
Rational numbers
Numbers that can be represented as a/b,
where a and b are integers and b 0;
decimal representations are repeating or
terminating
I
Irrational numbers
Numbers that can be represented as
nonrepeating and nonterminating decimal
numbers
R
Real numbers
Rational numbers and irrational numbers
2
4, 0, 1, 25, 3
5 , 3 , 3.67, 0.333,*
5.272727
3
2, ,7, 1.414213 . . . ,
2.71828182 . . .
*The overbar indicates that the number (or block of numbers) repeats indefinitely.
• The Real
Number Line
A one-to-one correspondence exists between the set of real numbers and the set of
points on a line. That is, each real number corresponds to exactly one point, and each
point to exactly one real number. A line with a real number associated with each point,
and vice versa, as in Figure 2, is called a real number line, or simply a real line.
Each number associated with a point is called the coordinate of the point. The point
with coordinate 0 is called the origin. The arrow on the right end of the line indicates a positive direction. The coordinates of all points to the right of the origin are
called positive real numbers, and those to the left of the origin are called negative
real numbers. The real number 0 is neither positive nor negative.
Figure 2 A real number line.
27
10
• Basic Real
Number Properties
4
3 Origin
5
0
7.64
5
10
We now take a look at some of the basic properties of real numbers. (See the box on
the next page.)
You are already familiar with the commutative properties for addition and multiplication. They indicate that the order in which the addition or multiplication of two
numbers is performed doesn’t matter. For example,
4554
and
4554
Is there a commutative property relative to subtraction or division? That is, does
x y y x or does x y y x for all real numbers x and y (division by 0
excluded)? The answer is no, since, for example,
7557
and
6336
1-1
Algebra and Real Numbers
5
Basic Properties of the Set of Real Numbers
Let R be the set of real numbers, and let x, y, and z be arbitrary elements of R.
Addition Properties
Closure:
x y is a unique element in R.
Associative:
(x y) z x (y z)
Commutative:
xyyx
Identity:
0 is the additive identity; that is, 0 x x 0 x for
all x in R, and 0 is the only element in R with this
property.
Inverse:
For each x in R, x is its unique additive inverse; that
is, x (x) (x) x 0, and x is the only element in R relative to x with this property.
Multiplication Properties
Closure:
xy is a unique element in R.
Associative:
(xy)z x(yz)
Commutative:
xy yx
Identity:
1 is the multiplicative identity; that is, for x in R,
(1)x x(1) x, and 1 is the only element in R with this
property.
Inverse:
For each x in R, x 0, 1/x is its unique multiplicative
inverse; that is, x(1/x) (1/x)x 1, and 1/x is the only
element in R relative to x with this property.
Combined Property
Distributive:
x(y z) xy xz
(x y)z xz yz
When computing
253
or
253
why don’t we need parentheses to indicate which two numbers are to be added or
multiplied first? The answer is to be found in the associative properties. These properties allow us to write
(2 5) 3 2 (5 3)
and
(2 5) 3 2 (5 3)
so it doesn’t matter how we group numbers relative to either operation. Is there an
associative property for subtraction or division? The answer is no, since, for example,
6
1 Basic Algebraic Operations
(8 4) 2 8 (4 2)
(8 4) 2 8 (4 2)
and
Evaluate both sides of these equations to see why.
Conclusion
Relative to addition, commutativity and associativity permit us to change the
order of addition at will and insert or remove parentheses as we please. The
same is true for multiplication, but not for subtraction and division.
What number added to a given number will give that number back again? What
number times a given number will give that number back again? The answers are 0
and 1, respectively. Because of this, 0 and 1 are called the identity elements for the
real numbers. Hence, for any real numbers x and y,
707
0 (x y) x y
0 is the additive identity.
166
1(x y) x y
1 is the multiplicative identity.
We now consider inverses. For each real number x, there is a unique real number x such that x (x) 0. The number x is called the additive inverse of x,
or the negative of x. For example, the additive inverse of 4 is 4, since 4 (4) 0. The additive inverse of 4 is (4) 4, since 4 [(4)] 0. It
is important to remember:
x is not necessarily a negative number; it is positive if x is negative
and negative if x is positive.
For each nonzero real number x there is a unique real number 1/x such that
x(1/x) 1. The number 1/x is called the multiplicative inverse of x, or the reciprocal of x. For example, the multiplicative inverse of 7 is 17, since 7(17) 1. Also
note that 7 is the multiplicative inverse of 17 . The number 0 has no multiplicative
inverse.
We now turn to a real number property that involves both multiplication and addition. Consider the two computations:
3(4 2) 3(6) 18
3(4) 3(2) 12 6 18
Thus,
3(4 2) 3(4) 3(2)
and we say that multiplication by 3 distributes over the sum (4 2). In general, multiplication distributes over addition in the real number system. Two more illustrations are given below:
2(x y) 2x 2y
(3 5)x 3x 5x
1-1
EXAMPLE 1
Algebra and Real Numbers
7
Using Real Number Properties
Which real number property justifies the indicated statement?
Statement
(A) (7x)y 7(xy)
(B) a(b c) (b c)a
(C) (2x 3y) 5y 2x (3y 5y)
(D) (x y)(a b) (x y)a (x y)b
(E) If a b 0, then b a.
Matched Problem 1*
Which real number property justifies the indicated statement?
(A) 4 (2 x) (4 2) x
(C) 3x 7x (3 7)x
(E) If ab 1, then b 1/a.
• Further Properties
DEFINITION 1
Property Illustrated
Associative ()
Commutative ()
Associative ()
Distributive
Inverse ()
(B) (a b) c c (a b)
(D) (2x 3y) 0 2x 3y
Subtraction and division can be defined in terms of addition and multiplication,
respectively:
Subtraction and Division
For all real numbers a and b:
Subtraction:
a b a (b)
Division:
ba a b (5) (3) (5) (3) 2
a
1
a
b
b
b0
12 323
Thus, to subtract b from a, add the negative of b to a. To divide a by b, multiply a
by the reciprocal of b. Note that division by 0 is not defined, since 0 does not have
a reciprocal. It is important to remember:
Division by 0 is never allowed.
The following properties of negatives can be proved using the preceding properties and definitions.
*Answers to matched problems in a given section are found near the end of the section, before the
exercise set.
8
1 Basic Algebraic Operations
Theorem 1
Properties of Negatives
For all real numbers a and b:
1. (a) a
2. (a)b (ab) a(b) ab
3. (a)(b) ab
4. (1)a a
5.
a
a
a
b
b b
6.
a
a
a
a
b
b
b b
b0
b0
We now state an important theorem involving 0.
Theorem 2
Zero Properties
For all real numbers a and b:
1. a 0 0
2. ab 0
EXAMPLE 2
if and only if
a 0 or b 0 or both
Using Negative and Zero Properties
Which real number property or definition justifies each statement?
Statement
(A) 3 (2) 3 [(2)] 5
(B) (2) 2
3 3
(C) 2
2
5
5
(D)
2
2
(E) If (x 3)(x 5) 0, then either
x 3 0 or x 5 0.
Property or Definition Illustrated
Subtraction (Definition 1 and
Theorem 1, part 1)
Negatives (Theorem 1, part 1)
Negatives (Theorem 1, part 6)
Negatives (Theorem 1, part 5)
Zero (Theorem 2, part 2)
1-1
Matched Problem 2
9
Which real number property or definition justifies each statement?
3
1
3
5
5
7
7
(D)
9
9
(A)
EXPLORE-DISCUSS 1
Algebra and Real Numbers
(B) (5)(2) (5 2)
(C) (1)3 3
(E) If x 5 0, then (x 3)(x 5) 0.
In general, a set of numbers is closed under an operation if performing the operation on numbers in the set always produces another number in the set. For example, the real numbers are closed under addition, multiplication, subtraction, and
division, excluding division by 0. Replace each ? in the following tables with T
(true) or F (false), and illustrate each false statement with an example. (See Table
1 for the definitions of the sets N, Z, I, Q, and R.)
Closed under Addition
Closed under Multiplication
N
?
?
Z
?
?
Q
?
?
I
?
?
R
T
T
Closed under Subtraction
Closed under Division*
N
?
?
Z
?
?
Q
?
?
I
?
?
R
T
T
*Excluding division by 0.
• Fraction Properties
Recall that the quotient a b, b 0, written in the form a/b is called a fraction.
The quantity a is called the numerator and the quantity b is the denominator.
10
1 Basic Algebraic Operations
Theorem 3
Fraction Properties
For all real numbers a, b, c, d, and k (division by 0 excluded):
1.
a c
b d
4 6
6 9
2.
ad bc
if and only if
since
ka a
kb b
4966
3.
73 3
75 5
5.
a c ac
b b
b
3 5 35
6 6
6
a c
ac
b d bd
4.
3 7 37
5 8 58
6.
a c a d
b d b c
2 5 2 7
3 7 3 5
a c ac
b b
b
7.
7 3 73
8 8
8
a c ad bc
b d
bd
2 3 2533
3 5
35
Answers to Matched Problems
1. (A) Associative ()
(B) Commutative ()
(C) Distributive
(D) Identity ()
(E) Inverse ()
2. (A) Division (Definition 1)
(B) Negatives (Theorem 1, part 2)
(C) Negatives (Theorem 1, part 4)
(D) Negatives (Theorem 1, part 5)
(E) Zero (Theorem 2, part 1)
EXERCISE
1-1
All variables represent real numbers.
13. Identity property (): 0 9m ?
14. Identity property (): 1(u v) ?
A
In Problems 1–8, indicate true (T) or false (F).
1. 4 {3, 4, 5}
2. 6 {2, 4, 6}
3. 3 {3, 4, 5}
4. 7 {2, 4, 6}
5. {1, 2} {1, 3, 5}
6. {2, 6} {2, 4, 6}
7. {7, 3, 5} {3, 5, 7}
8. {7, 3, 5} {3, 5, 7}
In Problems 15–26, each statement illustrates the use of one
of the following properties or definitions. Indicate which one.
Commutative (, )
Associative (, )
Distributive
Identity (, )
Inverse (, )
Subtraction
Division
Negatives (Theorem 1)
Zero (Theorem 2)
15. x ym x my
16. 7(3m) (7 3)m
17. 7u 9u (7 9)u
18. 1
19. (2)(2
)1
20. 8 12 8 (12)
10. Commutative property (): uv ?
21. w (w) 0
1
22. 5 (6) 5(6
)
11. Associative property (): x(yz) ?
23. 3(xy z) 0 3(xy z)
12. Associative property (): 3 (7 y) ?
24. ab(c d ) abc abd
In Problems 9–14, replace each question mark with an
appropriate expression that will illustrate the use of the
indicated real number property.
9. Commutative property (): x 7 ?
u
u
v v
1-1
25.
x x
y y
26. (x y) 0 0
B
Write each set in Problems 27–32 using the listing method;
that is, list the elements between braces. If the set is empty,
write .
{x x is a letter in “status”}
{x x is a letter in “consensus”}
{x x is a month starting with B}
{x x is a month with 32 days}
27. {x x is an even integer between 3 and 5}
28. {x x is an odd integer between 4 and 6}
29.
30.
31.
32.
33. The set S1 {a} has only two subsets, S1 and . How
many subsets does each of the following sets have?
(A) S2 {a, b}
(B) S3 {a, b, c}
(C) S4 {a, b, c, d}
34. Based on the results in Problem 33, how many subsets do
you think a set with n elements will have?
In Problems 35–42, each statement illustrates the use of one
of the following properties or definitions. Indicate which one.
Commutative (, )
Subtraction
Associative (, )
Division
Distributive
Negatives (Theorem 1)
Identity (, )
Zero (Theorem 2)
Inverse (, )
35. (y) 2x y 2x
36. (ab)(ba) (ab)(ab)
37. (wz)(zw) w[z(zw)]
Algebra and Real Numbers
11
46. Indicate which of the following are true:
(A) All integers are natural numbers.
(B) All rational numbers are real numbers.
(C) All natural numbers are rational numbers.
47. Give an example of a rational number that is not an integer.
48. Give an example of a real number that is not a rational
number.
In Problems 49 and 50, list the subset of S consisting of
(A) natural numbers, (B) integers, and (C) rational numbers.
49. S {3, 23, 0, 1, 3, 95, 12}
50. S {5 , 1,12, 2, 7 , 6, 25
3}
In Problems 51 and 52, use a calculator* to express each
number as a decimal fraction. Classify each decimal number
as terminating, repeating, or nonrepeating and nonterminating. Identify the pattern of repeated digits in any repeating
decimal numbers.
51. (A)
8
9
52. (A)
13
6
(B)
3
11
(B) 21
(C) 5
(C)
(D)
7
16
11
8
(D)
29
111
53. Indicate true (T) or false (F), and for each false statement
find real number replacements for a and b that will provide
a counterexample. For all real numbers a and b:
(A) a b b a
(B) a b b a
(C) ab ba
(D) a b b a
54. Indicate true (T) or false (F), and for each false statement
find real number replacements for a, b, and c that will provide a counterexample. For all real numbers a, b, and c:
(A) (a b) c a (b c)
(B) (a b) c a (b c)
(C) a(bc) (ab)c
(D) (a b) c a (b c)
38. s (t 2) (s t) 2
39. (n 2)(m 3) n(m 3) 2(m 3)
C
40. p(r 1) q(r 1) (p q)(r 1)
55. If A {1, 2, 3, 4} and B {2, 4, 6}, find:
(A) {x x A or x B}
(B) { x x A and x B}
41. (2x 3)(3x 5) 0 if and only if 2x 3 0 or
3x 5 0
42.
y
y
(1 y) 1 y
43. If ab 0, does either a or b have to be 0?
56. If F {2, 0, 2} and G {1, 0, 1, 2}, find:
(A) {x x F or x G}
(B) { x x F and x G}
44. If ab 1, does either a or b have to be 1?
45. Indicate which of the following are true:
(A) All natural numbers are integers.
(B) All real numbers are irrational.
(C) All rational numbers are real numbers.
*Later in the book you will encounter optional exercises that require
a graphing calculator. If you have such a calculator, you can certainly use it here. Otherwise, any scientific calculator will be sufficient for the problems in this chapter.
12
1 Basic Algebraic Operations
57. If c 0.151515 . . . , then 100c 15.1515 . . . and
59. To see how the distributive property is behind the mechanics of long multiplication, compute each of the following
and compare:
100c c 15.1515 . . . 0.151515 . . .
99c 15
Long
Multiplication
23
12
5
c 15
99 33
Proceeding similarly, convert the repeating decimal
0.090909 . . . into a fraction. (All repeating decimals are rational numbers, and all rational numbers have repeating
decimal representations.)
60. For a and b real numbers, justify each step using a property
in this section.
58. Repeat Problem 57 for 0.181818. . . .
SECTION
1-2
Use of the
Distributive Property
23 12
23(2 10)
23 2 23 10 Statement
1. (a b) (a) (a) (a b)
2.
[(a) a] b
3.
0b
4.
b
Reason
1.
2.
3.
4.
Polynomials: Basic Operations
•
•
•
•
•
•
•
Natural Number Exponents
Polynomials
Combining Like Terms
Addition and Subtraction
Multiplication
Combined Operations
Application
In this section we review the basic operations on polynomials, a mathematical form
encountered frequently throughout mathematics. We start the discussion with a brief
review of natural number exponents. Integer and rational exponents and their properties will be discussed in detail in subsequent sections.
• Natural Number
The definition of a natural number exponent is given below:
Exponents
DEFINITION 1
Natural Number Exponent
For n a natural number and a any real number:
an a a · · · a
agfgbggfc
n factors of a
24 2 2 2 2
4 factors of 2