Transcript Slide 1

Real Numbers
and Their Basic Properties
Copyright © Cengage Learning. All rights reserved.
1
Section
1.7
Properties of Real Numbers
Copyright © Cengage Learning. All rights reserved.
Objectives
1
Apply the closure properties by evaluating an
expression for given values for variables.
2
Apply the commutative and associative
properties.
3
Apply the distributive property of multiplication
over addition to rewrite an expression.
3
Objectives
4
Recognize the identity elements and find the
additive and multiplicative inverse of a nonzero
real number.
5
Identify the property that justifies a given
statement.
4
1. Apply the closure properties by evaluating
an expression for given values for variables
5
Closure Property
The closure properties guarantee that the sum,
difference, product, or quotient (except for division by 0) of
any two real numbers is also a real number.
Closure Properties
If a and b are real numbers, then
a + b is a real number.
ab is a real number.
a – b is a real number.
is a real number (b  0).
6
Example 1
Let x = 8 and y = –4. Find the real-number answers to
show that each expression represents a real number.
a. x + y b. x – y c. xy d.
Solution:
We substitute 8 for x and –4 for y in each expression and
simplify.
a. x + y = 8 + (–4)
=4
7
Example 1 – Solution
cont’d
b. x – y = 8 – (–4)
=8+4
= 12
c. xy = 8(–4)
= –32
d.
=
= –2
8
2.
Apply the commutative and
associative properties
9
Commutative Property of Real Numbers
The commutative properties (from the word commute,
which means to go back and forth) guarantee that addition
or multiplication of two real numbers can be done in either
order.
Commutative Properties
If a and b are real numbers, then
a + b = b + a commutative property of addition
ab = ba
commutative property of multiplication
Comment
Since 5 – 3  3 – 5 and 5  3  3  5, the commutative
property cannot be applied to a subtraction or a division.
10
Example 2
Let x = –3 and y = 7. Show that a. x + y = y + x b. xy = yx
Solution:
a. We can show that the sum x + y is the same as the sum
y + x by substituting –3 for x and 7 for y in each
expression and simplifying.
x + y = –3 + 7 = 4 and y + x = 7 + (–3) = 4
b. We can show that the product xy is the same as the
product yx by substituting –3 for x and 7 for y in each
expression and simplifying.
xy = –3(7) = –21 and yx = 7(–3) = –21
11
Associative Property of Real Numbers
The associative properties guarantee that three real
numbers can be regrouped in an addition or multiplication.
Associative Properties
If a, b, and c are real numbers, then
(a + b) + c = a + (b + c) associative property of addition
(ab)c = a(bc) associative property of multiplication
Because of the associative property of addition, we can
group (or associate) the numbers in a sum in any way that
we wish.
12
Apply the commutative and associative properties
For example,
(3 + 4) + 5 = 7 + 5
= 12
3 + (4 + 5) = 3 + 9
= 12
The answer is 12 regardless of how we group the three
numbers.
The associative property of multiplication permits us to
group (or associate) the numbers in a product in any way
that we wish.
13
Apply the commutative and associative properties
For example,
(3  4)  7 = 12  7
= 84
3  (4  7) = 3  28
= 84
The answer is 84 regardless of how we group the three
numbers.
Comment
Since (2 – 5) – 3  2 – (5 – 3)
and (2  5)  3  2  (5  3),
the associative property cannot be applied to subtraction or
division.
14
3. Apply the distributive property of multiplication
over addition to rewrite an expression
15
Distributive Property of Real Numbers
The distributive property shows how to multiply the sum
of two numbers by a third number. Because of this
property, we can often add first and then multiply, or
multiply first and then add.
For example, 2(3 + 7) can be calculated in two different
ways. We will add and then multiply, or we can multiply
each number within the parentheses by 2 and then add.
2(3 + 7) = 2(10) 2(3 + 7) = 2  3 + 2  7
= 20
= 6 + 14
= 20
Either way, the result is 20.
16
Apply the distributive property of multiplication over addition
to rewrite an expression
In general, we have the following property.
Distributive Property of Multiplication Over Addition
If a, b, and c are real numbers, then
a(b + c) = ab + ac
Because multiplication is commutative, the distributive
property also can be written in the form
(b + c)a = ba + ca
17
Apply the distributive property of multiplication over addition
to rewrite an expression
We can interpret the distributive
property geometrically. Since the
area of the largest rectangle in
Figure 1-25 is the product of its
width a and its length b + c, its
area is a(b + c).
Figure 1-25
The areas of the two smaller rectangles are ab and ac.
Since the area of the largest rectangle is equal to the sum
of the areas of the smaller rectangles, we have
a(b + c) = ab + ac.
18
Apply the distributive property of multiplication over addition
to rewrite an expression
The previous discussion shows that multiplication
distributes over addition. Multiplication also distributes over
subtraction.
For example, 2(3 – 7) can be calculated in two different
ways. We will subtract and then multiply, or we can multiply
each number within the parentheses by 2 and then
subtract.
2(3 – 7) = 2(–4)
2(3 – 7) = 2  3 – 2  7
= –8
= 6 – 14
= –8
Either way, the result is –8. In general, we have
a(b – c) = ab – ac
19
Example 3
Evaluate each expression in two different ways:
a. 3(5 + 9) b. 4(6 – 11) c. –2(–7 + 3)
Solution:
a. 3(5 + 9) = 3(14)
= 42
b. 4(6 – 11) = 4(–5)
= –20
3(5 + 9) = 3  5 + 3  9
= 15 + 27
= 42
4(6 – 11) = 4  6 – 4  11
= 24 – 44
= –20
20
Example 3 – Solution
c. –2(–7 + 3) = –2(–4)
=8
cont’d
–2(–7 + 3) = –2(–7) + (–2)(3)
= 14 + (–6)
=8
21
Apply the distributive property of multiplication over addition
to rewrite an expression
The distributive property can be extended to three or more
terms.
For example, if a, b, c, and d are real numbers, then
a(b + c + d) = ab + ac + ad
22
4.
Recognize the identity elements and find the additive
and multiplicative inverse of a nonzero real number
23
Additive Identity (0) and Multicative Identity
(1)
The numbers 0 and 1 play special roles in mathematics.
The number 0 is the only number that can be added to
another number (say, a) and give an answer that is the
same number a:
0+a=a+0=a
The number 1 is the only number that can be multiplied by
another number (say, a) and give an answer that is the
same number a:
1a=a1=a
24
Additive Inverse (-a) and Multicative Inverse
(1/a)
• Given a, its additive inverse is –a, because
a + (-a) = 0
3 and -3 are additive inverses of each other
• Given a, its multicative inverse is (1/a), because
a ∙ (1/a) = 1
3 and (1/3) are multicative inverses of each other.
(Also called the reciprocal of each other)
25
Your Turn
Find the additive and multiplicative inverses of
Solution:
The additive inverse of
is
The multiplicative inverse of
.
because
is
because
.
.
26
5.
Identify the property that
justifies a given statement
27
Example 6
The property in the right column justifies the statement in
the left column.
a. 3 + 4 is a real number
closure property of addition
b.
closure property of division
is a real number
c. 3 + 4 = 4 + 3
commutative property of addition
d. –3 + (2 + 7) = (–3 + 2) + 7
associative property of addition
e. (5)(–4) = (–4)(5)
commutative property of multiplication
28
Example 6
cont’d
f. (ab)c = a(bc)
associative property of multiplication
g. 3(a + 2) = 3a + 3  2
distributive property
h. 3 + 0 = 3
additive identity property
i. 3(1) = 3
multiplicative identity property
j. 2 + (–2) = 0
additive inverse property
k.
multiplicative inverse property
29
Identify the property that justifies a given statement
The properties of the real numbers are summarized as
follows.
Properties of Real Numbers
30