Download Sections 2.7/2.8 – Real Numbers/Properties of Real Number

Section 2.7/2.8
Real Numbers/Properties of Real Number Operations
A real number is any number that belongs to the set of rational numbers or the set of irrational
numbers.
Each real number corresponds to a point on the number line. Each real number is either
negative, zero, or positive.
http://math.tutorvista.com/number-system/operations-using-number-line.html
The set of all real numbers is denoted by  .
The subsets of real numbers are:
Sections 2.7/2.8 – Real Numbers/Properties of Real Number Operations
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Example 1: Classify each number below as rational, irrational, integer, whole number, and/or
natural number.
a. -2
rational, irrational, integer, whole number, natural number
b.
3
rational, irrational, integer, whole number, natural number
c. −
7
5
rational, irrational, integer, whole number, natural number
d. 0
rational, irrational, integer, whole number, natural number
e.
84
28
rational, irrational, integer, whole number, natural number
f.
9
rational, irrational, integer, whole number, natural number
1
g.  
3
−2
rational, irrational, integer, whole number, natural number
h. 3.552 ×10−3
rational, irrational, integer, whole number, natural number
i. 4.24
rational, irrational, integer, whole number, natural number
j. 3.121221222…
rational, irrational, integer, whole number, natural number
Sections 2.7/2.8 – Real Numbers/Properties of Real Number Operations
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Comparing Real Numbers
Given two real numbers x and y,
•
if x – y is negative, then x < y.
•
if x – y is positive, then x > y.
•
if x – y = 0, then x = y.
Example 2: Use <, > or = to make each statement true.
a. -2
1.9
b. -2.1
-1.9
c. 2.1
-1
d. 2.0010002
2.00099999
e.
1.4
2
f. 1.0023 ×10−2
9.915 ×10−3
g. 3.2285 ×1015
3.2301×1015
Sections 2.7/2.8 – Real Numbers/Properties of Real Number Operations
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Comparing Fractions
To compare two fractions, first find a common denominator. The larger fraction has the larger
numerator.
OR
a
c
and
first make sure if any of the denominators are negative,
b
d
a −a
rewrite the fraction so that the numerator is negative. For example,
.
=
−b b
Given two fractions such as,
•
if ad < bc , then
a c
< .
b d
•
if ad > bc , then
a c
> .
b d
Example 3: Use <, > or = to make each statement true.
a.
2
9
1
4
b.
7
12
11
20
c.
3
−8
−2
5
d.
6
−7
9
−11
Sections 2.7/2.8 – Real Numbers/Properties of Real Number Operations
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Comparing Radicals
Given two natural numbers a and b, a < b if and only if
a< b.
Example 4: Which of the following is true?
a.
3
>
7
b.
10 >
2
c.
121 > 11
Example 5: Place the following numbers in order from smallest to largest.
a. 10,
122,
b. 7, 5,
c. 4,
11,
98
31
7
Sections 2.7/2.8 – Real Numbers/Properties of Real Number Operations
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Section 2.8
Properties of Real Number Operations
The Commutative Properties of Addition and Multiplication
Given two real numbers a and b,
a+b=b+a
a×b = b×a
Subtraction and Division are not commutative.
The Associative Properties of Addition and Multiplication
For any three real numbers a, b and c,
a + (b + c) = (a + b) + c
a × (b × c) = (a × b) × c
Subtraction and Division are not associative.
The Distributive Properties of Multiplication over Addition
For any three real numbers a, b and c,
a (b + c) = ab + ac
(a + b)c =ac + bc
Example 6: State the property used in each statement below.
a.
(3× 3 ) ×
b.
1
1
+2 = 2+
7
7
c.
(9 + 6) × 3 = 9 ⋅ 3 + 6 ⋅ 3
d.
( −5 + 11) + 9 =−5 + (11 + 9 )
3=
3×
(
3× 3
)
Sections 2.7/2.8 – Real Numbers/Properties of Real Number Operations
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Identity Elements
Given any real number a
a+0=a=0+a
(0 is the identity element for addition)
a × 1 = a = 1× a
(1 is the identity element for multiplication)
Example 7: Does the following statement illustrate the property of the identity element for
addition, multiplication or neither?
a. 1.23 ×1 = 1×1.23
b. 5 − 5 =
0
c. 8 − 7 =
1
d.
3 4
× =
1
4 3
e. 1.23 + 1 = 1 + 1.23
f. 0 + ( −5 ) = ( −5 ) + 0
g.
(8 + 0 ) + 1 = 8 + 1
Sections 2.7/2.8 – Real Numbers/Properties of Real Number Operations
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