1-1 Properties of Real Numbers
Objectives (What You'll Learn):
To graph and order real numbers
To identify and use properties of real numbers
Title: Jun 25 ­ 5:52 PM (1 of 10)
Check Skills You'll Need :
Simplify
1.
-(-7.2)
2.
1 - (-3)
3.
-9 + (-4.5)
4.
(-3.4)(-2)
5.
-15
3
6.
-2 + 3
5 5
*GO FOR HELP - Skills Handbook page 873
Title: Jun 25 ­ 6:00 PM (2 of 10)
Subsets of Real Numbers
Real Numbers
Examples: -5, -√3, - 1/2, 1, √5, 8/3
Rational Numbers
Examples: 1/2, 0.3, 1, 2 2/3, -5/4, -1.07
Integers ..., -2, -1, 0, 1, 2,...
Whole Numbers 0, 1, 2, 3,...
Natural Numbers
1, 2, 3,...
Title: Jun 25 ­ 6:07 PM (3 of 10)
Irrational
Numbers
Examples:
-√3, π, ∛40
Subsets of Real Numbers
n
e
t
it
r
w
e s!
b
an eger
c
t int
a
h
t of
s
r
e nts
b
m tie
u
N quo
as
Real Numbers
Examples: -5, - √3, -1/2, 1, √5, 8/3
Rational Numbers
Examples: 1/2, 0.3, 1, 2 2/3, -5/4, -1.07
Integers ..., -2, -1, 0, 1, 2,...
Whole Numbers
Irrational
Numbers
Examples:
-√3, π, ∛40
0, 1, 2, 3,...
Natural Numbers
1, 2, 3,...
Positive and Negative Natural
Numbers and zero!
Title: Jun 25 ­ 6:07 PM (4 of 10)
Example 1 : Many mathematical relationships involving variables are related to
amusemet parks. Which set of real numbers
best describes the values for each variable?
A)
The cost C in dollars of admission for n people.
C - rational numbers (such as $7.25)
n - whole numbers
B)
The park's profit (or loss) P in dollars for each week w of the year.
P - rational number
w - one of the first 52 natural numbers
Title: Jun 25 ­ 6:30 PM (5 of 10)
Example 2 : Graph the numbers - 3/2, 1.7 and √5 on the number line.
Hint : Re-write each number as a decimal first and it will be easier to graph.
-3/2 =
1.7 =
√5 =
­5
­4
­3
­2
­1
0
1
2
3
4
5
Example 3 : Compare - √0.08 and - √0.1. Use the symbols < <
and >. >
Pull from here!
Title: Jun 25 ­ 7:00 PM (6 of 10)
-√0.08
- √0.1
- √0.1
- √0.08
Definitions :
The opposite or additive inverse
of any number a is -a. The sum of opposites is 0.
The reciprocal or multiplicative inverse
of any nonzero number a is 1/a. The product of reciprocals is 1.
Example 4 : Find the opposite and the reciprocal of each number.
A) 4 1/5
B) -0.002
Opposite:
Opposite:
Reciprocal:
Reciprocal:
Title: Jun 25 ­ 7:00 PM (7 of 10)
Properties of Real Numbers
:
Let a, b, and c represent real numbers.
Property
Addition
Closure
a + b is a real number
Commutative
Multiplication
ab is a real number
a+b=b+a
Associative
ab = ba
(a + b) + c = a + (b + c)
(ab)c = a(bc)
Identity
a + 0 = a, 0 + a = a
a(1) = a, 1(a) = a
Inverse
a + (-a) = 0
a(1/a) = 1, a not = 0
Distributive
a(b + c) = ab + ac
Example 5 : Which property is illustrated?
A)
4 + (-5) = (-5) + 4
Commutative Property of Addition
B)
6(xy) = 6x(y)
Associative Property of Multiplication
Title: Jun 25 ­ 7:00 PM (8 of 10)
Definition :
The absolute value of a real number
is its distance from zero on the number line.
Example 6 : Find the following absolute values.
A)
|-9| =
B)
|0| =
C)
|3 - 9| =
Title: Jun 25 ­ 8:06 PM (9 of 10)
HOMEWORK:
page 8 (3 - 66) multiples of 3
Title: Jun 25 ­ 8:17 PM (10 of 10)