Transcript 1.2 Properties of Real Numbers

1.2 Properties of Real Numbers
• Objectives: To graph and order real numbers
To identify properties of real numbers
• Vocabulary:
Opposite
Additive Inverse
Reciprocal
Multiplicative Inverse
Properties
Vocabulary Terms: Properties of Real Numbers
Let a, b, and c represent real numbers
Property
Closure
Commutative
Associative
Identity
Inverse
Distributive
Addition
Multiplication
REAL NUMBER CLASSIFICATIONS
Subsets of the Real Numbers
Rational
Irrational
Integers
Whole
Π, 2
Natural
EXAMPLES
• Use natural numbers to count - {1,2,3,4,5,6….}
• The whole numbers are the natural numbers
plus 0 - {0,1,2,3….}
• Integers are the natural numbers and their
opposites plus 0 - {...,-3,-2,-1,0,1,2,3…}
• Rational numbers are all numbers that can be
written as a quotient of integers. a/b, b≠0.
• Rational numbers include terminating decimals…1/8
= .0125
• Rational Numbers include repeating decimals…
• 1/3 = .3333333333333333333333333333333333,
or 0.3 with a hat over it
CLASSIFY EACH NUMBER
name ALL sets to which each belongs
-1 • real, rational, integer
3 • real, rational, integer, whole,
natural
√17
• real, irrational
⅜ • real, rational
0 • real, rational, integer, whole
-5.555 • real, rational
PROPERTIES OF REAL NUMBERS
COMMUTATIVE
• Think… commuting to school.
• Deals with ORDER. It doesn’t matter what
order you ADD or MULTIPLY.
• a+b = b+a
•4 • 6 = 6 • 4
PROPERTIES OF REAL NUMBERS
ASSOCIATIVE
• Think…the people you associate with; your
group. Are you the member of more than 1
club?
• Deals with grouping when you Add or
Multiply.
• Order does not change.
Additive (a
•
+ b) + c = a + ( b + c)
Multiplicative
•
(nm)p = n(mp)
PROPERTIES OF REAL NUMBERS
IDENTITY
Additive Identity Property
•s+0=s
•0 is the additive identity.
Multiplicative Identity Property
• 1(b) = b
•1 is the multiplicative identity
PROPERTIES OF REAL NUMBERS
INVERSE
• Multiplicative
Inverse Property
• Product = 1
• Additive Inverse
Property
• Sum = Zero
• a ∙ 1/a = 1, a ≠ 0
• a + (-a) = 0
• 8(1/8) = 1
• -5(-1/5)=1
• 12 + (− 12 ) = 0
• −7 + 7 = 0
Properties of Real Numbers
Distributive
Distributive Property
•
a(b + c) = ab + ac
•
9(r + s) = 9r + 9s
Name the Property
•5=5+0
• 5(2x + 7) =10x + 35
•8•7=7•8
• 24(2) = 2(24)
• (7 + 8) + 2 =2 + (7 + 8)
Name the Property
•5=5+0
• 5(2x + 7) =10x + 35
•8•7=7•8
• 24(2) = 2(24)
• (7 + 8) + 2 =2 + (7 + 8)
Additive Identity
Distributive
Commutative
Commutative
Commutative
Name the Property
• 7 + (8 + 2) = (7 + 8) + 2
• 1 • v + -4 = v + -4
• (6 - 3a)b =
6b - 3ab
• 4(a + b) =
4a + 4b
Name the Property
• 7 + (8 + 2) = (7 + 8) + 2
• 1 • v + -4 = v + -4
• (6 - 3a)b =
6b - 3ab
• 4(a + b) =
4a + 4b
• Associative
• Multiplicative
Identity
• Distributive
• Distributive
Vocabulary Terms: Properties of Real Numbers
Let a, b, and c represent real numbers
Property
Addition
Multiplication
Closure
The sum of a + b is a
real number
The product of ab is a
real number
Commutative a + b = b + a
ab = ba
Associative
(a + b) + c = a + (b + c)
(ab)c = a(bc)
Identity
a + 0 = a, 0 + a = a
0 is the additive
identity
a●1=a, 1●a=a
1 is the multiplicative
identity
Inverse
a + (-a) = 0 (opposite
sign)
Distributive
a(b + c)=ab + ac
reciprocal
1
a   1, a  0
a