Download Section 1.5 Properties of Real Numbers

Properties of Real Numbers
Commutative
Associative
Identity + ×
Inverse + ×
Zero Property
Reflexive
Distributive
Commutative Properties
Changing the order of the numbers in
addition or multiplication will not
change the result.
Commutative Property of Addition
states: 2 + 3 = 3 + 2 or a + b = b + a.
Commutative Property of
Multiplication states: 4 • 5 = 5 • 4 or
ab = ba.
Verbal Hints for Commutative
Property
Commute
Switch Places
Move to a new location
Associative Properties
Changing the grouping of the
numbers in addition or multiplication
will not change the result.
Associative Property of Addition
states: 3 + (4 + 5)= (3 + 4)+ 5 or
a + (b + c)= (a + b)+ c
Associative Property of Multiplication
states: (2 • 3) • 4 = 2 • (3 • 4) or
(ab)c = a(bc)
Verbal Hints for Associative
Property
Regroup
They simply group with a new friend
( ) change places but numbers stay the
same
Additive Identity Property
There exists a unique number 0 such
that zero preserves identities under
addition.
a + 0 = a and 0 + a = a
In other words adding zero to a
number does not change its value.
Multiplicative Identity Property
There exists a unique number 1 such
that the number 1 preserves identities
under multiplication.
a ∙ 1 = a and 1 ∙ a = a
In other words multiplying a number
by 1 does not change the value of the
number.
Verbal Hints for Identity
Property
The value that returns the input
unchanged
Remember “I” in the word identity
Additive Inverse Property
For each real number a there
exists a unique real number –a
such that their sum is zero.
a + (-a) = 0
In other words opposites add to
zero.
Verbal Hints for Additive Inverse
Opposite numbers
Same number but different sign
Zero Pairs
Multiplicative Inverse Property
For each real number a there exists a
unique real number
product is 1.
1
a 1
a
1
a
such that their
Verbal Hints for Multiplicative
Inverse
Flip the number
Reciprocal
Zero Property of Multiplication
Any number multiplied by 0 is equal to 0
A●0=0
-23 ● 0 = 0
½●0=0
0.25 ● 0 = 0
“zero times
any value
is 0
Reflexive Property
A number is always equal to itself
A=A
-2 = -2
5=5
A real number is always equal to itself
Distributive Property
Multiplication distributes over addition.
ab  c   ab  ac
32  5  3  2  3  5
Verbal Hints for Distributive
Property
Multiplication distributes across addition or
subtraction.
Multiply across the parentheses!!!!!
Let’s play “Name that property!”
State the property or properties that
justify the following.
3+2=2+3
Commutative Property
State the property or properties that
justify the following.
10(1/10) = 1
Multiplicative Inverse Property
State the property or properties that
justify the following.
3(x – 10) = 3x – 30
Distributive Property
State the property or properties that
justify the following.
3 + (4 + 5) = (3 + 4) + 5
Associative Property
State the property or properties that
justify the following.
(5 + 2) + 9 = (2 + 5) + 9
Commutative Property
2.
3+7=7+3
Commutative
Property of Addition
3.
8+0=8
Identity Property of
Addition
5.
6•4=4•6
Commutative Property
of Multiplication
11.
5•1=5
Identity Property of
Multiplication
25.
1
5/
+
0
=
7
1
5/
7
Identity Property of
Addition
40.
a + (-a) = 0
Inverse Property of
Addition
Properties of Real Numbers
Commutative
Associative
Distributive
Identity + ×
Inverse + ×