Properties of Numbers
We’ll learn 4 properties:
•Commutative Property•Associative Property•Distributive Property•Identity
CommutativeProperty
We commutewhen we go back and forth
from work to home.
Algebra terms commute when they trade places
x y
y x
This is a statement of thecommutative property
for addition:
x y y x
It also works for multiplication:
xy yx
Associative
Property
To associate with someone means that we like to
be with them.
The tiger and the pantherare associating with eachother.
They are leaving thelion out.
( )
In algebra:
( )x y z
The panther has decided tobefriend the lion.
The tiger is left out.
( )
In algebra:
( )x y z
This is a statement of theAssociative Property:
( ) ( )x y z x y z
The variables do not change their order.
The Associative Propertyalso works for multiplication:
( ) ( )xy z x yz
DistributiveProperty
We have already used thedistributive property.
Sometimes executives askfor help in distributing
papers.
The distributive property onlyhas one form.
Not one foraddition . . .and one for
multiplication
. . .because both operations areused in one property.
4(2x+3)
We multiply here:
We add here:
4(2x+3)=8x+12
This is an exampleof the distributive
property.
8x 124
2x +3
Here is the distributiveproperty using variables:
( )x y z xy xz
xy xz
y +z
x
The identity
property makes
me thinkaboutmy
identity.
The identity property for addition asks,
“What can I add to myselfto get myself back again?
_x x0
The above is the identity propertyfor addition.
_x x0
is the identity elementfor addition.0
The identity property for multiplication
asks, “What can I multiply to myself
to get myself back again?
(_ )x x1
The above is the identity propertyfor multiplication.
1
is the identity elementfor multiplication.1
(_ )x x
Here are the 3 propertiesthat have to do with
addition:x + y = y + x
x + (y + z)= (x + y) + z
x + 0 = x
Commutative
Associative
Identity
Here are the 3 propertiesfor multiplication:
xy = yx
x(yz)= (xy)z
Commutative
Associative
Identity 1x x
The distributive propertycontains both addition
and multiplication:
Distributive
( )x y z xy xz