Properties
of Numbers
We’ll learn 4 properties:
• Commutative Property
• Associative Property
• Distributive Property
• Identity
Commutative
Property
We
commute
when 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 the
commutative property
for addition:
xy  yx
It also works for
multiplication:
xy  yx
To associate with someone
means that we like to
be with them.
The tiger and the panther
are associating with each
other.
They are leaving the
lion out.
(
)
In algebra:
( x  y)  z
The panther has decided to
befriend the lion.
The tiger is left out.
(
)
In algebra:
x  (y  z)
This is a statement of the
Associative Property:
( x  y)  z  x  (y  z)
The variables do not change
their order.
The Associative Property
also works for multiplication:
( xy)z  x(yz )
We have already used the
distributive property.
Sometimes executives ask
for help in distributing
papers.
The distributive property only
has one form.
Not one for
.
.
.and
one
for
addition
multiplication
. . .because both operations are
used in one property.
We add here:
4(2x+3)
We multiply
here:
This is an example
of the distributive
property.
4(2x+3) =8x+12
4
2x
+3
8x
12
Here is the distributive
property using variables:
x(y  z )  xy  xz
y
x
+z
xy
xz
The
identity
property
makes
me
think
about
my
identity.
The identity property for addition
asks,
“What can I add to myself
to get myself back again?
0 x
x_
0 x
x_
The above is the identity property
for addition.
0
is the identity element
for addition.
The identity property for
multiplication
asks,
“What can I multiply to myself
to get myself back again?
1 x
x (_)
1 x
x (_)
The above is the identity property
for multiplication.
1
is the identity element
for multiplication.
Here are the 3 properties
that have to do with addition:
Commutative
x+y=y+x
Associative x + (y + z)= (x + y) + z
Identity
x+0=x
Here are the 3 properties
for multiplication:
Commutative
Associative
Identity
xy = yx
x(yz)= (xy)z
x 1   x
The distributive property
contains both addition
and multiplication:
Distributive
x(y  z )  xy  xz