Transcript Properties of Numbers

Properties
of Numbers
In
Algebra County
We’ll learn 5 properties:
• Commutative Property
• Associative Property
• Distributive Property
• Identity
• Inverse
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
Here is the distributive
property using variables:
x( y  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.
We learned about the inverse
property when we did zero pairs.
2  (2)  0
The inverse property
is related
This
is
the
to the
identity
element
identity property.
for addition.
2  (2)  0
The whole thing is
the inverse property.
This is the
inverse element
for addition.
2  (2)  0
A statement of the inverse
property for addition is:
x  ( x )  0
What is the identity
To keep the
element for
same pattern,
multiplication?
it would
1
go here.
2  (2)  0
Therefore. . .
To keep the
same pattern,
it would
go here.
1
2(_)
2 
1
A statement of the inverse
property for multiplication is:
1
x   1
x
 
Some examples of the inverse
property for multiplication are:
1
5   1
2 3
5

1


3 2 
Here are the 4 properties
that have to do with addition:
Commutative
x+y=y+x
Associative x + (y + z)= (x + y) + z
Identity
x+0=x
Inverse
x + (-x) = 0
Here are the 4 properties
for multiplication:
Commutative
Associative
Identity
Inverse
xy = yx
x(yz)= (xy)z
x 1   x
1
x   1
x
The distributive property
contains both addition
and multiplication:
Distributive
x( y  z )  xy  xz