► Number properties state a relationship between numbers.
1) The Commutative Property
The commutative property states that the order of the numbers when adding or
multiplying does not change the sum or product:
Subtraction and division are not commutative. In other words, the order of the numbers
when subtracting or dividing does change the answer:
2) The Associative Property
The associative property states that the grouping of numbers when adding or
multiplying does not change the sum or the product:
Again, subtraction and division don’t work with this property. Subtraction and division
are not associative because the grouping of the numbers does change the answer:
3) The Distributive Property
The distributive property relates the operations of multiplication and addition:
example 1
example 2
Not understanding the distributive property can lead to a great number of problems when
solving algebraic equations, so it is very important to know and understand this property!
example of common mistake
A common mistake is to distribute a number or variable but to forget to distribute the
sign of that number or variable:
► When an identity is combined with another number, that number stays the same.
4) Additive Identity Property
Zero is called the additive identity because adding zero to a number does not change it
― the number stays the same:
(Note: Notice the commutative property of addition in the above equations. The order of
the “0” and the “a” does not change the sum.)
5) Multiplicative Identity Property
One is called the multiplicative identity because multiplying any number by one does
not change it ― the number stays the same:
(Note: Notice the commutative property of multiplication in the above equations.)
► An inverse of something is its opposite, so the inverse of a mathematical operation
undoes that operation: subtraction undoes addition, and division undoes multiplication.
6) Additive Inverse Property
When the additive inverse of a number is added to that number, the sum will be zero:
So just as any number added to zero equals that number, adding the additive inverse of
that number brings you back to zero.
example 3
0+7=7
7 + (−7) = 0
To find the additive inverse of a number, simply change the sign of that number.
7) Multiplicative Inverse Property
When a number is multiplied by its multiplicative inverse, the product will be one:
The multiplicative inverse of a number is its reciprocal (where the numerator and
denominator switch places).
example 4
What is the multiplicative inverse of 7?
The number 7 can be thought of as the fraction:
If you switch the numerator and denominator,
So the multiplicative inverse of 7 is
7
1
7
1
becomes .
1
7
1
.
7
Checking the answer: If these two numbers are multiplied together, the product is
1
one: 7 × = 1.
7
Another way to think of a multiplicative inverse is how division is the opposite (or inverse)
of multiplication. Just as 1 multiplied by any number equals that number, dividing by that
number (or multiplying by the multiplicative inverse) brings you back to 1.
example 5
► Closure refers to numbers within a set. (For example, the set of all positive numbers, set
of all even numbers, set of all rational numbers, etc.)
8) Property of Closure
If a mathematical operation is performed on two numbers from a particular set, and the
answer is also from that same set of numbers, then that set of numbers is closed under the
operation performed.
example 6
If any two even numbers are added, the answer will always be an even number.
This means that the set of all even numbers is closed under addition.
example 7
Is the set of all even numbers closed under division?
Pick an even number and divide it by other even numbers to see if you can find a
quotient that isn’t even;
It just takes one occurrence to say that a set of numbers is not closed under a
particular operation.
Therefore, the set of all even numbers is not closed under division.
► You saw that anything added to zero equals itself (the additive identity property), but
what happens when something is multiplied by zero?
9) The Multiplicative Property of Zero
The multiplicative property of zero states that multiplying any number by zero equals
zero:
In multiplication, one number times another number can be thought of as having that first
number a certain number of times (such as 4 × 3 means you have the number 4 three
times: 4 + 4 + 4 = 12).
If you have a number times zero, that means you have that number zero times.
► When two negative numbers are multiplied together, the answer is the same as
multiplying the same positive numbers together.
10) (−x)(−y) = xy
To phrase it another way, a negative times a negative equals a positive.
This can be proven by using the additive inverse property, the multiplicative property of
zero, and the distributive property:
This identity is mainly used when distributing and making sure you get the signs right,
positive or negative.
example 8
Distribute: (−2)(−x + 2)
Using the distributive property, this can be re-written as: (−2)(−x) + (−2)(2)
Using the identity that was proven above, (−2)(−x) = 2x and (−2)(2) = −4
Therefore, (−2)(−x + 2) = 2x − 4
Name ______________________________
Match each statement shown with the appropriate property:
1) 0 + d = d
A) Commutative Property of Addition
2) x + (y + z) = (x + y) + z
B) Commutative Property of Multiplication
1
=1
6
1
1
=0
4)
+
2
2
3) 6 ·
C) Associative Property of Addition
D) Associative Property of Multiplication
5) 2 · (3 × 4) = (2 × 3) · 4
E) Distributive Property
6) 6x + 3y = 3y + 6x
F) Additive Identity Property
7) 7 × 0 = 0
G) Multiplicative Identity Property
a 3
· =1
3 a
9) a(b + c) = a(c + b)
8)
10) 8 × 1 = 8
11) (2xy)3 = 2x(y · 3)
12) ab + cd = ba + cd
13) a(b + c) = ab + ac
14) 4(xy) = 4(yx)
15) (2x + y) + 3 = 2x + (y + 3)
16) 4 + 0 = 4
17) c + (−c) = 0
18)
1
×0=0
b
19) ab × 1 = ab
20) −7(x – 1) = −7x + 7
H) Additive Inverse Property
I) Multiplicative Inverse Property
J) Multiplicative Property of Zero
Simplify the following expressions using the distributive property:
21) 2(a + b)
22) −4(3m – 2n)
23) x – (y + 4)
24) −2(m + 2)
25) (a – 8)b
26) 3x + 4y – 2(x – y)
27) −k(2k + 5)
28) a + 4(a + 2)
Name the multiplicative inverse of each number or variable (assume that no variable
represents zero):
29) 9
30)
1
3
31)
2
x
32) 5m
33)
a+2
3
34) 2
1
4