Basic Properties of Mathematics: Distributive, Associative, Commutative and Identity

The Distributive Property is written as follows:
 • a(b + c) = ab + ac
The Distributive Property is easy, if you remember that "multiplication distributes over addition".
example: 2(3 + 5) = 2 x 3 + 2 x 5

The Distributive Property either requires that you take an operation through the parentheses(distribute) or factor the equation out. The following example uses the Distributive Property to factor out.
example: 6x - 12 = 6(x - 2)

Note: that the property states 'multiplication distributes over addition'. This statement is meant to include addition of negative numbers, ie 6[x + (-2)].

The Associative Property is written as follows:
 • a + (b + c) = (a + b) + c   for addition
 • a(bc) = (ab)c  for multiplication
The Associative Property is the 'regrouping' property. This property shows us that with a
single operation, changing the grouping does not change the answer.

example: 2 + (3 + 5) = (2 + 3) + 5 is the rule for addition
example: 2(3 x 5) = (2 X 3)5 is the rule for multiplication

In order to rearrange or regroup 2(5x), you will show that (2 x 5)x is the same as
2(5x). For example 2[5(3)] = (2 x 5)3;   2 x 15 = 30 as 10 x 3 = 30.

The Commutative Property is written as follows:
 •  a + b = b + a   for addition
 •  ab = ba   for multiplication
The Commutative Property simply means to move around, swap or change the position of
the numbers or variables in the equation when adding or multiplying and get the same answer.

example: 4 + 6 = 6 + 4 is the rule for addition
example: 4 x 6 = 6 x 4 is the rule for multiplication

Using the Commutative Property, to restate 5 x 3 x b you could use any of the following:
5 x b x 3,  5 x 3 x b,  b x 5 x 3,   b x 3 x 5,   3 x 5 x b,  3 x b x 5

The Identity Property of Addition is written as follows:
 •  a + 0 = a   for additive Identity
The Identity Property of addition states that the sum of zero and any number or
variable is the number or variable itself.

examples: 5 + 0 = 5, -10 + 0 = -10, x + 0 = x
examples: 0 + 9 = 9, 0 + -12 = -12, 0 + b = b

The Identity Property of Multiplication is written as follows:
 •  a x 1 = a   for multiplicative Identity
The Identity Property of multiplication states that the product of one and any number or
variable is the number or variable itself.

examples: 5 x 1 = 5, -10 x 1 = -10, y x 1 = y
examples: 1 x 9 = 9, 1 x -12 = -12, 1 x b = b