Properties
1. Commutative Property
Order doesn’t make any difference in multiplication or addition.
Examples: 4 + 3 = 3 + 4
a+b=b+a
ab = ba
Note: This property was especially useful when you were memorizing your addition and
multiplication facts.
2. Associative Property
When the operations of addition or multiplication are grouped using parentheses, the placement
of the parentheses doesn’t matter.
Examples: (6 + 2) + 3 = 6 + (2 + 3)
(a + b) +c = a + (b + c)
Note: When multiplying or adding a group of numbers, you can group them any way you want
(or add them in any order)--this is often helpful when you want to add or multiply to multiples
of ten.
___________________________
3. Identity Property of Addition
The sum of any number and zero is that number.
Examples: 4 + 0 = 4 -2 = -2 + 0
0+a=a
Note: This property will prove useful when solving equations (while maintaining equality).
4. Identity Property of Multiplication
The product of any number and one is that number.
Examples: ___________________________
5. Equality Property of Addition
You can add the same number to both sides of an equation, and the statement will still be true,
equality will be
maintained.
Example: 6. Equality Property of Subtraction
You can subtract the same number from both sides of an equation, and the statement will still
be true, equality will be maintained.
Example: 7. Equality Property of Multiplication
You can multiply by the same number on both sides of an equation, and the statement will still
be true, equality will be maintained.
Example: 8. Equality Property of Division
You can divide by the same number on both sides of an equation, and the statement will still be
true, equality will be maintained.
Example:
4 ⋅ 5 = 20
4
20
⋅5 =
2
2
2 ⋅ 5 = 10
___________________________
9. Inverse Property of Addition
Any number added to its opposite (called the additive inverse) equals zero
Examples: 22 + (-22) = 0
a + (-a) = 0 a–a=0
10. Inverse Property of Multiplication
Any non-zero number multiplied by its reciprocal (also called the multiplicative inverse) equals
one.
Examples: or
as long as a ≠ 0 and b ≠ 0
Properties
11. Distributive Property
When a number or variable (or both) is multiplied by a series of terms in parentheses, it is
multiplied by each of the terms in parentheses.
Examples: 6(99) = 6(100 − 1) = 6 ⋅100 − 6 ⋅1 = 600 − 6 = 594
5 (x + 2) = 5x + 10
5 (x – 2) = 5x – 10
-5 (x + 2) = -5x – 10
-3x (2x – 4) = -6x2 – 12x
a (b + c) = ab + ac
a (b - c) = ab - ac
12. Zero Property of Multiplication
Any number multiplied by zero equals zero.
Examples: 22 (0) = 0