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The identity property of addition states that when zero is added to any number, the value does not change. In other words, the identity property of addition tells us that adding zero does not change the value of a number. Generally, we can express this as:
EXAMPLE
A similar property applies to multiplication. What quantity, when multiplied with any number, does not change the value of that number? When any number is multiplied by 1, the value does not change. The identity property of multiplication states that any number multiplied by 1 does not change in value. Generally, we can express this as:
EXAMPLE
The inverse property of addition states that any number and its opposite sum to zero. We can refer to the opposite of a number as its additive inverse. We can write this generally as:
EXAMPLE
The inverse property of multiplication states that a number and its reciprocal multiply to 1. In the same way that a number and its opposite are additive inverses, a number and its reciprocal are multiplicative inverses. The reciprocal of a number can be found by creating a fraction, and flipping the numerator and denominator. Here is our general rule:
EXAMPLE
are multiplicative inverses, or reciprocals, of one another.
Addition and multiplication is commutative. This means that we can add in any order we wish, and we can multiply in any order we wish. It is important to note that we cannot mix addition and multiplication. These are separate properties, but they behave the same with both operations.
The commutative property of addition is expressed with the following formula:
EXAMPLE
is the same as the expression 
, because addition is commutative. It does not matter in which order you add, the sum will be 5 in either case.
The same property applies to multiplication as well. It does not matter in which order you multiply, because multiplication is commutative. The general rule for the commutative property of multiplication is:
EXAMPLE
is the same as the expression 
, because multiplication is commutative. It does not matter in which order you multiply, the product will be 12 in either case.
The associative property allows us to group terms for addition and multiplication in any way we wish. As with the commutative properties of addition and multiplication, the associative property applies to addition and multiplication separately, and can be expressed with the following formulas:
EXAMPLE
The associative property allows us to group addends or group factors in different ways. This is particularly helpful in mental math, where we might easily recognize that 4 + 6 is 10. In such cases, regrouping can help us recognize certain sums or products to make mental math easier.
The distributive property applies multiplication over addition. This property is applied when we have a factor multiplied by a sum. It got its name from the process of distributing the outside factor into each part of the sum. Here is the general rule of the distributive property:
EXAMPLE
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Distribute 2 into 4 and 3, separately |
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Multiply inside the parentheses |
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Add |
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Our solution |
Source: ADAPTED FROM "BEGINNING AND INTERMEDIATE ALGEBRA" BY TYLER WALLACE, AN OPEN SOURCE TEXTBOOK AVAILABLE AT www.wallace.ccfaculty.org/book/book.html. License: Creative Commons Attribution 3.0 Unported License
