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Properties of Addition and Multiplication

Properties of Addition and Multiplication

Author: Sophia Tutorial
Description:

Identify the commutative property, associative property, or distributive property.

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what's covered
  1. Identity Properties
  2. Inverse Properties
  3. Commutative Property of Addition and Multiplication
  4. Associative Properties of Addition and Multiplication
  5. Distributive Property

1. Identity Properties

The identity property of addition states that when zero is added to any number, the value does not change. Below is an example of this property using real numbers:

7 plus 0 equals 7

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:

formula
Identity Property of Addition
a plus 0 equals a

A similar property applies to multiplication. What quantity, when multiplied to a number, does not change its value? When any number is multiplied by 1, the value does not change. Let's take a look at a numerical example of the identity property of multiplication:

4 bullet 1 equals 4

The identity property of multiplication states that any number multiplied by 1 does not change in value. Generally, we can express this as:

formula
Identity Property of Addition
a • 1 equals a


2. Inverse Properties

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. A numerical example is illustrated below:

9 + ( -9 ) = 0

9 and –9 are opposites of each other. This means that their magnitudes are the same (9), but their signs are different; one is a positive number, while the other is a negative number. The sum of a number and its opposite is zero. We can write this generally as:

formula
Inverse Property of Addition
a plus left parenthesis negative a right parenthesis equals 0

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. Below is a numerical example of the inverse property of multiplication:

9 cross times 1 over 9 equals 1

9 and 1/9 are multiplicative inverses, or reciprocals, of one another. The inverse property of multiplication dictates that the product of a number and its reciprocal is equal to 1. Here is our general rule:

formula
Inverse Property of Multiplication
a cross times 1 over a equals 1


3. Commutative Property of Addition and Multiplication

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.

term to know
Commutative Property
A property of addition that allows terms to be added in any order; a property of multiplication that allows factors to be multiplied in any order.

Let's take a look at a numerical example of the commutative property of addition:

2 plus 3 equals 3 plus 2

The expression 2 + 3 is the same as the expression 3 + 2, because addition is commutative. It does not matter in which order you add, the sum will be 5 in either case.

formula
Commutative Property of Addition
a plus b equals b plus a

The same property applies to multiplication as well. It does not matter in which order you multiply, because multiplication is commutative. A numerical example is illustrated below:

3 bullet 4 equals 4 bullet 3

The expression 3 x 4 is the same as the expression 4 x 3, because multiplication is commutative. It does not matter in which order you multiply, the product will be 12 in either case.

formula
Commutative Property of Multiplication
a b equals b a


4. Associative Properties of Addition and Multiplication

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.

term to know
Associative Property
A property of addition that allows terms to be grouped in any order; a property of multiplication that allows factors to be grouped in any order.

Here are some numerical examples of the associative property:

(3 + 4) + 6 = 3 + (4 + 6)

 4 x (2 x 8) = (4 x 2) x 8

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.

formula
Associative Property of Addition
left parenthesis a plus b right parenthesis plus c space equals space a plus left parenthesis b plus c right parenthesis
formula
Associative Property of Addition
left parenthesis a b right parenthesis c space equals space a left parenthesis b c right parenthesis


5. Distributive Property

The distributive property applies multiplication over addition. This property is applied when we have a factor being multiplied by a sum. It got its name from the process of distributing the outside factor into each added of the sum.

term to know
Distributive Property
A property of multiplication that states that a sum multiplied by a factor can be expressed as a sum of the products of each individual addend and that factor.

Let's take a look at a numerical example:

2 (4 + 3) Distribute 2 into 4 and 3
(2 x 4) + (2 x 3) Multiply inside the parentheses
8 + 6 Add
14 Our solution

formula
Distributive Property
a left parenthesis b plus c right parenthesis space equals space a b space plus space a c
hint
The distributive property is especially useful when working with variables. Realistically, an easier approach with numerical examples is to evaluate what is inside the parentheses first, and then multiply the outside factor. (This also follows the Order of Operations). However, as we work with algebraic expressions containing variables, the distributive property is going to be very helpful.
summary
There are many different properties for addition and multiplication. We looked at an identity property, an inverse property, a commutative property, the associative property. Also, for multiplication, there is the distributive property.
Terms to Know
Associative Property

A property of addition that allows terms to be grouped in any order; a property of multiplication that allows factors to be grouped in any order.

Commutative Property

A property of addition that allows terms to be added in any order; a property of multiplication that allows factors to be multiplied in any order.

Distributive Property

A property of multiplication that states that a sum multiplied by a factor can be expressed as a sum of the products of each original addend and that factor. 

Formulas to Know
Associative Property of Addition

open parentheses a plus b close parentheses plus c equals a plus open parentheses b plus c close parentheses

Associative Property of Multiplication

open parentheses a b close parentheses c equals a open parentheses b c close parentheses

Commutative Property of Addition

a plus b equals b plus a

Commutative Property of Multiplication

a b equals b a

Distributive Property

a open parentheses b plus c close parentheses equals a b plus a c

Identity Property of Addition

a plus 0 equals a

Identity Property of Multiplication

a times 1 equals a

Inverse Property of Addition

a plus left parenthesis negative a right parenthesis equals 0

Inverse Property of Multiplication

a cross times 1 over a equals 1