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Associative Property & Identity Property


The basic Number Properties (or laws) that apply to arithmetic operations are Commutative Property, Associative Property, Identity Property and Distributive Property. In this lesson, we will learn the associative property and identity property of numbers.

Related Topics:
Commutative Property, Distributive Property

Associative Property

The associative property states that the sum or product of a set of numbers is the same, no matter how the numbers are grouped.
An operation is associative if a change in grouping does not change the results. This means the parenthesis (or brackets) can be moved.

Numbers that are added can be grouped in any order.

For example: (4 + 5) + 6 = 5 + (4 + 6)
  (x + y) + z = x + (y + z)

Numbers that are multiplied can be grouped in any order.

For example: (4 × 5) × 6 = 5 × (4 × 6)
  (x × y) × z = x × (y × z)

Numbers that are subtracted are NOT associative.

For example: (4 – 5) – 6 ≠ 5 – (4 – 6)
  (xy) – zx – (yz)

Numbers that are divided are NOT associative.

For example: (4 ÷ 5) ÷ 6 ≠ 5 ÷ (4 ÷ 6)
  (x ÷ y ) ÷ z ≠ y ÷ ( x ÷ z)

Summary of Number Properties

The following table summarizes which number properties are applicable to the different operations:

Number Properties × ÷ +
Commutative Yes No Yes No
Associative Yes No Yes No
Distributive Yes No No No

What is the Associative Property?

The following video explains: What is the associative property? Why does it have the name it does? How can you recognize it when you see it? How can you distinguish it from other situations that look very similar?


The Associative Property

This video explains the associative property and how it can be used.
The following video shows an example on the Associative Property of Addition.

The following video shows an example of the Associative Property of Multiplication.

Identity Properties

Identity Property (or Zero Property) of Addition

When you add 0 to any a number, the sum is that number.
For example: 325 + 0 = 325.

Identity Property (or One Property) of Multiplication

When you multiply any number by 1, the product is that number.
For example: 65, 148 × 1 = 65, 148

Zero Property of Multiplication

The product of any number and 0 is 0
For example: 874 × 0 = 0

Identity Property of Addition & Multiplication

The following video shows the commutative & identity properties of addition & multiplication.

Identity Property

What is the identity property? How can you recognize it and name it when you see it? Why does is have the name it has? Why do mathematicians give EVERYTHING, even something as seemingly simple as this a name?

Multiplication: Zero Property

This video shows the zero property.

Rotate to landscape screen format on a mobile phone or small tablet to use the Mathway widget, a free math problem solver that answers your questions with step-by-step explanations.

You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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