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Number Properties




 
In these lessons, we will learn three basic number properties (or laws) that apply to arithmetic operations: Commutative Property, Associative Property and Distributive Property.

Related Topics: More Arithmetic Lessons

Summary of Number Properties

The following table gives a summary of the commutative, associative and distributive properties. Scroll down the page for more examples and explanations of the number properties.
number properties

Commutative Property

An operation is commutative if a change in the order of the numbers does not change the results. This means the numbers can be swapped.

Numbers can be added in any order.

For example: 4 + 5 = 5 + 4
  x + y = y + x

Numbers can be multiplied in any order.

For example: 5 × 3 = 3 × 5
  a × b = b × a

Numbers that are subtracted are NOT commutative.

For example: 4 – 5 ≠ 5 – 4
  x – y ≠ y –x

Numbers that are divided are NOT commutative.

For example: 4 ÷ 5 ≠ 5 ÷ 4
  x ÷ y ≠ y ÷ x

Associative Property

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 ≠ 4 – (5– 6)
  (xy) – zx – (yz)

Numbers that are divided are NOT associative.

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



Distributive Property

Distributive property allows you to remove the parenthesis (or brackets) in an expression. Multiply the value outside the brackets with each of the terms in the brackets.

4(a+b)

For example: 4(a + b) = 4a + 4b
  7(2c – 3d + 5) = 14c – 21d + 35

What happens if you need to multiply (a – 3)(b + 4)?

You do the same thing but with one value at a time.

number properties - distributive

For example:

Multiply a with each term to get a × b + 4 × a = ab + 4a

distributive

Then, multiply 3 with each term to get “ –3b – 12” (take note of the sign operations).

Put the two results together to get “ab + 4a – 3b – 12”

Therefore, (a – 3)(b + 4) = ab + 4a – 3b – 12


 

Videos

The Commutative Property of addition and multiplication.
The addition of real numbers is commutative.
The multiplication of real numbers is commutative.
The Associative Property of addition and multiplication.


The following video shows more examples of the distributive property.
Associative and Distributive Properties of Multiplication and Addition
A look at the logic behind the associative and distributive properties of multiplication and addition.


 

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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