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

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) |

(x – y) – z ≠ x – (y – z) |

Numbers that are divided are NOT associative.

For example: | (4 ÷ 5) ÷ 6 ≠ 4 ÷ (5÷ 6) |

(x ÷ y ) ÷ z ≠ x ÷ ( y ÷ z) |

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.

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.

For example:

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

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

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

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

The following table summarizes which number properties are applicable to the different operations: multiplication, division, addition and subtraction.

Number Properties |
× |
÷ |
+ |
– |

Commutative | Yes | No | Yes | No |

Associative | Yes | No | Yes | No |

Distributive | Yes | No | No | No |

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.

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