Use the a and b sliders to change the numbers being added or multiplied.

Slide the "Move the squares" or "Rotate the array" slider to show the operation reversed.

Click the check boxes to change between addition and multiplication.

What do you notice when you change the order of the addition?

What do you notice when you change the order of the multiplication?

The commutative properties of real numbers are:

Property | Symbols | Example |
---|---|---|

\text{Commutative property of addition} | a+b=b+a | -3+6=6+\left(-3\right) |

\text{Commutative property of multiplication} | a \cdot b=b \cdot a | 6 \cdot \left(-3\right)=\left(-3\right) \cdot 6 |

The **commutative property** is the reason that we can add numbers in any order or multiply numbers in any order. Keep in mind that while addition and multipication are commutative, subtraction and division are not.

Let's see why the commutative property applied with subtraction does not work.

\displaystyle 7-4 | \displaystyle \neq | \displaystyle 4-7 | Apply the commutative property with subtraction |

\displaystyle 3 | \displaystyle \neq | \displaystyle -3 | Simplify |

Notice that if we applied the commutative property with subtraction, the left and right side are not equal. We can adjust this by turning the subtraction into a addition of a negative.

\displaystyle 7-4 | \displaystyle = | \displaystyle 7+\left(-4\right) | Rewrite subtraction as addition |

\displaystyle 7-4 | \displaystyle = | \displaystyle -4+7 | Apply the commutative property of addition |

\displaystyle 3 | \displaystyle = | \displaystyle 3 | Simplify |

We can see that when we convert the subtraction operation to an addition operation, we can still apply the commutative property of addition.

Let's see why the commutative property applied with division does not work.

\displaystyle 6 \div 3 | \displaystyle \neq | \displaystyle 3 \div 6 | Apply the commutative property with division |

\displaystyle 2 | \displaystyle \neq | \displaystyle \dfrac{1}{2} | Simplify |

Notice that if we applied the commutative property with division, the left and right side are not equal. We can adjust this by turning the division into a multiplication of it's reciprocal.

\displaystyle 6 \div 3 | \displaystyle = | \displaystyle 6 \cdot \left( \dfrac{1}{3} \right) | Rewrite division as multiplication |

\displaystyle 6 \div 3 | \displaystyle = | \displaystyle \left( \dfrac{1}{3} \right) \cdot 6 | Apply the commutative property of multiplication |

\displaystyle 2 | \displaystyle = | \displaystyle 2 | Simplify |

We can see that when we convert the division operation to an multiplication operation, we can still apply the commutative property of multiplication.

The commutative property can be applied to help us evaluate expressions more easily.

Find the value of: 6+\left(-5\right)+5

Worked Solution

Use the commutative property of addition to fill in the missing number:

19 + \left(-15\right) = \left(-15\right) + â¬š

Worked Solution

Find the value of: -20 \cdot 7 \cdot 5

Worked Solution

Which property is demonstrated by the following statement?

-3 + a + 13 = a + \left(-3\right) + 13

A

Commutative property of multiplication

B

Associative property of multiplication

C

Distributive property

D

Associative property of addition

E

Commutative property of addition

Worked Solution

Idea summary

Property | Symbols |
---|---|

\text{Commutative property of addition} | a+b=b+a |

\text{Commutative property of multiplication} | a \cdot b=b \cdot a |

The commutative property only works with addition and multiplication. You need to convert subtraction to addition and division to multiplication to apply these properties.

The associative properties of real numbers are:

Property | Symbols | Example |
---|---|---|

\text{Associative property of addition} | a+\left(b+c\right) \\ =\left(a+b\right)+c | 6+\left(\left(-3\right)+2\right)=\\\left(6+\left(-3\right)\right)+2 |

\text{Associative property of multiplication} | a\cdot(b\cdot c)=(a\cdot b)\cdot c | 6 \cdot \left(-3 \cdot 2\right)=\\ \left(6 \cdot (-3\right)) \cdot 2 |

The **associative property** is the reason that we can group sums or products of numbers differently and the result remains the same. While addition and multiplication are associative, subtraction and division are not.

Let's see why the associative property applied with subtraction does not work.

\displaystyle \left(7-4\right)+3 | \displaystyle \neq | \displaystyle 7-\left(4+3\right) | Apply the associative property with subtraction |

\displaystyle 3+3 | \displaystyle \neq | \displaystyle 7-7 | Simplify inside of the parentheses |

\displaystyle 6 | \displaystyle \neq | \displaystyle 0 | Simplify |

Notice that if we applied the associative property with subtraction, the left and right not equal. We can adjust this by turning the subtraction into addition by the opposite.

\displaystyle \left(7-4\right)+3 | \displaystyle = | \displaystyle \left(7+\left(-4\right)\right) +3 | Rewrite subtraction as addition |

\displaystyle \left(7-4\right)+3 | \displaystyle = | \displaystyle 7+\left( -4 + 3\right) | Apply the associative property of addition |

\displaystyle 3+3 | \displaystyle = | \displaystyle 7+\left(-1\right) | Simplify inside the parentheses |

\displaystyle 6 | \displaystyle = | \displaystyle 6 | Simplify |

We can see that when we convert the subtraction operation to an addition operation, we can still apply the associative property of addition.

Let's see why the associative property applied with division does not work.

\displaystyle 6 \div \left( 3 \div 2 \right) | \displaystyle \neq | \displaystyle \left( 6 \div 3 \right) \div 2 | Apply the associative property with division |

\displaystyle 6 \div \left(\dfrac{3}{2} \right) | \displaystyle \neq | \displaystyle \left( 2 \right) \div 2 | Simplify inside of the parentheses |

\displaystyle 4 | \displaystyle \neq | \displaystyle 1 | Simplify |

Notice that if we applied the associative property with division, the left and right side are not equal. We can adjust this by turning the division into a multiplication of it's reciprocal

\displaystyle 6 \div \left( 3 \div 2 \right) | \displaystyle = | \displaystyle 6 \div \left(3 \cdot \dfrac{1}{2} \right) | Rewrite division in the parentheses as multiplication |

\displaystyle 6 \div \left( 3 \div 2 \right) | \displaystyle = | \displaystyle 6 \cdot \left( \dfrac{1}{3} \cdot 2 \right) | Rewrite division outside the parentheses as multiplication |

\displaystyle 6 \div \left( 3 \div 2 \right) | \displaystyle = | \displaystyle \left(6 \cdot \dfrac{1}{3}\right) \cdot 2 | Apply the associative property of multiplication |

\displaystyle 6 \div \left( \dfrac{3}{2} \right) | \displaystyle = | \displaystyle \left(2\right) \cdot 2 | Simplify inside of the parentheses |

\displaystyle 4 | \displaystyle = | \displaystyle 4 | Simplify |

We can apply the associative properties to evaluate and simplify expressions.

Which property is demonstrated by the following statement?

4 \cdot \left(9 \cdot \left(-5\right)\right) = \left(4 \cdot 9\right) \cdot \left(-5\right)

A

Commutative property of multiplication

B

Associative property of multiplication

C

Distributive property

D

Associative property of addition

E

Commutative property of addition

Worked Solution

Find the value of: -27 + \left(27+38\right)

Worked Solution

Idea summary

Property | Symbols |
---|---|

\text{Associative property of addition} | a+\left(b+c\right) =\left(a+b\right)+c |

\text{Associative property of multiplication} | a\cdot(b\cdot c)=(a\cdot b)\cdot c |

Start by choosing one of each of the following types of numbers:

negative number

decimal

whole number

fraction

Do the following to each of your chosen numbers and write down what you notice:

Add 0.

Multiply by 0.

Multiply by 1.

Try to find a number that you can add to each number and get 0.

Try to find a number that you can multiply each number by and get 1.

What patterns did you observe?

The identity and inverse properties are:

Property | Symbols | Example |
---|---|---|

\text{Identity property of addition} | a+0=a \text{ and } 0+a=a | 6+0=6 \text{ and } 0+6=6 |

\text{Identity property of } \\ \text{multiplication} | a\cdot 1=a \text{ and }1 \cdot a =a | 9\cdot 1=9 \text{ and }1 \cdot 9 =9 |

\text{Inverse property of addition} | a+\left(-a\right)=0 \text{ and }\\ \left(-a\right)+a=0 | 3+\left(-3\right)=0 \text{ and }\left(-3\right)+3=0 |

\text{Inverse property of } \\ \text{multiplication} | {a \cdot \dfrac{1}{a} =1 \text{ and } \dfrac{1}{a} \cdot a=1} \\ \text{; where }a \neq 0 | 5 \cdot \dfrac{1}{5}=1 \text{ and } \dfrac{1}{5} \cdot 5 =1 |

\text{Multiplicative property of zero} | a \cdot 0 =0 \text{ and }0 \cdot a =0 | 4 \cdot 0 =0 \text{ and }0 \cdot 4 =0 |

All of these properties can be applied to help us evaluate expressions more easily.

Which property is demonstrated by the following statement?

\dfrac{1}{12} \cdot 12 = 1

A

Identity property of multiplication

B

Inverse property of multiplication

C

Multiplicative property of zero

D

Inverse property of addition

E

Identity property of addition

Worked Solution

Which property is demonstrated by the following statement?

\left(-\dfrac{3}{4}\right) + \dfrac{3}{4} = 0

A

Identity property of multiplication

B

Inverse property of multiplication

C

Multiplicative property of zero

D

Inverse property of addition

E

Identity property of addition

Worked Solution

Evaluate the expressions.

a

7\cdot 1 + \left(-7\right)\cdot 1

Worked Solution

b

29+\left(-29\right)+17-17

Worked Solution

Idea summary

Property | Symbols |
---|---|

\text{Identity property of addition} | a+0=a \text{ and } 0+a=a |

\text{Identity property of multiplication} | a\cdot 1=a \text{ and }1 \cdot a =a |

\text{Inverse property of addition} | a+\left(-a\right)=0 \text{ and }\\ \left(-a\right)+a=0 |

\text{Inverse property of multiplication} | {a \cdot \dfrac{1}{a} =1 \text{ and } \dfrac{1}{a} \cdot a=1} \\ \text{; where }a \neq 0 |

\text{Multiplicative property of zero} | a \cdot 0 =0 \text{ and }0 \cdot a =0 |