Topics
6. Equations & Inequalities
topic badge
United States of AmericaVirginia
Grade 6

6.02 Properties of real numbers

Lesson
Practice

Commutative properties

Exploration

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.

Loading interactive...

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

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

The commutative properties of real numbers are:

PropertySymbolsExample
\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 a6 \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 4-7Apply the commutative property with subtraction
\displaystyle 3\displaystyle \neq\displaystyle -3Simplify

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+7Apply the commutative property of addition
\displaystyle 3\displaystyle =\displaystyle 3Simplify

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 6Apply 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 6Apply the commutative property of multiplication
\displaystyle 2\displaystyle =\displaystyle 2Simplify

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.

Examples

Example 1

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

Worked Solution
Create a strategy

Since we can add numbers in any order, let's add -5 and 5 first to make the calculation easier.

Apply the idea
\displaystyle 6+\left(-5\right)+5\displaystyle =\displaystyle -5+5+6Rewrite using the commutative property of addition
\displaystyle =\displaystyle 0+6Evaluate -5 + 5
\displaystyle =\displaystyle 6Evaluate
Reflect and check

We could apply a different property to get the same result:

\displaystyle 6+\left(-5\right)+5\displaystyle =\displaystyle 6+\left(\left(-5\right)+5\right)Rewrite using the associative property of addition
\displaystyle =\displaystyle 6+0Evaluate -5 + 5
\displaystyle =\displaystyle 6Evaluate

Example 2

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

19 + \left(-15\right) = \left(-15\right) + ⬚

Worked Solution
Create a strategy

The commutative property of addition means that when we add two numbers, it does not matter what order we add them.

Apply the idea

We want to write 19 + \left(-15\right) the opposite way around. 19 + \left(-15\right) = \left(-15\right) + 19

Example 3

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

Worked Solution
Create a strategy

Since we can multiply numbers in any order, let's find the product of -20 and 5 first to make the calculation easier.

Apply the idea
\displaystyle -20 \cdot 7 \cdot 5\displaystyle =\displaystyle -20 \cdot 5 \cdot 7Rewrite using the commutative property of multiplication
\displaystyle =\displaystyle -100 \cdot 7Evaluate -20 \cdot 5
\displaystyle =\displaystyle -700Evaluate
Reflect and check

If we didn't apply the commutative property of multiplication, we would get the same answer, but it would be with numbers that are more difficult to multiply.

\displaystyle -20 \cdot 7 \cdot 5\displaystyle =\displaystyle -140 \cdot 5Multiply -20 \cdot 7
\displaystyle =\displaystyle -700Evaluate

Example 4

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
Create a strategy

Only the addition operation is used in the statement so we know that the answer must be a addition property. To figure out which property, consider how the addition differs on the two sides of the equation.

Apply the idea

The statement demonstrates an commutative property of addition.

When we add two or more numbers, we can reorder them and get the same answer. So, the correct answer is E.

Idea summary
PropertySymbols
\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.

Associative properties

The associative properties of real numbers are:

PropertySymbolsExample
\text{Associative property of addition} a+\left(b+c\right) \\ =\left(a+b\right)+c6+\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 c6 \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 7-7Simplify inside of the parentheses
\displaystyle 6\displaystyle \neq\displaystyle 0Simplify

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) +3Rewrite 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 6Simplify

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 2Apply the associative property with division
\displaystyle 6 \div \left(\dfrac{3}{2} \right)\displaystyle \neq\displaystyle \left( 2 \right) \div 2Simplify inside of the parentheses
\displaystyle 4\displaystyle \neq\displaystyle 1Simplify

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 4Simplify

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

Examples

Example 5

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
Create a strategy

Only the multiplication operation is used in the statement so we know that the answer must be a multiplication property. To figure out which property, consider how the multiplication differs on the two sides of the equation.

Apply the idea

The statement demonstrates an associative property of multiplication.

When we multiply three numbers, we can group the first two or the last two and get the same answer. So, the correct answer is B.

Example 6

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

Worked Solution
Create a strategy

Use the associative property to group-27 and 27 to make the calculation simpler.

Apply the idea
\displaystyle -27 + \left(27+38\right)\displaystyle =\displaystyle \left(-27 + 27\right)+38Rewrite using the associative property of addition
\displaystyle =\displaystyle 0+38Evaluate the parentheses
\displaystyle =\displaystyle 38Evaluate the addition
Idea summary
PropertySymbols
\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

Inverse and identity properties

Exploration

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:

  1. Add 0.

  2. Multiply by 0.

  3. Multiply by 1.

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

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

  6. What patterns did you observe?

The identity and inverse properties are:

PropertySymbolsExample
\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 =a9\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.

Examples

Example 7

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
Create a strategy

Only the multiplication operation is used in the statement so we know that the answer must be a multiplication property. To figure out which property, observe the relationship between the numbers on the left side of the equation and the value on the right side of the equation.

Apply the idea

The statement demonstrates an inverse property of multiplication.

When we multiply a number with its inverse, the answer would be 1. So, the correct answer is B.

Example 8

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
Create a strategy

Only the addition operation is used in the statement so we know that the answer must be an addition property. To figure out which property, observe the relationship between the numbers on the left side of the equation and the value on the right side of the equation.

Apply the idea

The statement demonstrates an inverse property of addition.

When we add a number to its inverse, the answer would be 0. So, the correct answer is D.

Example 9

Evaluate the expressions.

a

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

Worked Solution
Create a strategy

Notice two numbers are being multiplied by 1 so let's start by using the identity property of multiplication.

Apply the idea
\displaystyle 7\cdot 1 + \left(-7\right)\cdot 1\displaystyle =\displaystyle 7\ + \left(-7\right)Identity property of multiplication
\displaystyle =\displaystyle 0Inverse property of addition
b

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

Worked Solution
Create a strategy

Notice 29 is being added to its opposite, so let's start by using the inverse property of addition.

Apply the idea
\displaystyle 29+\left(-29\right)+17-17\displaystyle =\displaystyle 17-17Inverse property of addition
\displaystyle =\displaystyle 0Evaluate the subtraction
Idea summary
PropertySymbols
\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

What is Mathspace

About Mathspace