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By the end of this section, you will be able to:
  • Use the commutative and associative properties
  • Use the identity and inverse properties of addition and multiplication
  • Use the properties of zero
  • Simplify expressions using the distributive property

A more thorough introduction to the topics covered in this section can be found in the Prealgebra chapter, The Properties of Real Numbers .

Use the commutative and associative properties

Think about adding two numbers, say 5 and 3. The order we add them doesn’t affect the result, does it?

5 + 3 3 + 5 8 8
5 + 3 = 3 + 5

The results are the same.

As we can see, the order in which we add does not matter!

What about multiplying 5 and 3 ?

5 · 3 3 · 5 15 15
5 · 3 = 3 · 5

Again, the results are the same!

The order in which we multiply does not matter!

These examples illustrate the commutative property . When adding or multiplying, changing the order gives the same result.

Commutative property

of Addition If a , b are real numbers, then a + b = b + a of Multiplication If a , b are real numbers, then a · b = b · a

When adding or multiplying, changing the order gives the same result.

The commutative property has to do with order. If you change the order of the numbers when adding or multiplying, the result is the same.

What about subtraction? Does order matter when we subtract numbers? Does 7 3 give the same result as 3 7 ?

7 3 3 7 4 −4 4 4 7 3 3 7

The results are not the same.

Since changing the order of the subtraction did not give the same result, we know that subtraction is not commutative .

Let’s see what happens when we divide two numbers. Is division commutative?

12 ÷ 4 4 ÷ 12 12 4 4 12 3 1 3 3 1 3 12 ÷ 4 4 ÷ 12

The results are not the same.

Since changing the order of the division did not give the same result, division is not commutative . The commutative properties only apply to addition and multiplication!

  • Addition and multiplication are commutative.
  • Subtraction and Division are not commutative.

If you were asked to simplify this expression, how would you do it and what would your answer be?

7 + 8 + 2

Some people would think 7 + 8 is 15 and then 15 + 2 is 17 . Others might start with 8 + 2 makes 10 and then 7 + 10 makes 17 .

Either way gives the same result. Remember, we use parentheses as grouping symbols to indicate which operation should be done first.

( 7 + 8 ) + 2 Add 7 + 8 . 15 + 2 Add. 17 7 + ( 8 + 2 ) Add 8 + 2 . 7 + 10 Add. 17 ( 7 + 8 ) + 2 = 7 + ( 8 + 2 )

When adding three numbers, changing the grouping of the numbers gives the same result.

This is true for multiplication, too.

( 5 · 1 3 ) · 3 Multiply. 5 · 1 3 5 3 · 3 Multiply. 5 5 · ( 1 3 · 3 ) Multiply. 1 3 · 3 5 · 1 Multiply. 5 ( 5 · 1 3 ) · 3 = 5 · ( 1 3 · 3 )

When multiplying three numbers, changing the grouping of the numbers gives the same result.

You probably know this, but the terminology may be new to you. These examples illustrate the associative property .

Associative property

of Addition If a , b , c are real numbers, then ( a + b ) + c = a + ( b + c ) of Multiplication If a , b , c are real numbers, then ( a · b ) · c = a · ( b · c )

When adding or multiplying, changing the grouping gives the same result.

Let’s think again about multiplying 5 · 1 3 · 3 . We got the same result both ways, but which way was easier? Multiplying 1 3 and 3 first, as shown above on the right side, eliminates the fraction in the first step. Using the associative property can make the math easier!

Questions & Answers

divide 3x⁴-4x³-3x-1 by x-3
Ritik Reply
how to multiply the monomial
Ceny Reply
Two sisters like to compete on their bike rides. Tamara can go 4 mph faster than her sister, Samantha. If it takes Samantha 1 hours longer than Tamara to go 80 miles, how fast can Samantha ride her bike? Got questions? Get instant answers now!
Seera Reply
how do u solve that question
Seera
Two sisters like to compete on their bike rides. Tamara can go 4 mph faster than her sister, Samantha. If it takes Samantha 1 hours longer than Tamara to go 80 miles, how fast can Samantha ride her bike?
Seera
Speed=distance ÷ time
Tremayne
x-3y =1; 3x-2y+4=0 graph
Juned Reply
Brandon has a cup of quarters and dimes with a total of 5.55$. The number of quarters is five less than three times the number of dimes
ashley Reply
app is wrong how can 350 be divisible by 3.
Raheem Reply
June needs 48 gallons of punch for a party and has two different coolers to carry it in. The bigger cooler is five times as large as the smaller cooler. How many gallons can each cooler hold?
Susanna Reply
Susanna if the first cooler holds five times the gallons then the other cooler. The big cooler holda 40 gallons and the 2nd will hold 8 gallons is that correct?
Georgie
@Susanna that person is correct if you divide 40 by 8 you can see it's 5 it's simple
Ashley
@Geogie my bad that was meant for u
Ashley
Hi everyone, I'm glad to be connected with you all. from France.
Lorris Reply
I'm getting "math processing error" on math problems. Anyone know why?
Ray Reply
Can you all help me I don't get any of this
Jade Reply
4^×=9
Alberto Reply
Did anyone else have trouble getting in quiz link for linear inequalities?
Sireka Reply
operation of trinomial
Justin Reply
y=2×+9
Jacob Reply
Keshad gets paid $2,400 per month plus 6% of his sales. His brother earns $3,300 per month. For what amount of total sales will Keshad’s monthly pay be higher than his brother’s monthly pay?
Hector Reply
Practice Key Terms 4

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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