<< Chapter < Page Chapter >> Page >
By the end of this section, you will be able to:
  • Use the commutative and associative properties
  • Evaluate expressions using the commutative and associative properties
  • Simplify expressions using the commutative and associative properties

Before you get started, take this readiness quiz.

  1. Simplify: 7 y + 2 + y + 13 .
    If you missed this problem, review Evaluate, Simplify and Translate Expressions .
  2. Multiply: 2 3 · 18 .
    If you missed this problem, review Multiply and Divide Fractions .
  3. Find the opposite of 15 .
    If you missed this problem, review Subtract Integers .

In the next few sections, we will take a look at the properties of real numbers. Many of these properties will describe things you already know, but it will help to give names to the properties and define them formally. This way we’ll be able to refer to them and use them as we solve equations in the next chapter.

Use the commutative and associative properties

Think about adding two numbers, such as 5 and 3 .

5 + 3 3 + 5 8 8

The results are the same. 5 + 3 = 3 + 5

Notice, the order in which we add does not matter. The same is true when multiplying 5 and 3 .

5 · 3 3 · 5 15 15

Again, the results are the same! 5 · 3 = 3 · 5 . The order in which we multiply does not matter.

These examples illustrate the commutative properties of addition and multiplication.

Commutative properties

Commutative Property of Addition : if a and b are real numbers, then

a + b = b + a

Commutative Property of Multiplication : if a and b are real numbers, then

a · b = b · a

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

Use the commutative properties to rewrite the following expressions:

−1 + 3 = _____

4 · 9 = _____

Solution

−1 + 3 = _____
Use the commutative property of addition to change the order. −1 + 3 = 3 + ( −1 )
4 · 9 = _____
Use the commutative property of multiplication to change the order. 4 · 9 = 9 · 4
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Use the commutative properties to rewrite the following:

  1. −4 + 7 = _____
  2. 6 · 12 = _____

  1. −4 + 7 = 7 + (−4)
  2. 6 · 12 = 12 · 6
Got questions? Get instant answers now!

Use the commutative properties to rewrite the following:

  1. 14 + ( −2 ) = _____
  2. 3 ( −5 ) = _____

  1. 14 + (−2) = −2 + 14
  2. 3(−5) = (−5)3
Got questions? Get instant answers now!

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
The results are not the same. 7 3 3 7

Since changing the order of the subtraction did not give the same result, we can say 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
The results are not the same. So 12 ÷ 4 4 ÷ 12

Since changing the order of the division did not give the same result, division is not commutative.

Addition and multiplication are commutative. Subtraction and division are not commutative.

Suppose you were asked to simplify this expression.

7 + 8 + 2

How would you do it and what would your answer be?

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 .

Both ways give the same result, as shown in [link] . (Remember that parentheses are grouping symbols that indicate which operations should be done first.)

Questions & Answers

what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
y=10×
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
is it 3×y ?
Joan Reply
J, combine like terms 7x-4y
Bridget Reply
im not good at math so would this help me
Rachael Reply
how did I we'll learn this
Noor Reply
f(x)= 2|x+5| find f(-6)
Prince Reply
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Prealgebra. OpenStax CNX. Jul 15, 2016 Download for free at http://legacy.cnx.org/content/col11756/1.9
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Prealgebra' conversation and receive update notifications?

Ask