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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

a perfect square v²+2v+_
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algebra 2 Inequalities:If equation 2 = 0 it is an open set?
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or infinite solutions?
Kim
y=10×
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if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
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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
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No. 7x -4y is simplified from 4x + (3y + 3x) -7y
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is it 3×y ?
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J, combine like terms 7x-4y
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f(x)= 2|x+5| find f(-6)
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f(n)= 2n + 1
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Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
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. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
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Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
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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.
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the Beer law works very well for dilute solutions but fails for very high concentrations. why?
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Smarajit Reply
4^×=9
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Did anyone else have trouble getting in quiz link for linear inequalities?
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operation of trinomial
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y=2×+9
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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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