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This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses properties of multiplication of whole numbers. By the end of the module students should be able to understand and appreciate the commutative and associative properties of multiplication and understand why 1 is the multiplicative identity.

Section overview

  • The Commutative Property of Multiplication
  • The Associative Property of Multiplication
  • The Multiplicative Identity

We will now examine three simple but very important properties of multiplication.

The commutative property of multiplication

Commutative property of multiplication

The product of two whole numbers is the same regardless of the order of the factors.

Sample set a

Multiply the two whole numbers.

6 and 7.

6 7 = 42 size 12{6 cdot 7="42"} {}

7 6 = 42 size 12{7 cdot 6="42"} {}

The numbers 6 and 7 can be multiplied in any order. Regardless of the order they are multiplied, the product is 42.

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Practice set a

Use the commutative property of multiplication to find the products in two ways.

15 and 6.

15 6 = 90 and 6 15 = 90

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432 and 428.

432 428 = 184,896 and 428 432 = 184,896

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The associative property of multiplication

Associative property of multiplication

If three whole numbers are multiplied, the product will be the same if the first two are multiplied first and then that product is multiplied by the third, or if the second two are multiplied first and that product is multiplied by the first. Note that the order of the factors is maintained.

It is a common mathematical practice to use parentheses to show which pair of numbers is to be combined first.

Sample set b

Multiply the whole numbers.

8, 3, and 14.

( 8 3 ) 14 = 24 14 = 336 size 12{ \( 8 cdot 3 \) cdot "14"="24" cdot "14"="336"} {}

8 ( 3 14 ) = 8 42 = 336 size 12{8 cdot \( 3 cdot "14" \) =8 cdot "42"="336"} {}

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Practice set b

Use the associative property of multiplication to find the products in two ways.

The multiplicative identity

The multiplicative identity is 1

The whole number 1 is called the multiplicative identity , since any whole num­ber multiplied by 1 is not changed.

Sample set c

Multiply the whole numbers.

12 and 1.

12 1 = 12 size 12{"12" cdot 1="12"} {}

1 12 = 12 size 12{1 cdot "12"="12"} {}

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Practice set c

Multiply the whole numbers.

Exercises

For the following problems, multiply the numbers.

For the following 4 problems, show that the quantities yield the same products by performing the multiplications.

( 4 8 ) 2 size 12{ \( 4 cdot 8 \) cdot 2} {} and 4 ( 8 2 ) size 12{4 cdot \( 8 cdot 2 \) } {}

32 2 = 64 = 4 16 size 12{"32" cdot 2="64"=4 cdot "16"} {}

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( 100 62 ) 4 size 12{ \( "100" cdot "62" \) cdot 4} {} and 100 ( 62 4 ) size 12{"100" cdot \( "62" cdot 4 \) } {}

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23 ( 11 106 ) size 12{"23" cdot \( "11" cdot "106" \) } {} and ( 23 11 ) 106 size 12{ \( "23" cdot "11" \) cdot "106"} {}

23 1, 166 = 26 , 818 = 253 106 size 12{"23" cdot 1,"166"="26","818"="253" cdot "106"} {}

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1 ( 5 2 ) size 12{1 cdot \( 5 cdot 2 \) } {} and ( 1 5 ) 2 size 12{ \( 1 cdot 5 \) cdot 2} {}

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The fact that ( a first number a second number ) a third number = a first number ( a second number a third number ) size 12{ \( "a first number " cdot " a second number" \) cdot " a third number "=" a first number " cdot \( "a second number " cdot " a third number" \) } {} is an example of the property of mul­tiplication.

associative

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The fact that 1 any number = that particular number size 12{"1 " cdot " any number "=" that particular number"} {} is an example of the property of mul­tiplication.

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Use the numbers 7 and 9 to illustrate the com­mutative property of multiplication.

7 9 = 63 = 9 7 size 12{"7 " cdot " 9 "=" 63 "=" 9 " cdot " 7"} {}

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Use the numbers 6, 4, and 7 to illustrate the asso­ciative property of multiplication.

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Exercises for review

( [link] ) In the number 84,526,098,441, how many millions are there?

6

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( [link] ) Replace the letter m with the whole number that makes the addition true. 85 +   m ̲ 97 alignr { stack { size 12{"85"} {} #size 12{ {underline {+m}} } {} # size 12{"97"} {}} } {}

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( [link] ) Use the numbers 4 and 15 to illustrate the commutative property of addition.

4 + 15 = 19 size 12{"4 "+" 15 "=" 19"} {}

15 + 4 = 19 size 12{"15 "+" 4 "=" 19"} {}

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( [link] ) Find the product. 8, 000 , 000 × 1, 000 size 12{8,"000","000" times 1,"000"} {} .

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( [link] ) Specify which of the digits 2, 3, 4, 5, 6, 8,10 are divisors of the number 2,244.

2, 3, 4, 6

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Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
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Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
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Adin
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Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
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sciencedirect big data base
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Introduction about quantum dots in nanotechnology
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what does nano mean?
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nano basically means 10^(-9). nanometer is a unit to measure length.
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it is a goid question and i want to know the answer as well
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characteristics of micro business
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for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
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Do you know which machine is used to that process?
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for screen printed electrodes ?
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s. Reply
of graphene you mean?
Ebrahim
or in general
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in general
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Graphene has a hexagonal structure
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Source:  OpenStax, Fundamentals of mathematics. OpenStax CNX. Aug 18, 2010 Download for free at http://cnx.org/content/col10615/1.4
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