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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

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
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
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
7hours 36 min - 4hours 50 min
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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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