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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

show that the set of all natural number form semi group under the composition of addition
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The denominator of a certain fraction is 9 more than the numerator. If 6 is added to both terms of the fraction, the value of the fraction becomes 2/3. Find the original fraction. 2. The sum of the least and greatest of 3 consecutive integers is 60. What are the valu
SABAL Reply
1. x + 6 2 -------------- = _ x + 9 + 6 3 x + 6 3 ----------- x -- (cross multiply) x + 15 2 3(x + 6) = 2(x + 15) 3x + 18 = 2x + 30 (-2x from both) x + 18 = 30 (-18 from both) x = 12 Test: 12 + 6 18 2 -------------- = --- = --- 12 + 9 + 6 27 3
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2. (x) + (x + 2) = 60 2x + 2 = 60 2x = 58 x = 29 29, 30, & 31
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×/×+9+6/1
Debbie
Q2 x+(x+2)+(x+4)=60 3x+6=60 3x+6-6=60-6 3x=54 3x/3=54/3 x=18 :. The numbers are 18,20 and 22
Naagmenkoma
Mark and Don are planning to sell each of their marble collections at a garage sale. If Don has 1 more than 3 times the number of marbles Mark has, how many does each boy have to sell if the total number of marbles is 113?
mariel Reply
Mark = x,. Don = 3x + 1 x + 3x + 1 = 113 4x = 112, x = 28 Mark = 28, Don = 85, 28 + 85 = 113
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4
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x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
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Solve for the first variable in one of the equations, then substitute the result into the other equation. Point For: (6111,4111,−411)(6111,4111,-411) Equation Form: x=6111,y=4111,z=−411x=6111,y=4111,z=-411
Brenna
(61/11,41/11,−4/11)
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x=61/11 y=41/11 z=−4/11 x=61/11 y=41/11 z=-4/11
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Need help solving this problem (2/7)^-2
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x+2y-z=7
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-1
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A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place.
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Jeannette has $5 and $10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
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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?
Chris Reply
In the number 779,844,205 how many ten millions are there?
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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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