# 2.5 Properties of multiplication

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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.

## 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.

$6\cdot 7=\text{42}$

$7\cdot 6=\text{42}$

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

## Practice set a

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

$\mathrm{15}\cdot 6=\mathrm{90}$ and $6\cdot \mathrm{15}=\mathrm{90}$

$\mathrm{432}\cdot \mathrm{428}=\mathrm{184,896}$ and $\mathrm{428}\cdot \mathrm{432}=\mathrm{184,896}$

## 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.

$\left(8\cdot 3\right)\cdot \text{14}=\text{24}\cdot \text{14}=\text{336}$

$8\cdot \left(3\cdot \text{14}\right)=8\cdot \text{42}=\text{336}$

## Practice set b

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

168

165,564

## 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.

$\text{12}\cdot 1=\text{12}$

$1\cdot \text{12}=\text{12}$

## Practice set c

Multiply the whole numbers.

843

## Exercises

For the following problems, multiply the numbers.

234

4,032

326,000

252

21,340

8,316

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

$\left(4\cdot 8\right)\cdot 2$ and $4\cdot \left(8\cdot 2\right)$

$\text{32}\cdot 2=\text{64}=4\cdot \text{16}$

$\left(\text{100}\cdot \text{62}\right)\cdot 4$ and $\text{100}\cdot \left(\text{62}\cdot 4\right)$

$\text{23}\cdot \left(\text{11}\cdot \text{106}\right)$ and $\left(\text{23}\cdot \text{11}\right)\cdot \text{106}$

$\text{23}\cdot 1,\text{166}=\text{26},\text{818}=\text{253}\cdot \text{106}$

$1\cdot \left(5\cdot 2\right)$ and $\left(1\cdot 5\right)\cdot 2$

The fact that $\left(\text{a first number}\cdot \text{a second number}\right)\cdot \text{a third number}=\text{a first number}\cdot \left(\text{a second number}\cdot \text{a third number}\right)$ is an example of the property of mul­tiplication.

associative

The fact that $\text{1}\cdot \text{any number}=\text{that particular number}$ is an example of the property of mul­tiplication.

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

$\text{7}\cdot \text{9}=\text{63}=\text{9}\cdot \text{7}$

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

## Exercises for review

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

6

( [link] ) Replace the letter m with the whole number that makes the addition true.

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

$\text{4}+\text{15}=\text{19}$

$\text{15}+\text{4}=\text{19}$

( [link] ) Find the product. $8,\text{000},\text{000}×1,\text{000}$ .

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

2, 3, 4, 6

show that the set of all natural number form semi group under the composition of addition
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