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# 1.7: Properties of Multiplication

Difficulty Level: At Grade Created by: CK-12

## Properties of Multiplication of Real Numbers

Objectives

The lesson objectives for The Properties Addition of Real Numbers are:

• The commutative property of multiplication
• The closure property of multiplication
• The associative property of multiplication
• The identity property of multiplication
• The inverse property of multiplication
• Property of negative one

Introduction

In this concept you will learn the properties of multiplication that apply to real numbers. These properties are the commutative property, the closure property, the associative property, the identity property, the inverse property and the property of negative one. You will learn what is meant by each of these properties and how each one applies to the multiplication of real numbers. You will learn to apply the properties to multiplication statements, how to recognize the property as it is applied to multiplication and to name the property that justifies a given multiplication statement.

Guidance

In the lesson that dealt with the multiplication of integers, the multiplication statement (2)×(3)\begin{align*}(-2)\times(-3)\end{align*} is represented by using color counters. The statement means to remove two groups of 3 yellow counters. The yellow counters are negative counters. The result is six positive (red) counters. Would the result be the same if the statement were (3)×(2)\begin{align*}(-3)\times(-2)\end{align*}?

(3)×(2)\begin{align*}(-3)\times(-2)\end{align*}? This statement means to remove 3 groups of two yellow counters.

The result is six positive (red) counters.

The order in which you multiplied the numbers did not affect the answer.

Example A

Does (3)×(+2)=(+2)×(3)\begin{align*}(-3)\times(+2)=(+2)\times(-3)\end{align*}?

Let’s use color counters to determine the answer.

(3)×(+2)\begin{align*}(-3)\times(+2)\end{align*}

This statement means to remove 3 groups of two red counters.

The result is 6 negative counters. Therefore (3)×(+2)=6\begin{align*}(-3)\times(+2)=-6\end{align*}.

(+2)×(3)\begin{align*}(+2)\times(-3)\end{align*}

This statement means to add two groups of three yellow counters.

The result is 6 negative counters. Therefore (+2)×(3)=6\begin{align*}(+2) \times (-3) = -6\end{align*}.

The result was the same regardless of the order in which the multiplication was performed. This is called the commutative property of multiplication. In general, the commutative property of multiplication states that the order in which two numbers are multiplied, does not affect the sum. If a\begin{align*}a\end{align*} and b\begin{align*}b\end{align*} are real numbers, then a×b=b×a\begin{align*}\boxed{a\times b=b\times a}\end{align*}.

Example B

Does (6)+(+3)=\begin{align*}(-6)+(+3)=\end{align*} a real number?

(6)×(+3)\begin{align*}(-6)\times(+3)\end{align*}

This multiplication statement means to remove six groups of 3 red counters.

(6)×(+3)=18\begin{align*}(-6)\times(+3)=-18\end{align*}

The result is -18. This is an integer. An integer is a real number. The product of any two real numbers will result in a real number. This is known as the closure property of multiplication. The result will always be a real number. In general, the closure property states that the product of any two real numbers is a unique real number. If a,b\begin{align*}a, b\end{align*} and c\begin{align*}c\end{align*} are real numbers, then a×b=c\begin{align*}\boxed{a\times b=c}\end{align*}.

Example C

Does (3×2)×2=3×(2×2)\begin{align*}(-3\times 2)\times 2=-3 \times (2 \times 2)\end{align*}?

(3×2)×2\begin{align*}(-3\times 2)\times 2\end{align*}

To begin the problem, we must do the multiplication inside the parenthesis.

This statement means to remove 3 groups of two red counters.

The result is 6 negative counters. Therefore (3)×(+2)=6\begin{align*}(-3)\times(+2)=-6\end{align*}.

Now the multiplication must be continued to represent (6)×(2)\begin{align*}(-6)\times(2)\end{align*}. This statement means to remove 6 groups of two red counters.

The result is 12 negative counters. Therefore (6)×(+2)=12\begin{align*}(-6)\times(+2)=-12\end{align*}.

3×(2×2)\begin{align*}-3\times(2 \times 2)\end{align*}

To begin the problem, we must do the multiplication inside the parenthesis.

This statement means to add 2 groups of two red counters.

The result is 4 positive counters. Therefore (+2)×(+2)=+4\begin{align*}(+2)\times(+2)=+4\end{align*}.

Now the multiplication must be continued to represent (3)×(4)\begin{align*}(-3)\times(4)\end{align*}. This statement means to remove 3 groups of four red counters.

(3×2)×2=3×(2×2)\begin{align*}(-3 \times 2) \times 2=-3 \times(2 \times 2)\end{align*}?

The numbers in the problem were the same but on the left side of the equal sign, the numbers -3 and +2 were grouped in parenthesis. The multiplication in the parenthesis was completed first and then -6 was multiplied by +2 to determine the final product. The result was -12.

On the right side of the equal sign, the numbers +2 and +2 were grouped in parenthesis. The multiplication in the parenthesis was completed first and then (+4) was multiplied by -3 to determine the final product. The result was -12.

The order in which three or more real numbers are grouped for multiplication, will not affect the product. This is known as the associative property of multiplication. The result will always be the same real number. In general, the associative property states that order in which the numbers are grouped for multiplication does not change the product. If a,b\begin{align*}a, b\end{align*} and c\begin{align*}c\end{align*} are real numbers, then (a×b)×c=a×(b×c)\begin{align*}\boxed{(a\times b)\times c=a \times(b \times c)}\end{align*}.

Example D

Does 8×1=8\begin{align*}8 \times 1=8\end{align*}?

The statement means to add 8 groups of one positive (red) counter.

The result is eight positive counters. Therefore the result 8×1=8\begin{align*}8 \times 1=8\end{align*} is correct.

Does 6×1=6\begin{align*}-6 \times 1=-6\end{align*}?

The statement means to remove 6 groups of 1 positive counter.

The result is six negative counters. Therefore the result 6×1=6\begin{align*}-6 \times 1 = -6\end{align*} is correct.

When any real number is multiplied by the number one, the real number does not change. In the above example, this was true whether the real number was positive or negative. The number 1 is called the multiplicative identity or the identity element of multiplication. The product of a number and one is the number. This is called the identity property of multiplication. If a\begin{align*}a\end{align*} is a real number, then a×1=a\begin{align*}\boxed{a \times 1 =a}\end{align*}.

Example E

Does 7×17=1\begin{align*}7 \times \frac{1}{7}=1\end{align*}?

You have already learned that multiplication can be thought of in terms of repeated addition.

To show this multiplication, a number line can be used. The number line must be divided into intervals of 7.

When 17\begin{align*}\frac{1}{7}\end{align*} was added seven times, the result was one. The fraction 17\begin{align*}\frac{1}{7}\end{align*} is the reciprocal of 7.

If a\begin{align*}a\end{align*} is a nonzero real number, then the reciprocal or multiplicative inverse of a\begin{align*}a\end{align*} is 1a\begin{align*}\frac{1}{a}\end{align*}. The product of any nonzero real number and its reciprocal is always one. The number 1 is called the multiplicative identity or the identity element of multiplication. Therefore, the product of a\begin{align*}a\end{align*} and its reciprocal is the identity element of multiplication (one). This is known as the inverse property of multiplication. If a\begin{align*}a\end{align*} is a nonzero real number, then a×1a=1\begin{align*}\boxed{a \times \frac{1}{a}=1}\end{align*}.

Example F

Does 8×1=8\begin{align*}8 \times -1=-8\end{align*}?

The statement means to add 8 groups of one negative (yellow) counter.

The result is eight negative counters. Therefore the result 8×1=8\begin{align*}8 \times -1=-8\end{align*} is correct.

Does 6×1=+6\begin{align*}-6 \times -1=+6\end{align*}?

The statement means to remove 6 groups of 1 negative counter.

The result is six positive counters. Therefore the result 6×1=+6\begin{align*}-6 \times -1=+6\end{align*} is correct.

When any real number is multiplied by the number negative one, the result is the opposite of the number. The product of a real number and -1 changes the sign of the real number. This is called the property of negative one. If a\begin{align*}a\end{align*} is a real number, then a×1=a\begin{align*}a \times -1=-a\end{align*}.

Vocabulary

Multiplicative Identity
The multiplicative identity for multiplication of real numbers is one.
Multiplicative Inverse
The multiplicative inverse of multiplication is the reciprocal of the nonzero real number and the product of the real number and its multiplicative inverse is one. If \begin{align*}a\end{align*} is any nonzero real number, its multiplicative inverse is \begin{align*}\frac{1}{a}\end{align*}.
Associative Property
The associative property of multiplication states the order in which three or more real numbers are grouped for multiplication, will not affect the product. If \begin{align*}a, b\end{align*} and \begin{align*}c\end{align*} are real numbers, then \begin{align*}\boxed{(a \times b)\times c=a \times(b \times c)}\end{align*}.
Closure Property
The closure property of multiplication states that the product of any two real numbers is a unique real number. If \begin{align*}a, b\end{align*} and \begin{align*}c\end{align*} are real numbers, then \begin{align*}a \times b = c\end{align*}.
Commutative Property
The commutative property of multiplication states that the order in which two numbers are multiplied, does not affect the product. If \begin{align*}a\end{align*} and \begin{align*}b\end{align*} are real numbers, then \begin{align*}\boxed{a \times b= b \times a}\end{align*}.
Identity Element of Multiplication
The identity element of multiplication is another term for the multiplicative identity of multiplication. Therefore, the identity element of multiplication is one.
Identity Property
The identity property of multiplication states that the product of a number and one is the number. If \begin{align*}a\end{align*} is a real number, then \begin{align*}\boxed{a \times 1=a}\end{align*}.
Inverse Property
The inverse property of multiplication states that the product of any real number and its multiplicative inverse is one. If \begin{align*}a\end{align*} is a nonzero real number, then \begin{align*}a \times \left(\frac{1}{a}\right)=1\end{align*}.

Guided Practice

1. Multiply using the properties of multiplication: \begin{align*}\left(6 \times \frac{1}{6} \right)\times(3 \times -1)\end{align*}

2. What property of multiplication justifies the statement \begin{align*}(-9 \times 5)\times 2= -9 \times (5 \times 2)\end{align*}?

3. Apply the negative one property of multiplication to the following problem: \begin{align*}-176 \times -1\end{align*}?

1. \begin{align*}& \left(6 \times \frac{1}{6}\right) \times (3 \times -1)\\ & (1) \times (3 \times -1) \ \rightarrow \ \text{Inverse Property}\\ & (1) \times (-3) \ \rightarrow \text{Negative One Property}\\ & = -3 \ \rightarrow \text{The Product of two numbers with unlike signs is always negative.}\end{align*}

2. \begin{align*}(-9 \times 5) \times 2 = -9 \times (5 \times 2)\end{align*}

The numbers on each side of the equal sign are the same but they are not grouped the same.

\begin{align*}&(-9 \times 5)\times 2 && -9 \times(5 \times 2)\\ &=(-45) \times 2 && =-9 \times(10)\\ &=-90 && =-90\end{align*}

The order in which the numbers were grouped did not affect the answer. The property that is being used is the associative property of multiplication.

3. \begin{align*}-176 \times -1 &= ?\\ -176 \times -1 &= 176\end{align*}

The number -176 is being multiplied by -1. The number remains the same but its sign has changed. This is the negative one property.

Summary

The properties of multiplication of real numbers do not apply to subtraction or division. Many of the properties were represented using color counters. You learned that the order in which two numbers were multiplied did not change the product. This was the commutative property of multiplication. An extension to this property was the associative property which stated that the order in which three or more numbers were grouped for multiplication had not affect on the final product. The closure property simply stated that the product of any two real numbers is a unique real number. The term unique means that the product is a different real number than those being multiplied. The identity property stated the product of any real number and one is the real number. You also learned that one is the multiplicative identity or the identity element of multiplication. Another property that you learned was the inverse property of multiplication. This property stated that the sum of any nonzero real number and its reciprocal is always one. The final property that you learned was the property of negative one. This property stated that the product of any real number and negative one will be the opposite of the real number. In other words, multiplying a real number by negative one will change the sign of the real number.

The guided examples showed you how to apply the properties to a given multiplication statement. You were also shown how to use the properties to justify a multiplication statement.

Problem Set

Match the following multiplication statements with the correct property of multiplication.

A. \begin{align*}(-1)\times 15=-15\end{align*}

B. \begin{align*}9 \times \frac{1}{9}=1\end{align*}

C. \begin{align*}(-7 \times 4)\times 2 = -7 \times(4 \times 2)\end{align*}

D. \begin{align*}-8 \times (4) = -32\end{align*}

E. \begin{align*}6 \times(-3)=(-3) \times 6\end{align*}

F. \begin{align*}-7 \times 1=-7\end{align*}

a) Commutative Property

b) Closure Property

c) Inverse Property

d) Identity Property

e) Associative Property

f) Negative One Property

In each of the following, circle the correct answer.

1. What does \begin{align*}-5(4)\left(-\frac{1}{5}\right)\end{align*} equal?
1. -20
2. -4
3. +20
4. +4
2. What is another name for the reciprocal of any real number?
2. the multiplicative identity
3. the multiplicative inverse
3. What is the multiplicative identity?
1. -1
2. 1
3. 0
4. \begin{align*}\frac{1}{2}\end{align*}
4. What is the product of a nonzero real number and its multiplicative inverse?
1. 1
2. -1
3. 0
4. there is no product
5. Which of the following statements is NOT true?
1. The product of any real number and negative one is the opposite of the real number.
2. The product of any real number and zero is always zero.
3. The order in which two real numbers are multiplied does not affect the product.
4. The product of any real number and negative one is always a negative number.

Name the property of multiplication that is being shown in each of the following multiplication statements:

1. \begin{align*}(-6\times 7)\times 2=-6 \times(7 \times 2)\end{align*}
2. \begin{align*}-12 \times 1 =-12\end{align*}
3. \begin{align*}25 \times 3 = 3 \times 25\end{align*}
4. \begin{align*}10 \times \frac{1}{10}=1\end{align*}
5. \begin{align*}-12 \times -1=12\end{align*}

Match the following...

A. \begin{align*}(-1)\times 15=-15\end{align*} \begin{align*}\rightarrow\end{align*} f) Negative One Property

B. \begin{align*}9 \times \frac{1}{9}=1\end{align*} \begin{align*}\rightarrow\end{align*} c) Inverse Property

C. \begin{align*}(-7 \times 4)\times 2 = -7 \times(4 \times 2)\end{align*} \begin{align*}\rightarrow\end{align*} e) Associative Property

D. \begin{align*}-8 \times (4) = -32\end{align*} \begin{align*}\rightarrow\end{align*} b) Closure Property

E. \begin{align*}6 \times(-3)=(-3) \times 6\end{align*} \begin{align*}\rightarrow\end{align*} a) Commutative Property

F. \begin{align*}-7 \times 1=-7\end{align*} \begin{align*}\rightarrow\end{align*} d) Identity Property

In each of the following...

1. D
2. C
3. B
4. A
5. D

Name the property...

1. \begin{align*}(-6\times 7)\times 2=-6 \times(7 \times 2)\end{align*} Associative Property
1. \begin{align*}25 \times 3 = 3 \times 25\end{align*} Commutative property
1. \begin{align*}-12 \times -1=12\end{align*} Negative One Property

## Summary

In these lessons you have learned the properties of real numbers for both addition and multiplication. The properties for each of these are very closely related. The properties that you have learned for the real numbers \begin{align*}a, b\end{align*} and \begin{align*}c\end{align*} are:

Commutative \begin{align*}a+b=b+a\end{align*} \begin{align*}a \times b=b \times a\end{align*}
Associative \begin{align*}(a+b)+c=a+(b+c)\end{align*} \begin{align*}(a \times b) \times c=a \times (b\times c)\end{align*}
Closure \begin{align*}a+b=c\end{align*} \begin{align*}a\times b = c\end{align*}
Identity \begin{align*}a+0=a\end{align*} \begin{align*}a \times 1= a\end{align*}
Inverse \begin{align*}a+(-a)=0\end{align*} \begin{align*}a \times \frac{1}{a}=1\end{align*}
Negative One Not Applicable \begin{align*}a \times -1=-a\end{align*}

The properties were represented using color counters and/or the number line. In the problem sets for each lesson, you had the opportunity to apply your understanding of the properties. In algebra, these properties are very important. In addition to the properties themselves being important, it is also important to understand the terms that were introduced. You should know that the additive inverse of any real number is the opposite of the real number. You should also know that the additive identity of a real number is zero. For multiplication, the multiplicative inverse is the reciprocal of the real number and the multiplicative identity is one. You will apply the negative one property infinitely when you are multiplying.

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