1.10: Properties of Real Number Multiplication
The multiplication statement \begin{align*}(-2)\times(-3)\end{align*} can be 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 \begin{align*}(-3)\times(-2)\end{align*}?
Guidance
There are five properties of multiplication that are important for you to know.
Commutative Property
The commutative property of multiplication states that the order in which two numbers are multiplied does not affect the sum. If \begin{align*}a\end{align*} and \begin{align*}b\end{align*} are real numbers, then \begin{align*}a\times b=b\times a\end{align*}.
Closure Property
The product of any two real numbers will result in a real number. This is known as the closure property of multiplication. In general, the closure property 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*}.
Associative Property
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 \begin{align*}a, b\end{align*} and \begin{align*}c\end{align*} are real numbers, then \begin{align*}(a\times b)\times c=a \times(b \times c)\end{align*}.
Multiplicative Identity
When any real number is multiplied by the number one, the real number does not change. This is true whether the real number is 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 \begin{align*}a\end{align*} is a real number, then \begin{align*}a \times 1 =a\end{align*}.
Multiplicative Inverse
If \begin{align*}a\end{align*} is a nonzero real number, then the reciprocal or multiplicative inverse of \begin{align*}a\end{align*} is \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 \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 \begin{align*}a\end{align*} is a nonzero real number, then \begin{align*}a \times \frac{1}{a}=1\end{align*}.
Example A
Does \begin{align*}(-3)\times(+2)=(+2)\times(-3)\end{align*}?
You can use color counters to determine the answer.
\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 \begin{align*}(-3)\times(+2)=-6\end{align*}.
\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 \begin{align*}(+2) \times (-3) = -6\end{align*}.
This is an example of the commutative property of multiplication.
Example B
Does \begin{align*}(-6)+(+3)=\end{align*} a real number?
\begin{align*}(-6)\times(+3)\end{align*}
This multiplication statement means to remove six groups of 3 red counters.
\begin{align*}(-6)\times(+3)=-18\end{align*}
The result is -18. This is an integer. An integer is a real number. This is an example of the closure property.
Example C
Does \begin{align*}(-3\times 2)\times 2=-3 \times (2 \times 2)\end{align*}?
\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 \begin{align*}(-3)\times(+2)=-6\end{align*}.
Now the multiplication must be continued to represent \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 \begin{align*}(-6)\times(+2)=-12\end{align*}.
\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 \begin{align*}(+2)\times(+2)=+4\end{align*}.
Now the multiplication must be continued to represent \begin{align*}(-3)\times(4)\end{align*}. This statement means to remove 3 groups of four red counters.
\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.
Example D
Does \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 \begin{align*}8 \times 1=8\end{align*} is correct.
Does \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 \begin{align*}-6 \times 1 = -6\end{align*} is correct.
This is an example of the identity property of multiplication.
Example E
Does \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 \begin{align*}\frac{1}{7}\end{align*} was added seven times, the result was one. The fraction \begin{align*}\frac{1}{7}\end{align*} is the reciprocal of 7. This is an example of the inverse property of multiplication.
Concept Problem Revisited
Here is \begin{align*}(-2)\times (-3)\end{align*}:
\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. This is an example of the commutative property of multiplication.
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*}(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*}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*}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*}?
Answers:
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) = -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.
Practice
Match the following multiplication statements with the correct property of multiplication.
1. \begin{align*}9 \times \frac{1}{9}=1\end{align*}
2. \begin{align*}(-7 \times 4)\times 2 = -7 \times(4 \times 2)\end{align*}
3. \begin{align*}-8 \times (4) = -32\end{align*}
4. \begin{align*}6 \times(-3)=(-3) \times 6\end{align*}
5. \begin{align*}-7 \times 1=-7\end{align*}
a) Commutative Property
b) Closure Property
c) Inverse Property
d) Identity Property
e) Associative Property
In each of the following, circle the correct answer.
- What does \begin{align*}-5(4)\left(-\frac{1}{5}\right)\end{align*} equal?
- -20
- -4
- +20
- +4
- What is another name for the reciprocal of any real number?
- the additive identity
- the multiplicative identity
- the multiplicative inverse
- the additive inverse
- What is the multiplicative identity?
- -1
- 1
- 0
- \begin{align*}\frac{1}{2}\end{align*}
- What is the product of a nonzero real number and its multiplicative inverse?
- 1
- -1
- 0
- there is no product
- Which of the following statements is NOT true?
- The product of any real number and negative one is the opposite of the real number.
- The product of any real number and zero is always zero.
- The order in which two real numbers are multiplied does not affect the product.
- 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:
- \begin{align*}(-6\times 7)\times 2=-6 \times(7 \times 2)\end{align*}
- \begin{align*}-12 \times 1 =-12\end{align*}
- \begin{align*}25 \times 3 = 3 \times 25\end{align*}
- \begin{align*}10 \times \frac{1}{10}=1\end{align*}
- \begin{align*}-12 \times -1=-1\times -12\end{align*}
Image Attributions
Here 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 and the inverse property. You will learn what is meant by each of these properties and how each one applies to the multiplication of real numbers.
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