Does give the same result as ?
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 and are real numbers, then .
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 and are real numbers, then .
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 the order in which the numbers are grouped for multiplication does not change the product. If and are real numbers, then .
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 is a real number, then .
Multiplicative Inverse
If is a nonzero real number, then the reciprocal or multiplicative inverse of is . 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 and its reciprocal is the identity element of multiplication (one). This is known as the inverse property of multiplication. If is a nonzero real number, then .
Example A
Does ?
Solution: .
This is an example of the commutative property of multiplication.
Example B
Does a real number?
Solution: , a real number. This is an example of the closure property of multiplication.
Example C
Does ?
Solution: . Even though the numbers are grouped differently, the result is the same. This is an example of the associative property of multiplication.
Example D
Does ?
Solution: Yes. This is an example of the identity property of multiplication.
Example E
Does ?
Solution: Yes. This is an example of the inverse property of multiplication.
Concept Problem Revisited
.
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 is any nonzero real number, its multiplicative inverse is .
- 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 and are real numbers, then .
- Closure Property
- The closure property of multiplication states that the product of any two real numbers is a unique real number. If and are real numbers, then .
- Commutative Property
- The commutative property of multiplication states that the order in which two numbers are multiplied does not affect the product. If and are real numbers, then .
- 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 is a real number, then .
- Inverse Property
- The inverse property of multiplication states that the product of any real number and its multiplicative inverse is one. If is a nonzero real number, then .
Guided Practice
1. Multiply using the properties of multiplication:
2. What property of multiplication justifies the statement ?
3. What property of multiplication justifies the statement ?
Answers:
1.
2. associative property of multiplication
3. identity property of multiplication
Practice
Match the following multiplication statements with the correct property of multiplication.
- Commutative Property
- Closure Property
- Inverse Property
- Identity Property
- Associative Property
In each of the following, circle the correct answer.
- What does 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
- 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: