In this explainer, we will learn how to identify the properties of the multiplication operation in the set of rational numbers.
We first recall that if , , , and and and are nonzero so that and are rational numbers, then we can add and multiply these numbers by using the following formulas:
These definitions allow us to show many properties that multiplication and addition have over rational numbers. For example, we can recall that the addition of rational numbers is a closed operation. In other words, if we add two rational numbers together, the result is also a rational number. The same is true for the product of two rational numbers. We see that since and .
We can also show that the multiplication of rational numbers is commutative. To do this, we note that , , , and are integers and the multiplication of integers is commutative, so and . Hence,
We can also recall that the addition of rational numbers is associative. Let’s consider whether the multiplication of rational numbers is associative. To do this, we first evaluate the product of three rational numbers as follows:
Next, we note that the product of integers is associative, so we can rewrite this as
Hence, the product of rational numbers is associative.
There are three more properties we can find by considering the properties of addition we already know. Let’s next consider whether there is a multiplicative identity. That is, a rational number that leaves all rational numbers unchanged under multiplication. We note that
So, 1 is the multiplicative identity for the rational numbers.
Next, we want to consider whether all rational numbers have an inverse. That is, a rational number that, when multiplied by our original rational number, will give the multiplicative identity, 1. To do this, we need to be careful. We note that if , then
We need to be nonzero; otherwise, the reciprocal is not rational and not defined. We cannot find a multiplicative inverse for 0 since
In other words, we cannot find a rational number that multiplies with 0 to give 1. The fact that multiplying any rational number by 0 gives 0 is called the zero-product property. Hence, all rational numbers (except 0) have a multiplicative inverse given by .
There is one final property that links the multiplication and addition of rational numbers. To see where this comes from, we can recall that, for integers , , and , we have
This is called the distribution of multiplication over addition, and it holds for rational numbers as well. We see that
We can cancel the shared factors of and since these are nonzero, giving
It is worth noting that we can apply this property in both directions. We can distribute the product of a rational number over addition, and we can also factor out rational numbers over a sum.
We have shown the following properties for the multiplication of rational numbers.
Properties: Multiplication of Rational Numbers
For any rational numbers , , and , we have the following:
The closure property
is a rational number.
In particular, .
The commutativity of multiplication property
Here, .
In other words, we can reorder the product of rational numbers.
The associativity of multiplication property
Here, .
In other words, we can evaluate the product of rational numbers in any order.
The multiplicative identity property
Here, .
The multiplicative identity is 1 since multiplying a rational number by 1 leaves it unchanged.
The multiplicative inverse property
If , then and .
The zero-product property
Here, .
The distributive property of multiplication over addition
Here, .
We can apply this property in both directions, either to distribute a product over a sum or to take out a multiplicative factor over a sum.
Let’s now see some examples of using these properties to answer questions involving the product of rational numbers.
Example 1: Understanding the Closure Property of Multiplication
If , , , and , which of the following expressions results in a rational number?
Answer
We first recall that the closure property for the multiplication of rational numbers tells us that the product of rational numbers is always rational. So, if both factors in any of the options are rational, then we know their product is rational.
In option A, we note that the denominator of the second factor evaluates to give 0 since
Since we cannot divide by zero, this expression is not defined for these values. It is worth noting that the same denominator also appears in option E, so this expression is not defined for these values either. Since an undefined value is not a rational number, we can eliminate options A and E.
In option B, we note that the denominator of the first factor evaluates to give 0 since
So, this expression is not defined for these values and we can eliminate option B.
In option C, we can substitute the values into the expression and evaluate to get
Since 3 is not a perfect square, we know that is irrational. The same is true for since dividing an irrational number by an integer leaves the number irrational. Thus, this expression is not rational. Hence, the answer cannot be option C.
In option D, we see that both factors are rational since
Hence, their product is rational by the closure property for the multiplication of rational numbers.
Hence, only the expression in option D is rational for these values.
In our next example, we will determine which of five given expressions correctly represents the commutative property of multiplication.
Example 2: Identifying the Commutative Property of Multiplication
Which equation shows the commutative property of multiplication?
Answer
We first recall that the commutative property of multiplication tells us that, for any rational numbers and , we have
Since 3.5 and 2 are both rational numbers, we can see that only option C includes the reordering of the product of numbers.
It is worth noting that the other options show uses of the other properties of the multiplication of rational numbers.
Option A shows that multiplying by 1 does not change the value, so this is an example of the multiplicative identity property.
Option B shows that multiplying by 3 can be thought of as repeated addition; this is a use of the definition of multiplication by a positive integer.
Option D shows that we can distribute 4 over a subtraction, so it is an example of the distributive property.
Option E shows that we can evaluate a product in any order; this is called the associativity property of multiplication.
Hence, option C shows the commutative property of multiplication of rational numbers.
In our next example, we will use the properties of the multiplication of rational numbers to find the value of an unknown in an equation.
Example 3: Determining the Value of an Unknown Using the Properties of Multiplication
Find in the equation .
Answer
We could answer this question by evaluating and rearranging. However, it is easier to solve this by using the properties of the multiplication of rational numbers. We recall that the associative property of multiplication tells us that, for any rational numbers , , and , we have . We can use this to rewrite the left-hand side of the equation, giving us
Both sides of the equation are now in the same form. So, for both sides of the equation to be equal, their factors must be equal.
Hence, we must have that .
In our next example, we will determine which property of the multiplication of rational numbers is used in a given equation.
Example 4: Finding a Missing Value Using the Properties of Multiplication
Fill in the blank: .
Answer
We note that, inside the parentheses on the left-hand side of the equation, we have two rational numbers, each being multiplied by :
We can take out this shared factor by recalling the distributive property of multiplication over addition; this rule states that, for any rational numbers , , and , we have
Substituting , , and into this property gives
Hence, the answer is .
The previous example showcased a use of the distributive property of multiplication over addition. It is worth noting that this same result allows us to distribute multiplication over subtraction. We can show this by using the same rule and considering the extra factor of .
We have
In our next example, we will determine which property of the multiplication of rational numbers is used in a given equation.
Example 5: Determining Which Property of Multiplication Is Used in an Equation
What property of multiplication from the following properties is used in ?
- Multiplicative identity property
- Multiplicative inverse property
- Associative property
- Commutative property
- Distributive property
Answer
We recall that we say 1 is the multiplicative identity of the rational numbers since multiplying any rational number by 1 does not change its value:
Since the equation shows this property, we can say that this is the multiplicative identity property, which is option A.
In the final example, we will determine the value of the multiplicative inverse of a rational number given as a decimal.
Example 6: Solving an Equation Using the Properties of Multiplication
Fill in the blank: If , then .
Answer
We recall that the multiplicative inverse of a nonzero rational number is the number since . We see that 1.6 is a rational number and so must be its multiplicative inverse. Thus, we can find the value of by first writing 1.6 as a fraction and then switching its numerator and denominator.
We have
Thus, and , so
Let’s finish by recapping some of the important points from this explainer.
Key Points
- The multiplication of rational numbers has several properties that can be used to
simplify expressions. We have the following properties for any rational numbers
, , and
:
- The closure property: The product of rational numbers is also rational:
- The multiplicative identity property: The rational number 1 acts as an identity of multiplication:
- The commutativity property: We can reorder any product of two or more rational numbers:
- The associativity property: We can evaluate the product of rational numbers in any order:
- The zero-product property: The product of any rational number with 0 gives 0:
- The multiplicative inverse property: All nonzero rational numbers have a multiplicative inverse that is also a rational number:
- The distributive property of multiplication over addition: We can distribute the multiplication of a rational number over the addition of rational numbers:
