In this explainer, we will learn how to find the reciprocal (the multiplicative inverse) of an integer, a fraction, or a mixed number.

### Definition: Reciprocal

The reciprocal of a number is 1 divided by that number.

- The reciprocal is often called the multiplicative inverse, since the product of a number and its reciprocal is 1.
- The reciprocal of a fraction, , is found by switching the numerator and denominator so that the reciprocal is .
- To find the reciprocal of a mixed-number fraction, we first convert it to an improper fraction before finding the reciprocal.
- All numbers except 0 have a reciprocal.

Let us now look at an example of how to find the reciprocal of an integer.

### Example 1: Finding the Reciprocal of an Integer

What is the reciprocal of 13?

### Answer

The reciprocal of a number is 1 divided by the number.

So, the reciprocal of 13 is , which we can write as

Let us now look some examples of finding the reciprocals of fractions and mixed numbers.

### Example 2: Finding the Reciprocal of a Fraction

What is the reciprocal of ?

### Answer

The reciprocal of a number is 1 divided by the number.

So, the reciprocal of is

To divide by a fraction, we change the division to a multiplication and invert the fraction, giving us

This is how we obtain the rule that the reciprocal of a fraction is .

So, the reciprocal of is

### Example 3: Finding the Reciprocal of a Mixed Number

What is the reciprocal of ?

### Answer

Before we can find the reciprocal, we must make the mixed number, , into an improper fraction.

Recalling that 6 is made up of 12 halves, , plus a remaining half, , we can write

To find the reciprocal of a fraction, , we switch the numerator and denominator so that the reciprocal is .

Therefore, the reciprocal of is

So, the reciprocal of is

In the next examples, we will see how the term “multiplicative inverse” is used. A multiplicative inverse is the same as a reciprocal and is used to refer to the fact that the product of a number and its reciprocal is 1.

### Example 4: Finding the Multiplicative Inverse of an Expression

Find the multiplicative inverse of .

### Answer

The first step in this problem is to find the value of .

Recall that, to add fractions with different denominators, we find a common denominator to create equivalent fractions to the original fractions. Then, we add the numerators.

Since both denominators here are factors of 6, we have

Next, we need to find the multiplicative inverse, also known as the reciprocal. The multiplicative inverse of a fraction is .

So, switching the numerator and denominator of gives us

Therefore, the answer is that the multiplicative inverse of is

In the next example, we will explore how having a negative value changes the value of the reciprocal.

### Example 5: Finding the Multiplicative Inverse of a Negative Number

Find the multiplicative inverse of .

### Answer

Before we can find the multiplicative inverse, we must make the mixed number, , into an improper fraction.

So, we have

To find the multiplicative inverse of a fraction , we switch the numerator and denominator so that the multiplicative inverse is .

Therefore, the multiplicative inverse of is noting that a negative number will also have a negative reciprocal value.

So, the multiplicative inverse of is

We could also have approached the question using a different method of finding the number which when multiplied by would give 1.

Let us write an equation to represent this relationship. Let represent the multiplicative inverse of , and we solve for .

So, we have

Changing our mixed number to an improper fraction gives us

Dividing both sides by , we have

Recall that, to divide by a fraction, we change the division to a multiplication and invert the fraction, giving us

Since was our multiplicative inverse, we can write our answer that the multiplicative inverse of is

### Example 6: Investigating the Multiplicative Inverse of an Algebraic Expression

Find the value of for which the rational number does not have a multiplicative inverse.

### Answer

The multiplicative inverse, or reciprocal, of a fraction is .

So, the multiplicative inverse of is

When considering the value that would have no multiplicative inverse, we can see that if is a positive number or a negative number, we will still have a reciprocal. However, if the denominator of a fraction is 0, then we would produce a mathematical error as it is not possible to divide a number by 0.

So, here we can say that the denominator of the multiplicative inverse, , must not equal 0. We can write this as

Adding 18 to both sides, we have

So, the value of for which the rational number does not have a multiplicative inverse is 18.

### Key Points

- The reciprocal, or multiplicative inverse, of a number is 1 divided by that number.
- To find the reciprocal of a fraction we switch the numerator and denominator to obtain the reciprocal .
- To find the reciprocal of a mixed number, we first need to change it into an improper fraction before inverting it to find the reciprocal.