Explainer: Reciprocals

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, π‘Žπ‘, is found by switching the numerator and denominator so that the reciprocal is π‘π‘Ž.
  • 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 1Γ·13, which we can write as 113.

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 2322?

Answer

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

So, the reciprocal of 2322 is 1Γ·2322.

To divide by a fraction, we change the division to a multiplication and invert the fraction, giving us 1Γ·2322=1Γ—2223=2223.

This is how we obtain the rule that the reciprocal of a fraction π‘Žπ‘ is π‘π‘Ž.

So, the reciprocal of 2322 is 2223.

Example 3: Finding the Reciprocal of a Mixed Number

What is the reciprocal of 612?

Answer

Before we can find the reciprocal, we must make the mixed number, 612, into an improper fraction.

Recalling that 6 is made up of 12 halves, 122, plus a remaining half, 12, we can write 612=132.

To find the reciprocal of a fraction, π‘Žπ‘, we switch the numerator and denominator so that the reciprocal is π‘π‘Ž.

Therefore, the reciprocal of 132 is 213.

So, the reciprocal of 612 is 213.

In the next examples, we will see how the term β€œmultiplicative 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 ο€Ό83+92.

Answer

The first step in this problem is to find the value of ο€Ό83+92.

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 83+92=166+276=16+276=436.

Next, we need to find the multiplicative inverse, also known as the reciprocal. The multiplicative inverse of a fraction π‘Žπ‘ is π‘π‘Ž.

So, switching the numerator and denominator of 436 gives us 643.

Therefore, the answer is that the multiplicative inverse of ο€Ό83+92 is 643.

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 βˆ’414.

Answer

Before we can find the multiplicative inverse, we must make the mixed number, βˆ’414, into an improper fraction.

So, we have βˆ’414=βˆ’174.

To find the multiplicative inverse of a fraction π‘Žπ‘, we switch the numerator and denominator so that the multiplicative inverse is π‘π‘Ž.

Therefore, the multiplicative inverse of βˆ’174 is βˆ’417, noting that a negative number will also have a negative reciprocal value.

So, the multiplicative inverse of βˆ’414 is βˆ’417.

We could also have approached the question using a different method of finding the number which when multiplied by βˆ’414 would give 1.

Let us write an equation to represent this relationship. Let π‘₯ represent the multiplicative inverse of βˆ’414, and we solve for π‘₯.

So, we have βˆ’414Γ—π‘₯=1.

Changing our mixed number to an improper fraction gives us βˆ’174Γ—π‘₯=1.

Dividing both sides by βˆ’174, we have ο€Όβˆ’174Γ—π‘₯οˆΓ·βˆ’174=1Γ·βˆ’174π‘₯=1Γ·βˆ’174.

Recall that, to divide by a fraction, we change the division to a multiplication and invert the fraction, giving us π‘₯=1Γ—βˆ’417π‘₯=βˆ’417.

Since π‘₯ was our multiplicative inverse, we can write our answer that the multiplicative inverse of βˆ’414 is βˆ’417.

Example 6: Investigating the Multiplicative Inverse of an Algebraic Expression

Find the value of π‘₯ for which the rational number π‘₯βˆ’1826 does not have a multiplicative inverse.

Answer

The multiplicative inverse, or reciprocal, of a fraction π‘Žπ‘ is π‘π‘Ž.

So, the multiplicative inverse of π‘₯βˆ’1826 is 26π‘₯βˆ’18.

When considering the value that would have no multiplicative inverse, we can see that if π‘₯ 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, 26π‘₯βˆ’18, must not equal 0. We can write this as π‘₯βˆ’18β‰ 0.

Adding 18 to both sides, we have π‘₯β‰ 18.

So, the value of π‘₯ for which the rational number π‘₯βˆ’1826 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 π‘Žπ‘ we switch the numerator and denominator to obtain the reciprocal π‘π‘Ž.
  • 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.

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