Video: Finding Reciprocals

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

14:26

Video Transcript

In this video, we’ll learn how to find the reciprocal, sometimes called the multiplicative inverse, of different types of values, for example, integers, fractions, and mixed numbers. There’s something interesting about this word, reciprocal. It has two Latin roots “re” and “pro.” The prefix “re” means back. And “pro” here means forward. We might think of it as a back and forth. For now, let’s put the idea of back and forth on hold while we look at the mathematical definition of the word reciprocal. We’ll revisit the back and forth idea after we do some investigating.

We need to know that the reciprocal of a number is one divided by that number. And that means if we say the reciprocal of two is equal to one divided by two. And one divided by two can be written as the fraction one-half. And so we say that one-half is the reciprocal of two. I mentioned briefly in the introduction that reciprocal is sometimes called the multiplicative inverse. A multiplicative inverse is what you multiply by a number to get one. If we consider the multiplicative inverse of two, we’re asking, what would we multiply by two to get one. The multiplicative inverse is the same as the reciprocal. Since the reciprocal of two is one-half, if we multiply two by one-half, we would get one.

How do we know that two times one-half equals one? Well, we rewrite two as a fraction two over one. And when we multiply fractions, we multiply their numerators together and their denominators together. Then, we have two times one over one times two, which is two over two. And we can simplify two over two to one. But what if we put some variables in place of the two and one? Is it still true that 𝑎 over 𝑏 times 𝑏 over 𝑎 equals one?

Well, first we need to multiply the numerators together, 𝑎 times 𝑏. We can write that as 𝑎𝑏. Then, we multiply the denominators together, 𝑏 times 𝑎. 𝑏 times 𝑎 is the same thing as 𝑎 times 𝑏. And so we can write the denominator as 𝑎𝑏 as well. And 𝑎𝑏 over 𝑎𝑏 equals one. And so we can say that the reciprocal of a fraction, 𝑎 𝑏, is found by switching the numerator and the denominator so that the reciprocal is 𝑏 over 𝑎. Knowing this helps us quickly identify the reciprocals of fractions. For example, the fraction three-fourths has a reciprocal of four-thirds.

Let’s consider some other examples.

What is the reciprocal of three?

We remember the rule that the reciprocal of a fraction 𝑎 over 𝑏 is found by switching the numerator and the denominator so that the reciprocal is 𝑏 over 𝑎. You might be thinking three is not a fraction of 𝑎 over 𝑏. And that’s true. We call three, written in this format, a whole number. But can we write the whole number three as a fraction? If we give three a denominator of one, we haven’t changed its value. And it’s now written as a fraction in the form 𝑎 over 𝑏. And that means its reciprocal will be in the format 𝑏 over 𝑎. It would be one over three.

At this point, there’s something we should notice. One-third is the reciprocal of three over one. And three over one is the reciprocal of one-third. Because three over one times one-third equals one. And one-third times three over one equals one. And this is where we see that idea from the beginning, the back and forth. Reciprocals come in pairs. We’ve answered our first question, “what is the reciprocal of three?” It’s one-third.

But what if someone asks us the question, what is the reciprocal of zero?

The reciprocal of zero would be one divided by zero. Or we could say one over zero. At which point, we recognise that we can’t divide by zero. And we come to the conclusion that zero does not have a reciprocal. Remember that a value times its reciprocal has to equal one. And there’s no value that you can multiply by zero that will give you one. So it’s worth making a note that all numbers except zero have a reciprocal. We’ve considered reciprocals of fractions and of whole numbers. So now let’s look at a mixed number.

What is the reciprocal of one and one eleventh. Give your answer in simplest form.

We have this value one and one eleventh. And we want to know what we could multiply by one and one eleventh to equal one. Whatever that value is will be the reciprocal or the multiplicative inverse. If we remember that, for fractions 𝑎 over 𝑏, the multiplicative inverse is 𝑏 over 𝑎. The reciprocal of 𝑎 over 𝑏 is 𝑏 over 𝑎. And that means our goal will be to write this mixed number one and one eleventh as a fraction in the form 𝑎 over 𝑏. We need to convert this mixed number to an improper fraction as our first step.

To convert a mixed number to a fraction, you first multiply the whole number portion by the denominator. For us, that’s one times 11. And then, you add whatever is in the numerator. We have one in the numerator. And the denominator of the improper fraction is the same denominator as the mixed number we started with. One times 11 equals 11. And 11 plus one equals 12. And now, we can say twelve elevenths times what is equal to one. Well, when we have a fraction 𝑎 over 𝑏, its reciprocal is 𝑏 over 𝑎. We flip the numerator and the denominator. And we get eleven twelfths. 12 over 11 times 11 over 12 is equal to one. And that means eleven twelfths is the reciprocal of one and one eleventh. Eleven twelfths is already in its most simplified form. And so that’s our final answer.

Our last example was finding the reciprocal of a mixed number. In this case, we’re finding the reciprocal of a negative mixed number.

Find the multiplicative inverse of negative four and one-fourth.

What do we know about multiplicative inverse? Multiplicative inverse is another word for reciprocal. And for the fraction 𝑎 over 𝑏, its reciprocal is 𝑏 over 𝑎 because 𝑎 over 𝑏 times 𝑏 over 𝑎 equals one. But negative four and one-fourth is not in the form 𝑎 over 𝑏. Negative four and one-fourth is a mixed number. Before we can find the multiplicative inverse, we need to take this mixed number and write it as an improper fraction.

The first thing we need to do is bring down this negative as this whole mixed number is negative. After that, we multiply the integer piece, the whole number piece, four, by the denominator four. And then, we add what’s in the numerator. Again, we need to be really careful here to notice that the negative applies to the entire numerator. And the denominator will be whatever the denominator of the mixed number was. Four times four is 16. 16 plus one is 17. So we have negative 17 over four. The mixed number negative four and one-fourth can be rewritten as the improper fraction negative 17 over four.

To find its multiplicative inverse, we flip the numerator and the denominator so that we have four over negative 17. If we multiply negative 17 times four, we get negative 68. And if we multiply four times negative 17, we get negative 68. And negative 68 over negative 68 is the same thing as one, which means the multiplicative inverse of negative four and one-fourth is equal to four over negative 17. We know a negative times a negative equals a positive. What we see here is that negative numbers have negative reciprocals. We usually keep the negative symbol in the numerator. So our final answer for the multiplicative inverse of negative four and one-fourth can be written as negative four over 17.

Let’s consider finding the multiplicative inverse of an expression.

Find the multiplicative inverse of eight over three plus nine over two.

We’re given this expression that is the sum of two fractions. And we want to find the multiplicative inverse. We first need to take this expression and find its sum. To do that, we need to find a common denominator for two and three. If we use the denominator of six, three times two equals six. But if we multiply the denominator by two, we need to multiply the numerator by two. Eight-thirds is equal to sixteen sixths. We can follow the same process for nine-halves. Two times three equals six. And nine times three equals 27.

When two fractions have a common denominator, you can add them by adding their numerators. 16 plus 27 equals 43. And the denominator doesn’t change. Eight-thirds plus nine-halves is equal to forty-three sixths. And this is what we need to find the multiplicative inverse of. We know that, for the fraction 𝑎 over 𝑏, its multiplicative inverse is 𝑏 over 𝑎. If we flip the numerator and denominator, we get six over 43. And 43 times six over 43 times six is the same thing as one. Our final answer for the inverse of eight-thirds plus nine-halves is six over 43.

Let’s consider this final example.

Find the value of 𝑥 for which the rational number 𝑥 minus 18 over 26 does not have a multiplicative inverse.

We need to think about what we know about a value that does not have a multiplicative inverse. We know that all numbers except zero have a reciprocal. And that multiplicative inverse is another word for reciprocal. And so we can say that when 𝑥 minus 18 over 26 equals zero, it does not have a reciprocal. We now need to find a value of 𝑥 that makes 𝑥 minus 18 over 26 equal to zero. We can get the 26 out of the denominator by multiplying both sides of the equation by 26. 26 times zero equals zero. And 𝑥 minus 18 equals zero.

To find 𝑥, we need to add 18 to both sides. Zero plus 18 equals 18. And 𝑥 minus 18 plus 18 just equals 𝑥. We can flip that around to say 𝑥 equals 18. We’re saying that 18 minus 18 over 26 does not have a reciprocal because 18 minus 18 equals zero. And zero over 26 equals zero. Since zero does not have a multiplicative inverse, we can say that when 𝑥 is 18, this expression does not have a multiplicative inverse.

Let’s wrap up by considering some key points. The reciprocal is often called the multiplicative inverse, since the product of a number and its reciprocal is one. The reciprocal of a fraction 𝑎 over 𝑏 is found by switching the numerator and the denominator so that the reciprocal is 𝑏 over 𝑎. To find the reciprocal of a mixed-number fraction, we first convert it to an improper fraction before finding the reciprocal. And all numbers except zero have a reciprocal.

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