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.
