Video Transcript
In this video, we’re going to learn
how to distinguish between rational and irrational numbers and represent real
numbers on number lines. We’ll simply begin by defining what
it means for a number to be real and for a number to be rational.
Nearly any number you can think of
is likely to be a real number. In fact, any number that you can
list on your usual number line is real. These include integers. That’s whole numbers, like the
number four. A number like three over two is a
real number. We could add 𝜋 to our number line
is being approximately equal to 3.14. And we’d also include negative
numbers, such as negative three. Any number that we can add to this
number line belongs in the set of real numbers.
The numbers we don’t include in
this set are ∞ or negative ∞ nor imaginary numbers. Now these might sound silly, but
they are a group of numbers based on the square root of negative one. Then we can split the set of real
numbers itself up into several groups. The two subsets we’re interested in
are rational numbers and irrational numbers. These types of numbers are all
real, but they’re also mutually exclusive. That is to say, a number can’t be
both rational and irrational. It’s just one or the other.
A diagram representing this might
look a little bit as shown. We have the set of real numbers,
which contains both rational and irrational numbers. Integers, which are whole numbers,
are included in the set of rational numbers, and then included within those are the
natural numbers. Those are the counting numbers,
one, two, three, and so on.
The definition of a rational number
is a real number that can be written as 𝑎 over 𝑏 where 𝑎 and 𝑏 are integers, for
example, two-fifths or 0.3 recurring, which is, of course, one-third. There will be some numbers which
might not feel like they should be rational, for example, 0.142857 recurring. This is actually the same as
one-seventh. And since this is a fraction made
up of two integers, it’s a rational number. In fact, any recurring decimal can
be written as the quotient of two integers. So recurring decimals are examples
of rational numbers.
An irrational number is simply the
opposite of this. It’s a number which is not
rational. Some key examples of this are 𝜋
and the square root of two. Now that we have our definitions,
let’s look on how to apply them.
Is 0.456 a rational or irrational
number?
Let’s begin by recalling the
definition of a rational number. A rational number is one that can
be written in the form 𝑎 over 𝑏, where 𝑎 and 𝑏 are integers. They’re whole numbers. It follows then that an irrational
number is one that can’t be written in this form. So let’s have a look at the number
we’ve been given. 0.456 is a terminating decimal. So let’s see if we’re able to write
it as a fraction. We spot that the number has four
tenths, five hundredths, and six thousandths. This means we can write it as the
sum of four-tenths, five hundredths, and six one thousandths. And then we remember that we can
add fractions when their denominators are the same.
Let’s create a common denominator
of 1000. To achieve this, we’re going to
multiply the numerator and denominator of our first fraction by 100 and of our
second fraction by 10. Four-tenths is equivalent to four
hundred one thousandth. And five one hundredths is equal to
fifty one thousandths. And once their denominators are
equal, we simply add the numerators. 400 plus 50 plus six is equal to
456. And in turn, 0.456 is equal to four
hundred and fifty-six thousandths.
We could also simplify this to 57
over 125. Though, this isn’t entirely
necessary. All we really needed to show was
that we could write our number as a fraction whose denominator and numerator are
both integers. Since we’ve shown that 0.456 can be
written as the quotient of two whole numbers, it’s 456 over 1000, we can say that
0.456 must indeed be a rational number. The answer is yes.
Now, in fact, we can generalize
this and say that any terminating decimal can be written in this form. So all terminating decimals are
rational. So we now have that terminating
decimals and recurring decimals are rational. We’re now going to look at what
happens if we combine rational and irrational numbers.
Is 𝜋 over three a rational or an
irrational number?
We begin by recalling the
definition of a rational number. It’s one that can be written in the
form 𝑎 over 𝑏 where 𝑎 and 𝑏 are integers. They’re whole numbers. Once we have this definition, we
can say that an irrational number is one that’s not rational. In other words, it can’t be written
in this form. We’re looking to identify whether
𝜋 divided by three is rational or irrational.
We actually know that 𝜋 is an
example of an irrational number. It’s one of the ones we’re most
familiar with. So, actually, we need to ask
ourselves what happens if we divide an irrational number by an integer? Well, it’s still irrational. There’s no way to write 𝜋 divided
by three in the form 𝑎 over 𝑏 where 𝑎 and 𝑏 are both integers. Three is an integer, 𝜋 is not. So this is irrational. 𝜋 divided by three is an
irrational number.
Now, in fact, we can generalize
this. We can say that multiplying or
dividing an irrational number by a rational number not equal to zero will give us an
irrational number. Of course, if the rational number
was equal to zero and we were to multiply by it, we’d get zero, which is
rational. And so it’s really important that
we specify that we’re multiplying our irrational number by a nonzero rational.
In our next example, we’ll practice
how to identify an irrational number from a list.
Which of the following is an
irrational number? Is it (A) the cube root of 70, (B)
the cube root of 64, (C) 59, (D) the square root of 144 over 81, or (E) 109.5?
Let’s begin by recalling what we
mean when we say a number is irrational. For a number to be rational, we
must be able to write it in the form 𝑎 over 𝑏 where 𝑎 and 𝑏 are integers. They’re whole numbers. Then if a number is not able to be
written in this form, it’s not rational. And in fact, we say it’s
irrational.
So to work out which of our numbers
is irrational, we’re going to go through them in turn. Let’s begin by looking at the cube
root of 70. If we list out the cube numbers we
know by heart, we see that we have three cubed is 27, four cubed is 64, and five
cubed is 125. None of the numbers on the
right-hand side are equal to 70. And this certainly tells us that
the cube root of 70 doesn’t have a whole number in integer solution. It will be somewhere between four
and five. It’s likely to be much closer to
four since 70 is only a little bit bigger than 64.
So rather than trying to work out
exactly what the cube root of 70 is equal to, we’ll look at the remaining four
numbers. The cube root of 64, that’s (B), is
actually in our list of cubic numbers. The cube root of 64 is equal to
four. So actually, we can write it as
four or four over one, meaning it is a rational number. And we can disregard (B) from our
list. Let’s now look at (C). 59 is the same as 59 over one. Once again, both 59 and one are
whole numbers. They’re integers. So we disregard (C). It’s also rational.
But what about (D)? Well, one of the rules we have for
working with square roots is that we can find the square root of a fraction by
finding individually the square root of the numerator and the square root of the
denominator. The square root of 144 is 12, and
the square root of 81 is nine. This means root 144 over 81 is
rational. It is written in the form 𝑎 over
𝑏. 𝑎 is 12, and 𝑏 is nine. They’re both whole numbers. So what about (E)? Well, this is a terminating
decimal. And in fact, we know that all
terminating decimals are examples of rational numbers. It, therefore, cannot be
irrational. And we’re going to disregard this
one.
And so, by a combination of looking
at the cube numbers and disregarding the other options, we see that the answer must
be (A). The number that is irrational is
the cube root of 70.
We briefly looked at how we might
estimate solutions to roots. Let’s now have a look at an example
that involves this process.
Which of the following is an
irrational number that lies between three and four?
We know that a rational number can
be written in the form 𝑎 over 𝑏 where 𝑎 and 𝑏 are integers. They’re whole numbers. An irrational number then can’t be
written in this form. And so we’re going to begin by
working through the numbers in our list and identifying which are rational. Let’s begin with (A), the number
3.9. 3.9 is a terminating decimal. And actually, all terminating
decimals are rational. In this example, 3.9 can be written
as 39 over 10. Both 39 and 10 are whole numbers,
so 3.9 must be rational.
We’ll consider the square root of
19, the square root of 13, and the square root of seven at the same time. If we list out the first few square
numbers, we have one squared is one, we have four, nine, 16, and 25. None of these are in this list. And that’s a good indicator to us
that when we find the square root of 19, 13, and seven, not only will we get a
nonwhole number, we’ll get a decimal of some description. That decimal will also be
irrational. And so we’ll come back to these in
a moment.
Before we do, let’s look at
(D). Seven over two is already in the
form that we identified. It’s in the form 𝑎 over 𝑏. 𝑎 is seven, and 𝑏 is two. And they’re both whole numbers. And so we can disregard (D) from
our list. So we know both (A) and (D) are
rational numbers. So let’s go back to our square
roots.
A use of a number line here might
help us. We have one squared, which is
one. And so the square root of one is
one. We know that two squared is equal
to four, so the square root of four is equal to two. We see that three squared is equal
to nine, so the square root of nine is three. Similarly, the squares of 16 is
four. And the square root of 25 is
five. We can also conversely say that one
squared is one, two squared is four, and so on.
We have the square root of 19. So let’s find that on our number
line. 19 is here. Notice that 19 lies between the
result of four squared and five squared. We can say that the square root of
19 is greater than the square root of 16 but less than the square root of 25. Or we can also alternatively say
that the square root of 19 is greater than four and less than five.
Let’s repeat this process with the
square root of 13. 13 is here. It lies between the result of three
squared and four squared, which means, of course, that the square root of 13 must be
greater than three and less than four. We’ll do this one more time for the
square root of seven. Now seven is all the way down
here. It’s between the result of two
squared and three squared. And of course, this means that the
square root of seven must be between two and three. It’s greater than two and less than
three.
We want to find the irrational
number that lies between three and four. And of course, we’ve just
identified that that’s the square root of 13. And so out of our list, the
irrational number that lies between three and four is the square root of 13. It’s (C).
In our final example, we’ll look at
how to use the laws of radicals to compare the size of irrational numbers.
Determine which has a larger value:
Is it two root three or three root two?
There are two ways to answer
this. One method is to estimate the value
of the square root part first and then multiply them by two and three,
respectively. It’s much easier, though, to undo
this simplification of the surd. Let’s see what that looks like. We begin by recalling that two is
equal to the square of four. And so, if we take our first
number, two root three, we can then write that as the square to four times the
square root of three. But of course, for real numbers 𝑎
and 𝑏, the square root of 𝑎 times the square root of 𝑏 is the same as the square
root of 𝑎𝑏.
This means that two root three is
the same as the square root of four times three, which is 12. Let’s now repeat this process with
our second number. That’s three root two. This time, we recall that three is
equal to the square root of nine. And so we’re going to rewrite three
root two as the square root of nine times the square root of two. That is, of course, equal to the
square root of nine times two, which is the square root of 18.
And so let’s compare the sizes of
these two numbers. We have the square root of 12 and
the square root of 18. Now it follows that since 18 is
bigger than 12, the square root of 18 must be bigger than the square root of 12. And so the square root of 18 has a
larger value than the square root of 12, which, of course, means that three root two
must have a larger value than two root three. The answer is three root two.
In this video, we learned that a
rational number is a real number that can be written as 𝑎 over 𝑏 where 𝑎 and 𝑏
are integers. These include all recurring and
terminating decimals. Those are examples of rational
numbers. We then say that an irrational
number is the opposite of this. It’s a number which is not
rational. And some key examples of this are
𝜋 or any radical or surd where the number inside the surd is not a square
number.
We saw that if we were going to
represent these sets as a Venn diagram, they are mutually exclusive. There is no overlap. A number cannot be rational and
irrational at the same time. And then all of these numbers make
up the set of real numbers. We saw that multiplying or dividing
an irrational number by nonzero rational number gives an irrational number. And we saw that we can estimate the
value of irrational numbers which are radicals or surds by considering the square
numbers that lie around them.
