Video Transcript
In this video, we will learn how to
identify and tell the difference between rational and irrational numbers. Let’s begin by thinking about the
types or sets of numbers that we should already know. The main set or classification of
numbers at this level will all fall within the set of real numbers. The smallest set within this is the
natural or counting numbers, and this refers to the numbers one, two, three, and so
on. The set of whole numbers
encompasses this set, but it also includes the value zero. The set of integer numbers
encompasses these sets and includes the negative counterparts of the whole
numbers.
The two sets that we’re looking at
today include the rational numbers and the irrational numbers. We can see that both of these sets
are still part of the set of real numbers. But in order to make this diagram
slightly more accurate, we could split the set of real numbers into either rational
or irrational numbers since there are no values that are real numbers that aren’t
included in the set of either rational or irrational numbers. We’re now going to have a look at
what it actually means to be a rational or an irrational number.
Starting with rational numbers, a
rational number is defined as a number that can be expressed as a fraction 𝑝 over
𝑞 where 𝑝 and 𝑞 are integers and 𝑞 is not equal to zero. We can break down that definition
and have a look at each part in turn to see how it works. In order to be rational, we have to
be able to write the number as a fraction. We’re told that 𝑝 and 𝑞 in our
fraction must be integers. We can recall that an integer is a
number that has no fractional or decimal part, but negatives and zero are allowed in
the set of integers. We’re told that 𝑞 cannot be equal
to zero. That’s our denominator, so we can’t
have zero on the denominator. We can now take a look at some
numbers and see if we can work out if these would be rational or not.
Starting with the integer value of
five, well, this is not a fraction 𝑝 over 𝑞, but could we write it as a
fraction? Well, recall that we could write
any integer as a fraction over one. And here we do have 𝑝 and 𝑞 as
integers. As an aside, we could also have the
equivalent fractions 100 over 20 and negative 20 over negative four, and these would
both still be rational numbers. And so, we can say that our value
five is a rational number. And how about the fraction
one-quarter? Would this be rational? We can check in our fraction of 𝑝
over 𝑞, that’s one over four, that one and four are both integers and four is not
equal to zero. So, one-quarter is rational.
Looking next at the decimal value
negative 3.75, this is not a fraction. But could we potentially write it
as a fraction? You may recall that we can write
this as the mixed number negative three and 75 over 100. We could further simplify this as
negative three and three-quarters. And we could then write this as the
top-heavy fraction negative 15 over four. And we can see that we know have
this in the fraction form 𝑝 over 𝑞. And since we have negative 15 and
four are both integers, then we could say that negative 3.75 would be a rational
number.
Our next example will be to look at
the decimal 0.3 repeating. We could write this as a fraction
in the form of one-third. In this case, we can see that both
our numerator and denominator are integers, and therefore 0.3 repeating would be a
rational number. And finally, let’s take a look at
the square root of 25. We can recall that since 25 is a
square number or a perfect square, that we could write this as five. And we have already seen that five
is a rational number.
So far, it looks like there are
lots of numbers that will be rational. We’ve seen that integers and
fractions are rational, as are terminating decimals like our negative 3.75. We saw that the repeating decimal
0.3 repeating was rational. And we also saw that the square
root of a perfect square is rational. So which values exactly are going
to be not rational numbers? We can get a bit of a clue from
these last three categories, but let’s have a look in more detail at numbers that
are not rational.
Numbers that are not rational are
called irrational numbers. That means that they cannot be
written as a fraction 𝑝 over 𝑞 where 𝑝 and 𝑞 are integers and 𝑞 is not equal to
zero. You may remember that diagram we
saw previously where we can split a real number into either a rational number or an
irrational number, meaning that a number can either be written in a fractional form
𝑝 over 𝑞 or it can’t be.
We can now look at some numbers and
see if these are irrational. Perhaps the most famous irrational
number is 𝜋. But why is it irrational? The decimal approximation of 𝜋 is
3.141592654 and so on, meaning that the decimal value doesn’t terminate or
repeat. And therefore, we can’t write it in
the fraction form 𝑝 over 𝑞 where 𝑝 and 𝑞 are integers, meaning that 𝜋 is
irrational. As an aside, you may have seen 22
over seven for 𝜋, but this is an approximation for the decimal value and isn’t the
exact value of 𝜋. Looking at another example, we have
the decimal number 0.3030030003 and so on. Although the decimal digits in this
have a nice pattern, it’s not a repeating pattern. And as the decimal doesn’t
terminate or repeat, then this means we can’t write it as a fraction 𝑝 over 𝑞. So, this decimal value would be
irrational.
For our next example, we’ll look at
the square root of 11. We saw previously that the square
root of a square number will give us an integer value. However, since 11 is not a square
number, we have the square root of a nonperfect square. The decimal value in this case
would be a value which doesn’t repeat and doesn’t terminate, which means it can’t be
written as a fraction. And therefore, the square root of
11 is an irrational number. So, how about the square root of
five over two? This looks quite good because we do
have a fraction. However, our numerator, the square
root of five, is not an integer, which doesn’t fit the rule that for our fraction 𝑝
over 𝑞, both 𝑝 and 𝑞 have to be integer values. And so, root five over two must be
irrational.
We’ll now look at some example
questions involving rational and irrational numbers. And each time, we’ll work through
the definition of a rational number. So hopefully, by the end of the
video, we’ll have a much greater understanding of each part of this definition.
Is 0.456 repeating a rational or an
irrational number?
We can recall that a rational
number can be expressed as a fraction 𝑝 over 𝑞, where 𝑝 and 𝑞 are integers and
𝑞 is not equal to zero. An irrational number is a number
that isn’t rational. So, in order to check if 0.456
repeating is a rational number, we need to check if we can write it as a fraction 𝑝
over 𝑞. Here, we’re going to use a neat
method to write this repeating decimal as a fraction. And it begins by defining a
variable 𝑥 which is equal to 0.456 repeating. We can say that 𝑥 is equal to
0.456456456 and so on. In the next step, we create another
value which has the same decimal digits as 𝑥 does. As we have three digits that
repeat, then if we multiply by 10 to the third power, that’s the same as multiplying
by 1000. And so, we’ll have 1000𝑥 equals
456.456456 and so on.
We now have two values that have
the same decimal digits. And therefore, if we were to
calculate 1000𝑥 subtract 𝑥, this would give us 456 as each decimal digit will be
subtracted from another one of equal value. Continuing our calculation then, we
can write that 999𝑥 is equal to 456. And rearranging by dividing both
sides by 999 will give us that 𝑥 equals 456 over 999. As we’ve already defined 𝑥 to be
0.456 repeating, then we have proved that this decimal can be written as a
fraction. As both the numerator and
denominator are integers and the denominator is not equal to zero, it fits with the
definition of a rational number. So, 0.456 repeating is a rational
number.
Is the square root of two a
rational or an irrational number?
Let’s begin by recalling what a
rational number is. A rational number is a number that
can be expressed as a fraction 𝑝 over 𝑞 where 𝑝 and 𝑞 are integers and 𝑞 is not
equal to zero. And an irrational number is a
number that is not rational. So, here we have the square root of
two. We know that the square root of two
lies between the square root of one and the square root of four since one and four
are the nearest square numbers. The positive value of the square
root of one is one, and the positive value of the square root of four is two.
Using a calculator, we can evaluate
the square root of two as a 1.414213562 and continuing. The decimal is not a repeating
decimal. And we can also see that this
decimal does not terminate. Therefore, we could not write a
fraction to represent this decimal that represents the square root of two, meaning
that the rational definition would not fit, which means that the square root of two
is an irrational number.
In the next example, we’ll see a
story problem, and we have to establish if the result of a calculation is rational
or irrational.
According to the US Mint, the
diameter of a quarter is 0.955 inches. The circumference of the quarter
would be the diameter multiplied by 𝜋. Is the circumference of a quarter a
whole number, a rational number, or an irrational number?
So, here we have the quarter. We’re told that the diameter is
0.955 inches. That’s the distance from one side
of the circle to the other through the center. And we’re told that the
circumference is equal to 𝜋 times the diameter. And so, we can say that the
circumference of this quarter will be 0.955𝜋. We’re asked to work out if this is
a whole number, a rational number, or an irrational number. We can recall that a decimal
approximation for 𝜋 begins 3.141592654 and so on. And therefore, when we multiply
that by 0.955, we’re definitely not going to get a whole number. So, let’s look at a rational
number.
A rational number can be expressed
as a fraction 𝑝 over 𝑞 where 𝑝 and 𝑞 are integers and 𝑞 is not equal to
zero. If we cannot express a number as a
fraction 𝑝 over 𝑞, then it would be irrational. We can simply say that an
irrational number is a number that is not rational. We may recall that 𝜋 is an
irrational number. And that’s because we cannot
express it as a fraction 𝑝 over 𝑞. We know this to be the case because
the decimal value for 𝜋 does not terminate, and it is not a repeating decimal. So, here we have 𝜋, an irrational
number, multiplied by 0.955, which is a rational number. We can say that it’s rational
because it’s equivalent to the fraction 955 over 1000.
And therefore, we’re multiplying a
rational number by an irrational number, which will give an irrational number, which
is true for all cases except for when the rational number is a zero. So, our answer is that the
circumference of the quarter 0.955𝜋 is an irrational number.
We’re now going to try a final
question, and you may want to pause the video after you’ve seen the question and
have a go at it first.
A square of side length 𝑥
centimeters has an area of 280 square centimeters. Which of the following is true
about 𝑥? It is an integer number. It is a natural number. It is a rational number. It is an irrational number. Or, it is a negative number.
Let’s begin this question by
visualizing our square. As it is a square, we know that all
four sides will be 𝑥 centimeters. We’re told that the area is 280
square centimeters. And since we find the area of a
square by multiplying the length by the length, this means that 𝑥 squared is equal
to 280. We could therefore work out the
value of 𝑥 by taking the square root of both sides, meaning that 𝑥 is equal to the
square root of 280. So, what can we say about 𝑥?
Well, let’s start by looking at the
value of 280. If we work out the value of some
square numbers that are close to 280, we would see that 16 squared is equal to 256,
and 17 squared is equal to 289. Therefore, 280 is not a perfect
square or a square number. And its square root won’t have an
integer value. If we used a calculator, we would
get a decimal approximation of 16.73320053 and so on.
So, let’s look at some of the
answer options. An integer number has no fractional
part and no digits after the decimal point, so the square root of 280 is not an
integer. A natural number is a positive
integer excluding zero. These are often called the counting
numbers as they begin one, two, three, and so on. The set of natural numbers is
included within the integer numbers. So, if it’s not an integer, it’s
not a natural number either.
We can recall that a rational
number can be written in the form 𝑝 over 𝑞 where 𝑝 and 𝑞 are integers and 𝑞 is
not equal to zero. So, could we write the square root
of 280, that’s the decimal 16.73320053 and so on, as a fraction 𝑝 over 𝑞? And the answer is no, we can’t. A quick way to check if a decimal
number is rational is to see if it terminates or repeats. Either these would mean that the
decimal is a rational number. But since our value does not have
this, then it’s not rational.
Exploring the next option then, we
can recall that an irrational number is a number which is not rational. It’s worth pointing out that this
is only within the set of real numbers in which we’re working here. As we’ve established that it’s not
a rational number, this means that our value must be irrational. It looks like we have our answer
here, but let’s double-check option (E).
So, could the square root of 280 be
a negative number? Well, in fact, it could because the
square root of 280 could be the positive value of 16.733 and so on or the negative
value of negative 16.733 and so on. But we can rule out the fact that
it’s a negative number simply due to the context of the question. We couldn’t actually have a square
that has a negative length, and therefore it can’t be option (E). So, our final answer for 𝑥 is that
it is an irrational number.
So, now let’s summarize what we’ve
learnt in this video. We saw that a rational number is a
number that can be expressed as a fraction 𝑝 over 𝑞 where 𝑝 and 𝑞 are integers
and 𝑞 is not equal to zero. For example, some rational numbers
would be two-thirds, negative 1.75, 4.22 repeating, or the square root of 25.
If we have a decimal number that we
want to check if it’s rational, then if it’s a repeating decimal or a terminating
decimal, then this would mean that it’s rational. If we have a square root value,
then we can check if it’s the square root of a perfect square. If it is, then it’s a rational
number.
And finally, we learnt that an
irrational number is a number that is not rational. Some examples of irrational numbers
would be 𝜋, the square root of two, or 0.303003 and so on. And so now, we can identify
rational and irrational numbers.
