In this video, we’ll learn how to
identify rational numbers and how to find the position of a rational number on a
number line. But let’s begin by thinking about
the definition of a rational number.
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. So let’s break this down a little
bit. We’re told that a rational number
must be able to be written as a fraction 𝑝 over 𝑞 where 𝑝 and 𝑞 are
integers. Remember that integers are the
natural numbers, including zero, and the negatives of the natural numbers. So we could say, for example, that
a fraction like two-thirds is a rational number because two and three are integers
and three is definitely not equal to zero. We could also say that a number
like negative three and a quarter is a rational number. This is because it can be written
as a top-heavy fraction negative 13 over four, and negative 13 and four are both
But what about an actual integer
value like 10? Would it be rational? Well, yes, because remember that we
can write any integer as a value over one, in which case we can see very clearly
that this is in the form 𝑝 over 𝑞 where 𝑝 and 𝑞 are integers. Although a value like 4.75, which
is a decimal, doesn’t immediately appear to be rational, we can, in fact, remember
that we could write this as a fractional form. 4.75 can be written as four and
seventy-five hundredths or, alternatively, four and three-quarters. 19 over four would be this value as
a top-heavy fraction, fitting the form of a rational number.
In addition to terminating decimals
like 4.75, we also have repeating or recurring decimals such as this one, 0.131313
and so on. Any recurring decimal can also be
written as a fraction. For example, here, this is the
fraction of 13 over 99. So terminating decimals and
repeating decimals will be rational numbers. You might begin to wonder “what we
would say if a number is not rational?” Well, a number that’s not rational
is called an irrational number. It’s one which can’t be written as
a fraction 𝑝 over 𝑞 where 𝑝 and 𝑞 are integers and 𝑞 is not equal to zero.
Although in this video we’re going
to be focusing on rational numbers, it can be helpful to get an idea of some numbers
which are irrational. Perhaps, the most famous irrational
number is 𝜋, which is a value which cannot be expressed as a terminating or
repeating decimal. It can also not be expressed as a
fraction, so that’s why 𝜋 is irrational. Decimals which don’t terminate or
repeat and also values such as the square root of numbers, which are not perfect
squares. But let’s have a look at some
questions. As we go through each question,
we’ll recall this definition of a rational number. So hopefully, by the end of this
video, you’ll be able to recall it easily.
In our first question, we’ll
identify if a number is rational or not.
Is 12 and five-sixths a rational
Let’s begin by remembering the
definition of a rational number, which is a number that can be expressed as 𝑝 over
𝑞 where 𝑝 and 𝑞 are integers and 𝑞 is not equal to zero. Although we have a mixed number
here, 12 and five-sixths, could we write this as a complete fraction 𝑝 over 𝑞? In other words, could we write this
as a top-heavy or improper fraction? If we consider our value of 12,
then how many sixths would we have in 12 whole ones? Well, we’d have seventy-two sixths
plus the fractional value of five-sixths, which would give us a fraction of 77 over
Another way, of course, to do this
is to take our denominator six and multiply it by 12 and then add on the value of
five. Either way, we can see that 12 and
five over six is equivalent to 77 over six. So, is this value a rational
number? And the answer is yes.
Let’s have a look at another
Is every rational number an
In this question, we have two very
mathematical terms, rational and integer. Let’s take the word integer
first. This will be defined as a number
that has no a fractional part. It includes the counting numbers,
for example, one, two, three, four, zero, and the negatives of the counting
numbers. A rational number is defined as
one, which can be expressed as 𝑝 over 𝑞 where 𝑝 and 𝑞 are integers and 𝑞 is not
equal to zero.
So let’s take some numbers which
are rational. For example, we could have this
fraction six over one, which fits the form of a rational number. Six and one are integers and the
one, this 𝑞-value on the denominator, is not equal to zero. This would be equivalent to the
value six, which is an integer. Let’s take another rational
number. Here, we have the example negative
two-fifths. We should ask ourselves if negative
two-fifths is an integer, a number that has no fractional part. And this would be no. We couldn’t write this in any way
as a number that has no fractional part. Therefore, the answer to the
question “is every rational number an integer?” is no.
If we consider the Venn diagram
where we have the set of rational numbers, then contained within this set will be
the set of integers. Using this diagram is helpful to
illustrate that every integer is a rational number, but not every rational number is
In the next question, we’ll see
more formal notation for the set of rational numbers.
Which of the following is true? Option (A) one is an element of the
rational numbers. Option (B) one is not an element of
the rational numbers.
The symbols in this question
represents some formal mathematical notation. The ∊ symbol can be read as an
element of or belongs to or is a member of, and this ℚ symbol represents the set of
rational numbers. So in order to establish if one is
a member of the set of rational numbers or one is not a member of the set of
rational numbers, we’ll need to recall what the rational numbers are.
A rational number is a number which
can be expressed as 𝑝 over 𝑞 where 𝑝 and 𝑞 are integers and 𝑞 is not equal to
zero. We might then look at this number
one and think, “Well, it’s already an integer. How could I express this as a
fraction?” Well, any integer value can be
written as that value over one. Looking at the definition, we can
confirm that the one on the numerator is an integer, and so is the one on the
denominator. And crucially, this number on the
denominator is not equal to zero. This means that one is a member of
the set of rational numbers. Therefore, our answer is that given
in option (A); one is a member of the set of rational numbers.
In the next question, we’ll find
the position of a rational number on a number line.
Which of the numbers 𝑙, 𝑚, 𝑛,
and 𝑜 is four-tenths?
If we take a look at this number
line, we can see that we’ve got measurements of negative one, zero, and one. We’ve also got these four letters,
which represent different values on this number line. We need to work out which of these
represents four-tenths. The first thing we might think
about is the fact that four-tenths is a positive value. We could, therefore, rewrite any
options given that are below zero. Secondly, we might recognize that
four-tenths must be between zero and one. If it was a fraction over 10 that’s
larger than one, then the numerator would be higher than 10. Let’s consider this section then
between zero and one.
One method of finding the position
is to consider that this is divided into five sections. Ideally, however, we would like 10
sections. So if we split this stretch into 10
sections, then four-tenths along would be at the value which is represented by
𝑛. As an alternative method, we could
have simplified the fraction of four-tenths by taking out the common factor of two
in the numerator and denominator. Out of the five sections between
zero and one, we’re looking for two-fifths, which would mean that it does confirm
that the value 𝑛 represents four-tenths. So, 𝑛 is the answer. Notice that this value of 𝑜 would
represent four-fifths on the number line.
In this question, we’ll need to
find the position of a number without being given a number line.
Find the rational number lying
halfway between negative two-sevenths and four thirty-fifths.
Here, we have two fractions,
negative two-sevenths and four thirty-fifths. And we’re asked to find the
rational number which lies halfway between. We can recall the rational number
is a number which can be expressed as 𝑝 over 𝑞 where 𝑝 and 𝑞 are integers and 𝑞
is not equal to zero. The best way to start a question
like this is to see if we can make the denominators the same value.
We should be able to write negative
two-sevenths as a fraction over 35. Observing that we multiply the
denominator by five, then our numerator will also be multiplied by five, which gives
a value of negative 10 over 35. It may help if we could visualize
negative 10 over 35 and four over 35 on a number line. So at the lower end of this number
line, we have got negative 10 over 35. And at the top end, we have four
thirty-fifths. This value of zero thirty-fifths
would also just be equivalent to zero.
If we were to think in terms of the
distance from zero, on the left-hand side, we have a distance of ten thirty-fifths
and on the right-hand side, a distance of four thirty-fifths. That’s equivalent to fourteen
thirty-fifths in total. Half of that would give us seven
thirty-fifths. Therefore, if we start from
negative 10 and count one, two, three, four, five, six, seven, we would get a value
of negative three thirty-fifths. As a check, counting down seven
thirty-fifths from four thirty-fifths would also give us negative three
thirty-fifths. Therefore, we could give the answer
as negative three thirty-fifths.
There is, of course, a different
method if we don’t want to or couldn’t draw it on a number line. Let’s go back to our two original
values: negative two-sevenths, which we could write as negative ten thirty-fifths,
and four thirty-fifths. The halfway point between these two
is equivalent to finding the median. We would begin by adding these two
fractions. To add fractions, we must have the
same denominator and we add the values on the numerator. In this case, negative 10 plus four
would give us negative six. So, remember, we’re finding the
median. So, we’ve added our values. And then, as there’s two values, we
need to divide by two.
In order to divide this fraction by
two, we can consider it as the fraction two over one, and then we multiply by the
reciprocal. So, we need to work out negative
six thirty-fifths multiplied by one-half. Before we multiply, we can simplify
this by taking out the common factor of two. So, we’ll have negative three
thirty-fifths multiplied by one over one. Multiplying the numerators and then
multiplying the denominators separately, we get the value of negative three
thirty-fifths, which confirms our earlier answer given by the first method.
Let’s have a look at one final
Which of the following expressions
is rational given 𝑎 equals one and 𝑏 equals 34? Option (A) negative 39 over 𝑎
minus one, option (B) 39𝑏 over 𝑏 minus 34, option (C) 39𝑏 over 𝑎 minus one, or
option (D) 𝑏 over 𝑎.
In order to answer this question,
let’s start by remembering the definition of a rational number. A rational number is a number that
can be expressed as 𝑝 over 𝑞 where 𝑝 and 𝑞 are integers and 𝑞 is not equal to
zero. In the four options here, we can
see that we’ve got four fractions that contain numerical values and also the
algebraic terms 𝑎 and 𝑏. However, we’re given numerical
values for 𝑎 and 𝑏. So, we’ll take each expression in
turn and plug in these values.
Let’s start with the first
expression in option (A). We’ll plug in the value that 𝑎 is
equal to one. We’ll still have 39 on the
numerator, and we’ll have one subtract one on the denominator. This, of course, simplifies to
negative 39 over zero. You might think that this looks
pretty good as a fraction. We’ve got a number on the numerator
and a number on the denominator. But in fact, if you’ve ever tried
to divide a number by zero on your calculator, you’ll get an undefined answer. And importantly, if we look at our
definition of a rational number, the 𝑞, the value on the denominator, cannot be
equal to zero. So, this value of negative 39 over
zero is not a rational number. And so, we can exclude option
In the expression in option (B),
we’ll need to substitute in the value 𝑏 equals 34 twice as 𝑏 occurs twice. We’ll, therefore, have the
calculation 39 times 34 over 34 minus 34. Before we rush to calculate 39
multiplied by 34, you might already notice what’s going to happen on this
denominator. Once again, we’re going to have a
denominator that has a value of zero. So, we know that this expression
when 𝑏 is equal to 34 would not be rational.
We can use the same method of
plugging in the 𝑎- and 𝑏-values into option (C). So, it have 39 times 34 over one
minus one. You may have already noticed that
this denominator will also give a value of zero. So, option (C) is not a rational
number when 𝑎 is one and 𝑏 is 34.
The expression in option (D) is 𝑏
over 𝑎 which will be is 34 over one. Let’s check if this fits the
definition of a rational number. We have it as a fraction 𝑝 over 𝑞
where 𝑝 and 𝑞 are integers. And 34 and one are integers. And of course, the denominator one
is not equal to zero. Therefore, 34 over one is
rational. And so, our answer is (D). 𝑏 over 𝑎 is rational when 𝑎
equals one and 𝑏 equals 34.
Before we finish with this
question, let’s just take a quick look at this expression in option (A). We saw that this expression
negative 39 over 𝑎 minus one is not rational, but that’s not always the case. If we had any other value other
than 𝑎 equals one, for example, 𝑎 equals two, then we’ll work out negative 39 over
two minus one, which would give us a rational value of negative 39 over one. The expressions (A), (B), and (C)
are only irrational because they had a denominator of zero, as they did in fact have
integer values on the numerator and denominator.
Let’s now summarize what we’ve
learned in this video. Firstly, we began with the
definition of a rational number, which 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. Rational numbers include integers,
fractions and mixed-number fractions, terminating decimals, and repeating
decimals. Numbers that are not rational are
called irrational numbers. And finally, we saw this more
formal notation that this symbol, which looks like a ℚ with an extra line,
represents the set of rational numbers.