Lesson Video: The Set of Rational Numbers Mathematics • First Year of Preparatory School

Video Transcript

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 integers.

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 number?

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 six.

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 question.

Is every rational number an integer?

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 an integer.

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 question.

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 (A).

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.