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Lesson Video: Real Numbers Mathematics • Second Year of Preparatory School

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In this video, we will learn how to distinguish between rational and irrational numbers and represent real numbers on number lines.

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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.

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