Lesson Video: Rational and Irrational Numbers Mathematics • 8th Grade

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.