# Lesson video

In progress...

Hello, and welcome to this lesson with me, Dr.

Where we are going to learn about rational and irrational numbers.

For today's lesson, you'll need a pen and a paper.

Please take a moment to clear away any distractions that you may have and find a quiet place, where you will not be disturbed during this lesson.

Okay, when you're ready, let's begin.

Okay, for the try this, I want you to categorise these numbers into two groups.

Try categorising in different ways.

If not, hold on, I will give you some support.

Okay, for support, let's look at all the numbers here.

I can see if I have some positive, some negative numbers.

I can start by splitting them into those two groups.

So write positive numbers and list all the positive numbers that you can see here and write negative numbers and write down all the negative numbers that you can see from the list.

Okay, pause the video and see if he can do this.

Well done if you've done this correctly, I have put all the positive numbers together here.

And then I put all the negatives, including negative five, negative 25, and negative 2/3.

I left the square root of negative four on its own.

This is quite different.

We know that we square root, usually positive numbers, and we don't usually do that with negative numbers.

In fact, we can, but that's something that you will learn about a lot later in maths.

Where you start looking at imaginary numbers.

Okay, so if you've done this really good, well done.

How about splitting those into slightly different groups instead of positives and negatives.

Can you think about maybe trying to split them into integers, decimals, fractions.

Pause the video and have a go at this now.

Excellent, well done for having go at this.

So this is how I've done it, I've put all whole numbers, all of my integers together.

That's the two, nine, negative five and negative 25.

For all the fractions together.

It's 2/5, 2/9 and negative 2/3.

And for anything that had a square root or a cube root together.

So that's a cube root of seven, the cube root of negative four and the square root of two.

I've put the decimals together, 0.

6, 3.

657 and 0.

6 recurring.

I wonder where you put PI on the list.

Did you put it with the decimals? Did you put it with the fractions? Okay, I left Pi on its own, one because it's a very special number and we'll be talking about it in today's lesson later on.

And because it is a decimal, it has a value of roughly 3.

14, but actually it's not exact, it's a fraction that never ends.

And that's what today's lesson is going to be about.

Also, I left the square root of 81 on the side here.

And I want to have a chat with you about it.

Where did you put the square root of 81?.

Did you put it with the integers, the whole numbers, Because the square root of 81 is nine? If you've done that, that's really good, well done.

If you were splitting them into positive and negative numbers, I wonder where would you put it? Would you put it with positives or negatives? Or does it apply to both? Really good well done.

You can put it with positives because the square root of 81 is nine and the square root of 81 is also negative nine.

So it can be in both places.

Also well done if you just put it with the square roots there, under the square root of two, that's also correct, because you would have been looking at how the numbers look like.

Okay, really good job on the try this.

Now let's move on to the main part of today's lesson.

Okay, and now we're going to look at rational numbers.

What are rational numbers? Rational numbers are numbers that can be written as the ratio of two integers.

So just by looking at the word rational comes from ratio.

So it is a number that can be written as the ratio of two integers.

What does that mean? Let's look at the definition together and perhaps you can copy it down.

Rational numbers are numbers of can be written in the form, a/b, where a and b are integers.

Remember integers mean whole numbers.

Pause the video and copy the definition.

Well done.

Now let's look at some examples of rational numbers.

All integers can be written as integers out of one.

For example, 12 can be written as 12/1.

So we can write it as a fraction, where both the numerator and the denominator are integers they're whole numbers, and therefore all integers are rational numbers.

All terminating decimals, terminating decimals means decimals that end.

They do not just keep going forever.

Are also rational numbers, for example, 13.

61 is the same as 13 and 61/100 or 1,361/100.

So we can want it as a fraction with both the numerator and the denominator being whole numbers.

And therefore all terminating decimals are also rational numbers.

All recurring decimals, yeah.

So 0.

3 recurring, remember this dot means it's 0.

33333.

This is the same as 1/3.

So we can still write it as a fraction with both the numerator and the denominator being whole numbers.

And therefore all recurring decimals are rational numbers.

313131 and so on.

Is that also going to be a rational number? Yes, it is because it has a pattern and it's so recurring decimal.

It's the same pattern gets repeated over and over again to me is we can express it as a fraction in the form of a/b.

And therefore it is a rational number.

So as long as that estimate is a recurring decimal, then it is rational number.

Okay, so let's recap really quickly.

Rational numbers can be written in the format of a fraction, a/b where both a and b are integers.

All integers are rational numbers because they can be written out of one.

All terminating decimals, if the decimal has an end, then they can be written as a fraction.

And therefore they are rational numbers and all recurring decimals are also rational numbers because they can be written as a fraction too.

Okay, so what about numbers that are not rational.

What do we call them? Well done.

We call them irrational numbers and irrational numbers, are numbers that can not be written in the form a/b.

So we can not write them as a fraction where a and b are integers.

Where the numerator and the denominator are whole numbers.

They can not be written as recurring or terminating decimals.

So we cannot write them as a decimal accurately because they keep going on and on and on and on that, you know, and we can not write them as a recurring decimal because there isn't a specific pattern that they follow that repeats every time.

So they're not recurring decimals, they're not terminating decimals.

Let's look at some examples.

Square root of two is equal to 1.

41213562 and it keeps going on and on.

So square root of two is an irrational number.

Second example, PI what is PI? PI tells us the ratio of the circumference to that of the circumference, to the diameter of the circle? Well done, PI is also an irrational number.

It has the value of 3.

141592653 and it keeps going on and on and on.

There are no recurring patterns in the digits for PI and for the square root of two.

And the decimals do not terminate, they do not end.

Therefore those numbers are irrational numbers.

You may think that are only few irrational numbers, but actually there are a lot of irrational numbers.

In fact, the square root of anything, any number that is not a square number is irrational number.

And that's what we'd be looking at in today's lesson.

Okay, let's have a look at some examples.

If I have square root of nine and square root of five, just because I have the square root, do they automatically become irrational numbers? What is the square root of nine? Well done it's three.

It's a whole number therefore, it's a rational number.

So the square root of nine is an rational number.

'Cause the answer is an integer and I can write that three as 3/1.

I can write it as a fraction.

What about this square root of five? If you use a calculator and type in square root of five, what do you get? Well done, 2.

236067977 and it does not end.

Does not terminate.

Square root of five can not be written in the form a/b where a and b are integers.

Therefore the square root of five is an irrational number.

Let's have a look at this question together.

Determine whether each of the numbers below are rational or irrational numbers.

These are the numbers.

I would like you to pause the video and have a little think about it.

Okay, well done.

17.

0 can be written as 17/1.

So it can be within as a fraction, a/b and therefore it's a rational.

1/9 is already written for us as a fraction.

Therefore it's a rational number.

What did you write down for 2.

6 recurring? 2.

6 recurring is equal to 8/3 and therefore it's rational.

What about the square root of 25? Really good.

The answer is five and therefore it's a rational number.

What did you write down for PI? PI is roughly 3.

14, and we keep going.

Does not terminate, it's not a recurring decimal therefore it's an irrational number.

0.

2 recurring, you can write that down as 2/9 as a fraction and therefore it's a rational number.

What about the square root of two? Square root of two is 1.

4142 and it keeps going.

Does not terminate and therefore it is irrational number.

Well done, last one is a little tricky.

What do you think it is? Square root of two out of five.

So I have the square root of two, which is an irrational number, divided by a rational number, divided by five.

Now irrational divided by rational will you give me an irrational number.

Irrational multiplied by a rational will give me an irrational number and irrational add it to a rational number.

So the sum, would also be an irrational number.

And this takes me back to what I said to you in the previous slide that we've got lots of irrational numbers there.

So if you have an irrational number multiply it by another rational number or divide it by a rational number or add it to a rational number, you always end up with an irrational number.

Really good, well done.

We've done lots of learning about rational and irrational numbers.

We looked at what they are.

We looked at examples and we looked at a combination of numbers and we had to classify them and decide whether they were rational or irrational numbers.

Now it's time for you to do some independent learning.

So what I would like you to do here is to pause the video and have a go at these three questions for me, please.

Once you finished, you press play again, and then I will go through the answers.

You can mark and correct your work as we go along.

Okay, let's go.

You've got question number one, write down a rational number between four and five.

You could have written any number that is decimal between four and five, as long as it terminates, so 4.

5.

A rational number that has a rational square root.

So we want a number that is rational, if we square root it, we want the answer to also be rational.

For example, 16 is a rational number.

If we square root it, it give us four, which is also rational, 1/4 will also work.

A rational number with an irrational square root.

So this time I want the rational number, if I square root it, I don't want it to be rational.

So you could have had five, two, 1/3.

Five is a rational number, square root it, square root of five is an irrational number.

Okay D, an irrational number between four and five.

This is a really, really good question.

So we know that four and five are both rational numbers.

The four is the same as square root of 16.

Five is the same as square root of 25.

Therefore an irrational number will be a square root of a number between these two.

For example, square root of 20.

Well done, question two.

Cara say 0.

1111 recurring is an irrational number because it goes on forever.

Cara is not right.

All recurring decimals are rational numbers 'cause we can write them as fractions.

In fact, 0.

111 recurring is equivalent to 1/9.

John says when you multiply two irrational numbers the answer is always an irrational number.

Give an example to show that John is wrong.

So here you need to think about different numbers and apply them to see what happens.

For example, square root of two multiply that by square root of two.

So we have two irrational numbers we're multiply them by each other.

The product, which is two is actually a rational number.

So this where we're showing that John is wrong.

I wonder which numbers you have used.

I know that some of you would have probably use something like this as well.

Square root of two multiply that by square root of eight.

If you type that into your calculator, you would get four.

So you have an irrational multiplied by irrational number and the product is a rational number.

Excellent job, well done.

We're nearly there.

Decide if each statement is true or false, justify your answer, pause the video and have a go at this.

Okay, let's go through these together.

An irrational number is the same as a recurring decimal.

What did you write down? Well done, it is false.

Recurring decimals are rational numbers.

Next one, all rational numbers can be written as fractions.

Good job, true.

For example, five can be written as 5/1.

Next one.

The square root of a non-square number are irrational.

Good job, true.

The square root of eight is irrational.

Next one, a repeating decimal pattern means the number is a rational number.

Is this true or false? Really good, it is true.

There is an irrational number between any two rational numbers.

What did you write down? True, square root of 15 is between three and four and it is irrational.

Really good.

Can you come up with your own statements and decide whether they are true or false? Okay, this brings us to the end of today's lesson.

A really big well done, on all the fantastic learning that you've achieved today.

I've got three final things I'd like you to do.

First, look back at your notes and identify three most important things that you've learned today.

It's totally up to you what they are.

Second, please do the exit quiz.

I'd love to see your work.