Lesson video

Hello, my name is Miss Parnham.

In this lesson we're going to identify rational and irrational numbers.

Let's begin by thinking about the difference between natural numbers and integers.

Natural numbers, as the name suggests, come naturally to us.

These are the numbers that we first came across when we were first learning to count, so they are the whole numbers starting at one.

Integers are also whole numbers, but as well as being positive, it could be negative, or zero.

Let's look at a couple of examples.

At first glance, this example looks like a fraction, it's an improper fraction of 36/4, but this will simplify and it will simplify to an integer.

So this is an integer.

What about negative 6/12? This will simplify to negative 1/2 or negative 0.

5, which is not an integer.

Here are some questions for you to try.

Pause the video to complete the task and restart the video when you're finished.

Here are the answers.

Three of the integers were originally expressed as fractions, but when simplified, give positive or negative whole numbers answers, and therefore, they're integers.

Up to now we've been looking at rational numbers.

The term rational comes from the word ratio, which means the number can be expressed as a fraction.

So this can include decimals that terminate, that means they've got a finite number of digits after the decimal point, or decimals that recur, which means there's a repeating pattern of numbers after the decimal point.

But it can also include natural numbers and integers, because as we saw in the previous example, integers can be expressed as fractions and natural numbers are part of the family of integers.

So irrational numbers cannot be expressed as fractions.

Examples could include pi, which you may have come across in work on circles, or the square roots of numbers which aren't perfect squares.

Here are some questions for you to try.

Pause the video to complete the task and restart the video when you're finished.

Here are the answers.

Root 36 over 12 may have seemed tricky, but 36 divided by 12 is three and root three is irrational, because three is not a perfect square.

We've already come across surds in this lesson, but you may not have heard that name before.

Surds are square roots of numbers which aren't perfect squares.

And we always write them in this form for accuracy.

They are irrational, so we can't write them as fractions.

And writing them as decimals means we would have to round them.

Look at these two numbers.

Only one of them is a surd.

Root seven is a surd, because seven is not a square number or a perfect square, whereas root four has got two solutions, two and negative two, which are integers, so root four is not a surd.

Here's a quick question for you to try.

Pause the video to complete the task and restart the video when you're finished.

Here are the answers.

Root six and root 11 are surds, because six and 11 are not perfect squares.

Here's a question for you to try.

Pause the video to complete the task and restart the video when you're finished.

Here are the answers.

Three root two and root six over two are surds as well, because they're multiples of surds.

Here are some more questions for you to try.

Pause the video to complete your task and restart the video when you're finished.

Here are the answers.

For question four a you may have a different answer from me, but it could be equally right.

As long as you have five in the units column, it can be followed by a terminating decimal, a recurring decimal, or a fraction.

It will be a rational number between five and six.

I chose my irrational number by thinking of five as root 25 and six as root 36.

So just choosing the root of a number between 25 and 36 will get you an irrational number between five and six.

The equation with a rational solution is in the middle, because when we divide by 2/5, x squared equals 400, and the square root of 400 is 20, which is an integer, so that one's rational.

That's all for this lesson.

Thank you for watching.