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Hello everyone, it's Mr. Miller here.

In this lesson, we're going to be looking at the base 10 and base 5 number systems. So, first of all I hope that you're all doing well.

In the last lesson, we had a look at representing number in our own number system, which we call base 10.

In this lesson, we're going to have a look at an alternative way of representing number using the base 5 number system.

So, let's have a look at the try this task to get us started.

In the base 5 number system, we group units in groups of 5.

How would we represent 23 in the base 5 system? So underneath that, we've got 2 ways of representing 23.

The first one, in the base 10 system where we can see we've got groups of 10.

And on the right hand side, the base 5 system.

So what I want you to do, is I want you to copy and complete this sentence here, the base 10 sentence.

And then think about how we would represent 23 if instead of grouping in groups of 10, we grouped in groups of 5.

So pause the video and have a think, how would we represent 23 in groups of 5 and groups of 1? Okay great, so first of all, let's just go over the base 10 one very, very quickly, and we can see it right in front of us that we've got 2 lots of 10 and 3 lots of 1.

And that's 23 because 2 lots of 10 is 2 times by 10, which is 20.

We saw this last time.

And 3 lots of 1 is 3 times by 1 which is 3.

And then add them together, nice and straight forward, we get 23.

How do we get 23 if we were not grouping it in groups of 10, but groups of 5? Well, if we had 4 groups of 5, and 3 lots of 1, that would get us 23.

Because 4 lots of 5, is 4 times by 5, which is 20.

And 3 lots of 1, is 3 times by 1, is 3.

Add them together, and again I get 23.

So, here is an alternative way of grouping numbers, the base 5 system.

And let's have a look in the next slide about a little bit more complicated way of doing this.

Okay, so let's have a look at the connect task.

So, in the base 5 number system, the third column is twenty-fives.

So, in the last example, we just had fives and ones.

But when we need a third column, we're going to have groups of twenty-fives.

So let's have a look at the number 2, 4, 1, in the base 5 system.

Can you think, what that might be in the base 10 system? I'll start you off 2 twenty-fives is going to be 2 lots of twenty-five, which is 50.

Can you do the rest? Pause the video for 30 seconds, have a think.

Okay, nice and straight-forward, 4 lots of 5, is 4 times by 5, which is 20 and then 1 lot of 1 is just the 1.

Add them together and they get 71.

So, 2, 4, 1, in the base 5 system is 71 in the base 10 system and that's really easy to write.

That is 7 and 1.

One more thing to note is that this 2, 4, 1 we would write as 241 and then a little 5 there to signify that we are in the base 5 system.

The 71, we can write as 71, little 10.

That's in the base 10 system.

Okay, what about going from the base 10 number system, to the base 5 number system? The number 113 in the base 10 number system, how would we group that as groups of twenty-five, groups of five and groups of one in the base 5 system? Pause the video for 60 seconds to see if you can think what this might be.

Okay, well it's a little bit trickier because we need to think how many twenty-fives can we put into 113.

And if you're thinking 4 lots of twenty-five is the most we can put, then that's absolutely right.

4 lots of twenty-five, that equals 100.

And now we got 13 left over.

How would we make 13 with fives and ones? Well, it's going to be 2 lots of five, that's going to be 10.

And then just the 3 left over.

3 lots of one left over.

Add them all up, we get 113.

So how do we write this? Well, I got 4 lots of twenty-five, 2 lots of five and 3 ones.

So I would write this as 4, 2, 3, base 5.

Compared to 113 base 10.

Okay, this is how we go from base 5 to base 10, and vice versa.

Now it's your turn.

Let's have a look at some more examples in the independent task.

Okay so here's the independent task, we have got 4 numbers in base 5 that we need to convert into base 10, and then 4 numbers we need to go the other way.

Let's just do the first one of each together, just so that we're really confident before you have a go yourself.

So question 1, going from base 5 to base 10, we just need to remember, really important, what the different columns mean.

The first column, if I use the final example to show you, is ones.

The next column is fives, and then the next column is twenty-fives.

So, if I was doing the first example, I can see I've got 3 lots of five, which is 15 and 2 lots of one which is 2.

So adding those together, is going to be 17.

So the first one is 17 in the base 10 system.

And in question 2, I'm going from base 10 to base 5, so if I wanted to write 16 in the base 5 system, I need to think, first of all, how many twenty-fives are there in 16.

There are none, so it's going to be a 2-digit number.

Then I need to think, how many fives are there in 15.

Well, there are 3 lots of five, to make 15.

And then one left over, 1 lot of one, and then that would make 16.

So how do I write this? Well, I've got 3 lots of five and 1 lot of one, so that is 31 in the base 5 system.

Okay, your turn to do the rest.

Pause the video, 5 or 6 minutes, have a go at these remaining examples.

Great, so well done having a go, let's go through these very, very quickly.

So, in question 1, which was more straight-forward, we've got 4 lots of five, which is 20.

And 1 lot of one which is 1.

Add them together, 21.

Next one, I've got 4 lots of twenty-five, which is 100.

I've got no fives, I've got 1 one, 1 times 1 is one and therefore I've got 101.

Finally, 3 lots of twenty-five, which is 75.

3 lots of five, which is 15.

And 1 lot- 2 lots of one, which is 2.

Altogether 92.

Okay, the remaining ones in question 2, done the first one already.

26, well the first thing we're thinking is are there any twenty-fives in 26? Well yes, there is 1 lot of twenty-five.

And then just the 1 left over.

So how will I write this in base 5? Well, I've got 1 lot of twenty-five, no fives and 1 one.

So it's 101 base 5.

46, well that is going to be 1 lot of twenty-five and I've got 21 left over.

So, 4 times by 5, 4 lots of 5 is 20.

And 1 lot of one, which is 1.

So I write this as 141 base 5.

Final one, 112 base 10.

For that, it's going to be 4 lots of twenty-five which is 100 and then I need to make 12.

So I've got 2 lots of five, which is 10.

And 2 lots of one, which is 2.

So that's going to be 4, 2, 2, base 5.

Really well done if you got all of these, 'cause it's a tricky concept to pick up so really well done if you've done it.

And let's move on, when you're ready, to the explore task.

Okay, in the explore task you have got two statements and you need to think, are they always, sometimes or never true? If they are sometimes true, find an example when it is true and find a counter-example when it is false.

So let's read these two statements.

The first one is 2-digit base 5 numbers are greater than 1-digit base 10 numbers.

Second, 2-digit base 10 numbers are greater than 3-digit base 5 numbers.

Okay, slightly tricky but pause the video, have a think, are these always, sometimes or never true, and if they're sometimes true, find an example and a counter-example.

Pause the video now.

Great, so, hope that you have a nice think about it, they are actually both sometimes true.

So well done if you got that.

Let's have a think.

How 'about the first one.

2-digit base 5 numbers are greater than 1-digit base 10 numbers.

Well, if I have a think about a big base 5 number, let's have think about 44 base 5.

Well that is 4 lots of five, which is 20.

And 4 lots of one, which is 4.

And that is going to be 24.

Note that in the base 5 system, the biggest number that you can have in any column is going to be a 4.

So that is a 2-digit base 5 number which is 24 in the base 10.

And that is bigger than any 1-digit base 10 number.

The biggest, of course, being 9.

Can you think of a 2-digit base 5 number which is smaller than a 1-digit base 10 number.

Well, if you're thinking, for example, 12 in base 5, that is 1 lot of five, which is 5, 2 lots of one, which is 2, that is 7.

And 7 is smaller than 9, so in that case, our 2-digit base 5 number is smaller than a 1-digit base 10 number.

Okay, so if you didn't manage to get the second one originally, sometimes true, pause the video now and see if you can use the first example to help you find an example when it is true and an example when it is false.

Okay great, let's go through it.

So, let's think about this in terms of 3-digit base 5 numbers.

If I took, for example, 434 in base 5, well that is going to be 4 lots of twenty-five, which is 100, 3 lots of five, which is 15 and 4 lots of one, which is 4.

And that adds up to make 119.

And that is bigger than any 2-digit base 10 number.

And what if I had something like 320? Well, that's going to be 3 lots of twenty-five, which is 75, 2 lots of five, which is 10, that is 85.

And of course, I can think of 2-digit base 10 numbers that are bigger than that, anything over 85.

86, 87, etc.

So interestingly, both of these are sometimes true and it does, it should give you some thought about the nature of base 5 so I hope you found that interesting, that is it for today.

Next time, we're going to be looking at the Indian number system.

So the system that they used for numbers in India.

So that is it for today, hope that you enjoyed this lesson and see you next time, have a great day, bye bye.