Number systems: Operations in different bases

Key Stage 3

Maths

Different number systems

﻿Hello, and welcome to this video on operations and different bases. My name is Mr Maseko. Now before you start this lesson, make sure you have a pen or pencil and something to write on. Okay, now that you have those things, let's get on with today's lesson. Firstly, try this activity. Are these approximations true or false? So each of these numbers has been rounded. So is this true or false? Pause the video here and give this a go. Okay, now that you've tried this, let's see what you've come up with. Well, the easiest thing to do when you look at this question is to try and figure out which place you've rounded to. Well, if we look at this, our two has become a three so we've rounded up. Now, what place value is that two in? We are in base four. So base four's place value charts, has the ones, the fours and the sixteens. So it looks that you rounded up to the next 16. Well, would that be the case? If you've rounded to the nearest 16, will we get three, zero, zero in base four? Well, let's see. What is this number, as a base 10 number? The one we're used to. Well this is two sixteens and then we have two fours and three ones. Two sixteens as 32, two fours is 8 that gives us 40 and that would give us what, 43. And then if we look at this number well this number is 48, well will that work? Rounding to the nearest 16. What's the nearest 16 to 43? Well, you've got 32, 48, so 16, 32, 48. So we are in between 32 and 48. So which one are we closer to? Well we are closer to 48. So yes we would round up to three, zero, zero. Now how else could he have shown? Well we round to the nearest 16, if we look at the place values before that that's 2,3, what's that worth? That two, three, is worth four plus three, which is 11. Well, it's 11 bigger than eight? Yes, which means we will round up. So this is true, two to three base four is approximately three, zero, zero in base four. Could we round up to the nearest 16. Three, four in base five is approximately four zero. Well, if you look at the units digits we've rounded to the nearest five, and we can see that the units digit is bigger than half of five. So yes, we round up, so yes, that is true. Two to three and base five is the same as three, zero, zero in base five. Well let's see what would happen if we round the nearest what's the place value chart for base five? We've got our ones, our fives and our 25s. So this looks like we've rounded to the 25th position. So you've rounded it to the nearest 25. Is three zero, zero, the nearest 25 to two two three? Well, if you look at this look at two, three, so two fives and three ones. What's that worth? Two fives and three ones. Well two fives is 10, 3 ones is three 13. When rounded to the nearest 25 is 13 bigger than a half of 25? Yes, so this would also be true, rounded to the nearest 25. Well, if we look at this last one, three, two, one, one in base four, round it to one zero, zero, zero, zero in base 4. Can that be true? Well, what is one zero, zero, zero in base four. Well, that's the ones position, that's the fours position, that's the 16s position. That's the 64s position. And this is the 64 times four. Oh, that's one two eight. That is two, five, six. So this is the two five, six position. So this is 256. This on the other hand, this is one, one, one, four, two sixteens and three 64s. So what is that? Well, one plus four that is five add 16 that is, good, that is 21. Add 64 times three or 64 times three is equal to, will that be equal to one, nine, two. So if you do one, nine, two, add 32 add four, add one, oh, what do you get? Two hundred and twenty nine. So yeah, if we're rounding to the nearest 256 we can tell that 229 is really to 256. So yes, that would also be the case. Now what base is easiest to round in that depends on what you've worked with most of your life. So a lot of us will find it much easier to round in base 10, because we've worked with worked in base 10 all of our life. Now we're going to be looking at subtraction and addition and different bases. So there are many different ways to do this. But if we look at these right here, so these are subtractions in base seven. Why is that the case? One zero take away one gives you six. One zero take away three gives you four. One zero, zero, take away two gives you five zero. It's all in base seven why is that the case? Well, let's write this numbers in the base we're used to. So base 10 and see what we get. Well, one zero that is seven that's seven take away one is just one, which gives you seven take away gives you six. And that's six ones so that makes sense. Then one zero take away three, well that is seven take away three. Again that gives you four. So yeah, that makes sense that's four ones. And the one zero zero take away two zero one zero zero. So one zero, zero, so that is base 49, take away two zero our two zero is 14. 49 takeaway 14 that gives you what? Thirty five, and we know that 35 is five sevens so that's why we got five zero. Can you do the same with these calculations? Pause the video here and give that a go. Okay, now that you've tried this, let's see what you've come up with. Well, if we look, we've got one zero, zero take away one. Well whats this that's 49 take away one, that gives us 48. So we have no more 49s. But then that means that we have what, 148 now. So now on six sevens. So instead of having 149, we have now 6 sevens, which is 42. And we have six ones, 42 plus six gives you 48. So one zero, zero take away one is six six. Can you see what's happening? Well, one zero, zero, zero take away one. Well, we are on 2, 4, 3. If we take away one, we have no more two, four threes, but then what would we have? Well, we will have six 49s, and we will have what? Six sevens and we will have six ones. Huh, okay, let's see if this continues. Well, the next one. One zero, zero take away one, one. So first take away one, one. So we're left with, so first we do one zero, zero take away one we know that is 66. So now we've got to take away one zero. So now we're taking away one seven. Well, that means that we're left with five sevens and six ones. Then what about here? You can do it the same way. When you take away one one you're left with 666, and take away one seven this becomes 656. Do you spot something? What patterns are you recognising? The highest we can have of each place value in base seven is six lots of it. So anytime we take away one, we are left with six lots of the place values before. See what happened when we took away one from one zero, zero. Well, the highest place value disappeared because we didn't have any more two four threes, but then we're left with six sevens and then we left with six ones. Patterns that you meant to be spotting. Let's try something else. Now here's a number bonds charts. Can you complete this number bonds for base four and base five. And then also draw one for base seven. Pause the video here and give this a go. Okay, now I'll let you try your best. Let's see what you come up with. Well, if we look at this, one add one in base three gives you two, one add two gives you one zero, 'cause one zero this is worth three. So it's the same here one add one that's two, one add two that's three. One add three column base four. If one add three is four, but then column base four that becomes, we move up at place value so it becomes one zero. We have normal ones and one zero. So we already know this that's three. One zero and then three add two is five. So it's one, four and one, one. So that's one zero one, one and this would be one, two. What about for base five always the same thing two, three, four, one zero. Four, five, three four one zero, one one. Four, one zero, one one, one two. One zero, one one, one two and one three. This is what you do to come up with a base seven. Now for this independent task, all these calculations are in base four. Can you complete them? Pause the video here and give this a go. Okay, now that you've tried this, let's see what you have come up with. Well, two, three add three one, if you write this the same way, two, three, add three one. Well, what do we have? Three add one that gives us four so that. What would happen, so three add one gives us four? So we know that four is one zero. So we put a zero in the unit column and we carry the one over. Well three add two, three add two gives us five and then add one gives a six. So we now have six fours. We know we can't write six fours. So what is six fours? You know that six fours is 24. Think of the fours place value chart so the ones, the fours, the sixteens, and then the 64s. So we can have a 16 in there. So, so this gives us, and we've seen this in the example, we end up with two fours, and then we carry all four fours over. So carry our four 4s over. And then that gives us what, one 16. So two three add three one is one, two, zero. Huh, okay, what about three three add two one? Well, this will look kind of the same. That is zero carry the one could go one four carried over to the next place. And then three or two, same thing. We've got six fours, so what do we have? Well, that's is two fours with four fours carried over, which makes our one 16. So those two answers are the same. One two three zero add three two one zero. So one two, three, zero and three, two, one zero. What does that give us? Oh, zero add zero that's zero nice and easy. Three fours add one four that gives you four fours. So when you have four fours, you carry what? The four four is over which is one 16. So we carry the one 16 over 'cause four fours is one 16, I guess carried over. And then we got two sixteens. Add two sixteens, well, that's four sixteens, add one 16 that's five sixteens. So five sixteens well that is one 16 and one 64 carried over. Then we've got one 64 add three 64s add one 64. Well, we've got four 64s, which is the next place value up, so we got one 64 left over, and then we have four 64s where four 64s is what? As one, two, five, six. Huh, can you see what's happening? Now this is a really hard thing. But you have to be thinking about what you have in your place value charts. And what you got to be carrying over. Just like when we were adding in base 10. Now let's do this one. We got one, three, three, one add two, zero three. Well, one add three is four, so we got one four. So you can carry over one four, three add one that's four fours. Well for four fours, we can carry that over as one 16. Three add two add one, well three add two add one that is six sixteens. Well, in six sixteens, we have four sixteens. Four sixteens can get carried over as one 64 and then you're left with two sixteens. And then you got one add one, which is two 64s. So ends up at two, two zero, zero. And then this one, three, two, three, add three, three, three. Oh, three add three that's six ones or six ones has one four in it so it can leave two ones and carry one four over. Two add three add one, well, that is six fours. Six fours has two fours in it. Then we can carry one 16 over that's four fours. So carry four fours over that's one 16. Then three add three add one well that is seven sixteens. In seven sixteens we have three sixteens, and we can carry a four sixteens over as one 64. So that gives us one three two two. And then we can do the same thing for this two zero three three three three add three one. Well, three add three at one, that's seven ones and seven ones with three ones and one four. Three add three add one that is seven fours. And seven fours, we have three fours and one 16, 'cause four fours is one 16 and two add one that is good, that is three sixteens. So just like the rules for adding in base 10, we can apply them to what we're adding in base four. 'Cause if you think about it, if you add 19 add 23. Well, let's do this, what is nine add three? Well, add three is 12. Well in 12, we have two ones and 10 ones makes one 10. So two ones and one 10 gets carried over. See, and then one add two add three well that makes four tens. If we got to 10 tens, then we just carry that over into the hundreds column. So addition rule whatever base we are in, they're same. Now, if you try this calculations in base 8 are really well done. I look forward to seeing what you came up with. Now for this explore task, these are the digits for base 12. 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B. Why do you think we're using A, B? Well, because if we're to use one zero and one one, well in base 12, one zero means 12 and one one means the 13. So that's why we can't use one zero and one one. So whenever you get into these two digit bases, this is what happens. We have to introduce, other symbols to represent our two digit unit values in this case we use A and B. Those represent 10 and 11. So if we look at this calculations, based on those stages for base 12, are they true or false? And in task two, work out those calculations in base 12. Pause the video head and give this a go. Okay, now that you've tried this, let's see what you've come up with. Well, seven add seven is one, two, well, seven add seven that gives us 14. Well, in 14 ones in 14 ones we have one 12 and then two ones left over. So that makes sense, one 12 and two ones left over. So that is true. A add B is 19 is one nine. So A and B is one nine, is that true? Or A represents 10 ones and B represents 11 ones or 10 ones add 11 ones that is 21 ones. And 21 ones we have one 12 and nine ones. We have one 12 and nine ones 'cause 21 takeaway 12 is nine. So we've got one 12 and nine ones. So yes, that is true. Well, now let's look at one A take away B. So what does this represent? For one A this is one 12 and 10 ones, one 12 and 10 ones with that represents 22 in base 10, and then one B whatever represents 11 ones so as to take away 11. 22 takeaway 11 is 11, 11 ones in base 12 is B. So yeah, that is true. Now, want to calculate the following, so five, three, six add ABA. So five, three, six add ABA. Well, six add A, six add 10 that is 16 and 16, you have four ones and one 12. Three add B so that's three add 11, that is, what is it? Three add 11 is 14 add one that's 15. In 15 twelves, well, what do you have in 15 twelves? In 15 twelves you have three twelves, and 12 twelves which makes one one four four. And then five add A so five add A is five add 10 is 15. Add one is 16 and 16 one four fours, well you have, four, one, four fours and 12 one four fours, which get carried over. So this becomes one, four, three, four. And this is how you'd have done the other calculations. So if you follow these basic rules, you'd have got answers in this way so the same way as we're doing for base seven. Now before in impure measures, we always used to base 12 seems to be very common. Now, why do you think base 12 was common? I won't tell you the answer, but think about the factors of back base 12. And if you think you know why base 12 was commonly used, ask your parent or carer to share your work on Twitter, tagging @OakNational and #LearnwithOak. Bye for now.

Number systems: Operations in different bases

Number systems: Operations in different bases

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UnitMaths / Different number systems

Maths