Numbers and numerals

Axioms and arrays

Factors and multiples

Order of operations

Positive and negative numbers

Expressions, equations and inequalities

Angles

Classifying 2-D shapes

Constructing triangles and quadrilaterals

Coordinates

Area of 2-D shapes

Transforming 2-D figures

Prime factor decomposition

Conceptualising and comparing fractions

Manipulating and calculating with fractions

Ratio

Percentages

Different number systems

Sequences

Forming and solving equations

Forming and solving inequalities

Linear graphs

Accuracy and estimation

Algebra and problem solving

Ratio (8.6a)

Ratio, real life graphs, and rates of change

Direct and indirect proportion

Univariate data

Bivariate data

Famous maths problems

Angles and parallel lines

Angles in polygons

Bearings

Volume and surface area of prisms

Area review

Circles

FDP review (9.1a)

Probability

Sets and Venn Diagrams

Solving linear simultaneous equations algebraically

Solving linear simultaneous equations graphically

Angle review (9.5a)

Constructions, congruence, and loci

Pythagoras's theorem

Ratio review (9.7a)

Similarity and enlargement

Surds and trigonometry

Quadratic expressions (9.9)

Quadratic equations (9.10)

Indices and standard form

Growth and decay

Finance

Maths

Y7 Maths

Key Stage 3

Year 7

UNIT

In this lesson, we will be looking at writing numbers in different bases and how to convert them.

Number systems: Writing numbers in different bases

﻿Hello, and welcome to this lesson. My name is Mr Maseko. In this lesson, we're looking at number systems. We're going to be looking at writing based ten numbers and different basis. Now we'll find out what all that means during the course of the lesson. Before you start the lesson, make sure you have a pen or a pencil and something to write on. Okay? Now that you have all those things, let's get on with today's lesson. First try this activity, there's two students wrote these two sentences. Antony says I can see that one ten is ten times greater than one one. And the second student says I can see that 1000 is 100 times greater than one ten. Now how many sentences can you write using that diagram? Pause the video here and give that a go. Okay. Now that you've tried this, now that you've tried this let's see what you have come up with. Well, what could we have said? Well, let's look at the relationship between one ten and 100. Well, 100 is ten times greater than one ten. Well, what else could you have said? Well, if we look at the relationship between a thousand and a one, well, 1000 is 1000 times greater than one one. Now what we're looking at today is what we call base ten. And other base systems for our number systems. Now what is base ten? It's all about place value in base ten, each place value is a power of ten. Cause if you look, we always start with our wants column so every number system has a ones column, cause we need to be able to count in singles. But the next place value up, depends on what base you're in. In base ten, we go from the ones to the tens and then from the tens to the hundreds and then from the hundreds to the thousands. So, to go from place to place on your place value chart, you always multiply by ten. And that's why this is called base ten. So if you look at a typical number like this, which is 2349, or what can you see? Well, in this, what we have is we have two thousands, three hundreds, four tens, and nine ones. Now Here we have 24 written in base ten. Now this stop scripts don't have, this is to show what base we are in. So if you look that just tells us when base ten, usually we don't have to write it, but because we're dealing with numbers written in different bases, we have to specify what base we had written our number in. And then if we look at this, this says this is 24 written in base five. Now what could that mean? We know about this means this number has two tens. This number has two tens because the two is in the tens column in base tens and four ones. Cause the four is in the ones column in base ten. But here we have four, four. Notice how I didn't say 44. We have four, four and base five, which also means 24. Well, let's look at a place value chart for base five what would we have? we'll have our ones column and then what will come after, what we multiply by it? Because the next place value, we have the ones it's base five so we multiply by? Good! So we multiply by five. So, this becomes a five column. So the next one will be our 25 column. And then the next one will be our 125 column, to go from place to place on your place value chart in base five, you multiply by a five. In base ten, you multiply by ten. Although in intuitive, so what does this mean? What does four four mean? Well, this means we have, well, that four is in the fives column. So we have four fives. And what else? And we have four ones. Well four fives, well four multiplied by five, what does that give you? That gives you 20 and four multiplied by one, that gives you four, 20 add four that gives you 24. So we've represented 24, which in base ten, we've represented it in base five cause this is four fives and four ones. Can you do the same thing for 28? Right back in base five. Pause the video here and give that a go. Okay. Now that you've tried this, let's see what you come up with. If you think of 28 how many fives do we have in 28? How many fives? well have five fives or five fives is what? five fives is 25. So we can see that in our place value chart, we have at 25 column for best five. So we know that we have at 25 and it was left over well, or we're left with three. So we have a 25 and three ones. So. Would I write it like this? No, I would invite this because just the same way as in base ten, if we are, if we have a place value that has no value in our number. We have to use our placeholder, which is a zero. So this is an edge we would write, we have three ones, we have zero fives and we have one 25. So if you look at this, that could be 28 written in base five. So let me write out, I'm going to to write a subscript five to show that that's us written in base five. How do we know? this has one 25, one times 25 is 25 and three ones, three times one is three, 25 and three gives you 28. Well, let's try this again. This time we are in a different base. So we have 24, which in base ten we've seen already. And what is written at base seven is written as three three. One of the plays values for base seven. Well, we have the ones column, the sevens column, the 49 column, and then the 243 column. So ones, sevens, 49s and then 243 column and then they'll still just keep going. So remember to go from place to place and base seven would multiply by Good. Seven. So what does three three mean in base seven? Well, three three in base seven. That means that we have three sevens. and we have three ones three sevens and three ones. What's three sevens? three sevens is 21, three ones is three, 21 add three gives you 24. And that's how you lost 24 and base 10. We'll come to why we can, why we would want to write different numbers in different basis in a later lesson in the series. But at this point, it's really interesting how we can identify what base we're in, based on the place value in our charts. Now, how would you write 28 in base seven? So if you think about it, how many sevens do we have in here? We have four sevens, exactly four sevens. So this has, would have zero ones and four sevens. So this would be 28, which in base seven is four zero cause four sevens. Now notice how don't say it's 40 because you have to be careful how you're staying your number when you are in a different base. The way we say our numbers is, is based on what based ten. Cause that's the one we use and why do we use base 10? Well, some people say it's because we have ten fingers. In other cultures, they use different bases like in the Mayan number system they would use, they use base 20. why do you think they use base 20? Well, they were just better with their toes than we are. Ten fingers, ten toes. I'm not going to show you my toes though. But let's move on with this lesson. So here's an independent task for you to try. So here's some numbers, which in different bases. Write them in base ten. 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, what do we have? We have two threes in base five so what does that mean? Or we've got two places, so I'm just going to use two place values of our base five. So we've got our ones and our fives. That's in base five. So we have three ones and two fives. What two times five, that is ten add three. So two three in base five is 13 in base ten. So this is the same as 13 in base ten or two one in base seven. Well, again, we have two place values so that is, the ones and the sevens and we have one one and two sevens. Two lots of seven is 14 add one this is 15 in base ten. Three five in base eight. Well, that is five ones and three eight, three eight gives you 24, five ones gives you five, So this would be 29 in base ten. Four Zero in base five, well we know of four fives, So this would be 20 in base ten. 123 in base four. Well base four, we haven't seen yet with their place values or base four with the ones, the fours and the 16s. So what do we have? We have one 16th, one 16. We have two fours and we have one one. One 16 and 16, add two fours, that is eight, add one so this is 25 in base ten. And then one zero, one zero in base tune well base tune is one two, four places. It's the ones, the twos, the fours and the eights. We have one eight, zero fours, one two and zero ones. So that is, would be equal to ten in base 10. Now looking at base two specifically, phase two is code binary. And these, the place values for base two. So you the ones, the twos, the fours, the eight, the sixteens of five to 60 fours. And it keeps going. So write each of the following numbers, these are all base ten numbers. Write them all in base two. And then you got to answer this question to write this number in base 10. Pause the video here and give that a go. Okay. Now that you've tried this, let's see what you come up with. Well, 24, what about the A written in base two? Well, if you think, how many twos do you have in 24? while you have. What? 12 twos, but we can't buy 12 twos. We know that in 24 we have one 16. So we know that we have one 16. 16, so we're left with eight and we have one eight and then we have a zero fours, zero twos and zero ones. So there should be 24 in base two over 32. But in 32 we have one 32 and then zero sixteens, zero eight, zero fours, zero twos and zero ones. See how base two takes really two digit numbers and they end up being what? Multiple digit numbers, much greater than two. Now 35 or 35 is one 32 zero sixteens, zero eight, zero fours, zero twos. and then Nope, not zero twos. One two and one one. Cause we can't have three ones, Cause three ones contains one two. So we have one two and one one. What about four? Well, four is one four and zero twos and zero ones. And then 64 or 64 is one 64, zero 32 zero 16, zero eight zero four zero twos and zero ones. And then 128. Well, if you increase, go up one more step. We have a 128 column. So you have 128, zero 64s, zero 32s, zero 16, zero eight, zero fours, zero twos, zero ones. Now why would anyone want to ride numbers like this? We will talk more about binary coding, it's uses in lesson four of this series. But why do you think base two is called the binary system? Like bicycle it's like bicycle has two wheels. Binary is because it only has two they're just like up here. It's all either zeroes and ones. And that's really important the fact that it's only zeros and ones is really important, as some really quick applications in the real world. Now, this number we're going to ride this in base ten. So let's look at our place volume this is all ones we got one one, zero twos and one four, so one, one, one four makes five and then zero eights, zero sixteens, zero 32s. So we got 64 add five, so this number is 69. Now really well done for getting through this lesson. And I hope you've learned something really interesting or writing numbers in different basis. If you want to share some numbers that you written down in different basis, ask your parent or carer to share your work on Twitter, tagging at Oak National and hashtag LearnwithOak. Bye for now.

In this lesson, we will be looking at rounding in different bases.

Number systems: Rounding in different bases

﻿Hello, and welcome to this lesson on rounding in 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. Now that you have this, let's get on with today's lesson. The first thing, try this activity. We are talking about rounding, and we're looking at rounding to nearest five, 50 and 500, and coming up with rules for those. So pause the video here and give this a go. Now that you've tried this, let's see what you've come up with. First, let's just think about the numbers that would round to five, 50 and 500. So we're going to start with some boundaries, so zero, five and 10, zero, 50 and 100, and then zero, 500, and 1000. Now the smallest number that rounds to five is at the halfway point, so it's 2.5. So that would round to five. And the largest number is less than the halfway point between five and 10. So less than, 7.5. And a number less than 7.5 would round to five. Now, if we look at that, what do you notice? The boundary is half of what? The value we are rounding to. Think back to rounding to the nearest 10, when rounding to the nearest 10, the smallest number that can round to the nearest 10 was five. And five is half of 10 and the largest number, that had to be less than 15. And so you added five to 10. So rounding is consistent, but we're doing that with our boundaries and our boundaries are always half of the value we're rounding to. And we take that away and we added on to find our largest value and we take it away to find our smallest value. Now for 50, this should now become really obvious, that's 25 and 75, these are boundaries. And then for 500, that's 250 and 750. Now for the upper boundary, remember it should be less than it. And then for the smaller boundary, we can be there and anything above. And those are similarities to rounding to the nearest 10, 100, 1000. That should say 1000, I apologise. Next, when rounding to the nearest 500, what's the smallest number that rounds to 10,000? Subtract 500, so that's 9,500 and then 10000s. So what's the smallest number that rounds, we are rounding to the nearest 500, so you've got to work from 9,500, you've got to add on 250, which would give us 9,750. Now at this boundary, if we're here, we round up to 10,000. So the smallest number is 9,750. So now we're going to be looking at rounding and rounding in different bases. This first example, we're looking at 34, which is written in base 10, and we want to round that in base seven. So 34 in base 10 is three tens and four ones. Now in base seven, we know that, before we have no 49s, but we know that we have what? Four sevens, which gives us 28. And then six ones, which gives us six. So 28 plus six that gives us 34. Now we want to round this number to the nearest one zero in base seven. Now what is one zero in base seven? Good, that is seven. So rounding to the nearest seven. So you've got to think, remember our boundaries for rounding, that we were talking about in the previous exercise. What's the smallest number that rounds to seven? Remember to think of your boundary between zero and seven, what's the smallest number that rounds to seven? It's 3.5. So anything above 3.5 would round up. So we have to look, when we are rounding to any number, we're looking at our units value. So if we're looking at units value, is it bigger than 3.5? Yes, so that means this rounds up. So that means that rounds up. Because we've bigger than the smallest number that rounds up to seven. So this rounds up, in that we're adding one more seven, so we're going to end up at five, zero in base seven. So four six in base seven rounds to five zero in base seven, which makes sense because 34, the nearest seven to 34 would be 35, and 35 is five zero in base seven. So when deciding to round up or down, we have to look at whether our unit value is bigger than the smallest number that rounds up to what we're rounding to. Now, let's try this again with a different example. We're looking at numbers in base five now. So we are through with 34, but I want to write this in base five. We have one 25, one five, that makes 30, and four ones. So one, one, four that is 34 in base five. So we want to round this to the nearest one zero in base five, which is five to the nearest five. So let's look at our units digits, and we've got to think a number that the smallest value that rounds up to five, that is 2.5. So is our unit value bigger than 2.5? Yes, so this rounds up. So this rounds up because we've bigger than 2.5, which means this becomes one two zero in base five. And that makes sense because 34, rounded to the nearest five is 35 and one, two, zero, this represents 35. Now this one I want you to try on your own. Pause the video here and give this a go. Now that you've tried this, let's see what you come up with. What do we have? We have the number 13. Now this rounded to the nearest 10 in base 10 would be 10. Because if you look at three, three is less than five, which means this rounds down. So rounds down. So if in base 10, this would round down, what about base eights? In base eight, what do we have? In base eight we have one eights and what? Five ones. In base eight, now we're looking at our units value, we're rounding to the nearest eights. Between zero and eight what's the smallest number? It's four. Are we bigger than four? Is our units value bigger than four? Yes, so this rounds up. So this becomes two zero in base eights. Look what happened. In base 10, 13 rounded down. But when we wrote it in base eight, it rounded up, when we rounds to the nearest eight. So the nearest eights to 13 is 16. And that makes sense 'cause two zero represents 16. So when we're deciding to round up and down, you're looking at that unit value and seeing whether it's bigger or smaller than the smallest value that rounds up to the value you're of rounding to. Now here's an independent task for you to try. Pause the video here and give this a go. Now that you've tried this, let's see what you come up with. Two three rounded to the nearest five. Three is bigger than half of five, so this rounds up to three zero in base five. One two in base seven rounded to the nearest seven. Two is less than half seven, so this becomes one zero in base seven. So that rounds down. One three in base six rounded to the nearest six. Three is half of six, so where there's a halfway point, we round up. So this becomes two zero in base six. Two one in base four rounded to the nearest four. one is less than half of four, so this becomes what? Two zero in base four. 'Cause it rounds down. One zero three in base eight rounded to the nearest eights. Three is less than half of eight, so this becomes one zero zero in base eight, 'cause it rounds down. What do we notice when rounding even and odd bases? With even bases, we have a halfway point that we can be. So you can have a number like three at the halfway point. But with odd bases, our halfway points are usually decimals. So we never have values in our units place that are decimals, so we're either always above or below, so it's not easier to tell whether we round up or down. And to round up or down, we have to be, if we're bigger than half the place value we're rounding to, then we round up. If we're less than half the place value we're rounding, there we round down. For this independent task we're looking at decimals written in base 10 and in base five. If you don't want to a clue, pause the video here and give this a go. If you want to a clue, keep watching. So those of you that want to try this on your own, pause the video in three, two, one. For your clue, you've got to think about the place value charts for base five. So we've got to keep dividing down, so you do one divided by five, that is 0.2. That is 0.2s position in base five. Then divide by five again and you end up at 0.04. Then divide down again, you end up at 0.08. So these are the decimal place values in base five. Now, can you use those to figure out decimals that you can write and decimals that you can't write. And why you can write them? So pause the video here and give this a go. Now let's see what we've come up with. Numbers that are possible. Values that are possible were any numbers we can make, 0.2s, 0.04s and 0.08s and so on and so forth. The next one is, would be 0.0016s and then divide that down again and so on and so forth. So any decimal values that are in the base five place value chart, we can make those in base 10. So we could make 0.8 base 10, how? Because 0.8 base 10 is zero ones and what? Four 0.2s. So 0.8 base 10, that's the same as 0.4 base five. Is there another one we can make? We could make 0.24 'cause it's a combination of a 0.2 and a 0.04. So 0.24, that should be the same as 0.11 in base five. And now can you figure out the reason why we can't write some decimals in base five and in base 10? And also should we be using the word decimal in base five? If you think that you've got an answer to this, ask you parent or carer to share your work on Twitter, tagging @OakNational and #LearnwithOak.

In this lesson, we will be looking at adding and subtracting numbers in different bases.

Number systems: Operations in different bases

﻿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.

In this lesson, we will be exploring the binary code and its use in computing.

Number systems: Binary - the language of computers

﻿Hello and welcome to another video. In this lesson we'll be talking about binary, the language of computers. My name is Mr Maseko. Before you start this lesson, please make sure you have a pen or a pencil and something to write on. Okay, now that you have those things, let's get on with today's lesson. So, first complete the following calculations which are all in base 2. And it's about doubling numbers in base 2. So 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. One add one is one zero in base 2. Well 1 add 1 is 2, so that makes sense because 2 is represented as one zero. Think of your place value chart for base 2, what do we have? We have our ones, our twos, our fours, our eights, our 16s, our 32s, and 64s, and so on and so forth. So it makes sense, 1 add 1, well 1 add 1 gives us one two, so that makes sense, that's one zero. One zero add one zero is, well two add two is 4, so that's one four, so that would give us one zero zero. Okay. What about one zero zero add one zero zero? That gives us 8. What's happening, what are you noticing so far? When you're doubling in base 2, what's happening? Keep that thought, okay, let's see, let's see if that thought you're thinking is the same thing that I'm thinking. What's one zero one add one zero one? Well, one zero one, that's 5 add 5. 5 add 5 is what? 10. 10 is the same as one zero one zero. Huh. Okay. Can you see what's happening? One add one is one zero. One zero add one zero is one zero zero. One zero zero add one zero zero is one zero zero zero. One zero one add one zero one is one zero one zero. One one one add one one one, well one one one is seven, because it's one four one two, and one one. Well if we do seven add seven, that gives us 14. And 14 is one 8, one 4, one 2, and zero ones. Huh. What's happening here? 1 0 1 0 add 1 0 1 0. Can you spot a pattern? Without even thinking about what the numbers are worth, what would this answer be. Good, it would be 1 0 1 0 0. Well, now check it. 1 0 1 0, well what's that? That is 10 add 10. 10 add 10 is 20. And 20 is one 16, zero 4, one 4, zero twos and zero ones. So looking at this, what does doubling in base 2 remind you of? Good, it reminds you of multiplying by 10 in base 10. Because any time you multiply by 10 in base 10, you shift your digits one space to the left. And that's the same thing that's happening here when we double in base 2, our digits just shift one space to the left. Huh. Now if you look at doubling in base 10, well doubling in base 10, 10 times two, 10 times two, well that is 20. 242 times two, well you know that's 4, that's 8, that's 4. Doubling in base 10 is not as easy as doubling in base 2, 'cause doubling in base 2 is just shifting digits one space to the left, whereas in base 10 you actually have to be thinking about it. Okay. Now, connection. The binary code. Now the binary code is called the binary code because it only uses two symbols. Now the reason base 2 is called a binary base, because in base 2 there are only two digits that you ever see, it's always zeros and ones. Now, binary is very very important in computing, because of this nature only either being on or off, zero or one, it's very useful in computing, which use electricity, because computers can't understand complex languages, all they understand is am I receiving electricity, or am I not? On or off, zero or one. And that's why the binary code is really useful in computing. Because it allows us to encode information in such a way that a computer can understand. So each little block of code is how much a computer can read at a time. And what humans have done is that they've assigned each little block to a symbol, and when a computer reads that block, it then translates it to whatever the humans have set that to. Now the way computers read this is beyond what we're learning here today, but it is quite interesting because it brings up some quite interesting maths, about the number of possible combinations you can have, depending on the length of the blocks you allow your computers to read. Now, like I was saying before, computers read the binary code in chunks. And each chunk codes for a specific symbol. Like for example, if you want the computer to write the letter "A", we would give it a chunk of binary code that the computer knows, oh that means that I need to draw this symbol on the screen. So, depending on how many pieces there are in the chunk, that effects how many different symbols you can code for. Now, let's say in a chunk there are a certain number of individual spaces. Now, each space in a chunk is called bit. So this chunk I've just drawn, will be a 4-bit chunk. And if my computer read in 4-bit chunks, it will look at every four blocks of zeros and ones, and each four blocks will code for a different thing. Now, let's look at 2-bit chunks. If I had 2-bit chunks, that my computer read in, how many different codes can I have? Remember a code is either zero or one. A code either has zero or one. So how many different codes can I make if my computer only reads in 2-bit chunks? So this is a 2-bit chunk. Well, what can I have? Well I could have zero zero, so that's one code it could have. I could have zero one, I could have one zero, or I could have one one. So on my 2-bit chunk, there are four codes that I can have. So, there's only four pieces of information that I can tell this really rubbish computer, 'cause it can only read every 2-bits. But, luckily for us, our computers read in much bigger chunks than this. And we'll start looking at how many different combinations there are, as we go forward. But the first thing that I want you to try is this. So, in this independent task, read the questions carefully, and you're thinking about the number of different combinations of zeros and ones you can have, depending on the size of your chunk. Pause the video here, and give this a go. Okay, now that you've tried this, let's see what you should have come up with. So, for 3-bit chunks, these are all the different codes you could have had, so for 3-bit chunks, you could have had eight different codes. So if yOur computer read in 3-bit chunks, you can code for eight different things. So what we were seeing before, we've seen that for 2-bit chunks, you had four different codes. For 3-bit chunks, you had eight different codes. What about for 4-bit chunks? How many different codes could you have? Can you see a pattern? Good, for 4-bit chunks, you could have had 16 different codes. So look at all the codes that you came up with, did you get to 16? Now these are some of the others you could have had, so you could have had. These are all the 16 codes you could have come up with for a 4-bit chunk. Now do you see what's happening? When you increase the number of bits you have in this chunk, what's happening? You are increasing the amount of information you can encode. And coincidentally, if your computer only read in 1-bit chunks, well there's only two codes you could have, 'cause either zero or one. Now look at this. 1-bit is two, 2-bits is four, 3 bits is eight, 4-bits is 16. What would 5-bit chunks be? Well 5-bit chunks would be 32 different codes. How else can you write 2, 4, 8, 16, 32? Well these are all powers of 2, 2 to the power of 1, 2 to the power of 2, 2 to the power of 3, 2 to the power of 4, 2 to the power of 5. Look at this. So what if you had an 8-bit chunk? How much information could you encode in an 8-bit chunk? Good, the pattern is in an 8-bit chunk, you could encode 2 to the power of 8, which would be 256 different codes. Now an 8-bit chunk, this turns out to be the smallest number of bits that computers read in. They code for 256 different things. And now that's the most basic binary code that there is. Now if you think about an iPhone for example, so if you think about an iPhone, what happens to an iPhone? An iPhone works on 64-bit processors. So how much information can you store in there. Well you can store 2 to the 64 different pieces of information. Now 2 to the power of 64 is a really big number. So, look at the number of different codes and think back to the base 2 place value chart. Go on look at it. You've got the twos, the fours, the eights, the 16s. The 32s. You see! The maths that we learn is very applicable in real world situations. Now let's look at some of these 8-bit codes that computers read and what they mean. Now, this is what your computer reads and translates to. So it read 8-bit codes and each code codes for a specific character. So here we have the alphabet, capitals and lowercase, and we also have some symbols and some numbers. Using this binary code, can you write your name? Remember to leave a space between each character. And use capitals at the start of your first name and your surname. 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 first letter on my name is "A", so I would have started with the code for "A". So, that would be 0100001. So 0100001. So that is "A". And I'm going to put a dot, going to be "A" dot, so my dot would be, there it is. "A", my dot would be, 00101110. And then, for the first letter of my surname is "M", and I want a capital "M" and that would be, so my capital "M" would be 01001101. And so on and so forth. Now, your computer just reads in this long string of numbers. But what it actually comes out with is these very simple symbols, and this is what we've told the computer those numbers mean. Now if you've written your name and you want to share how you've written your name in binary, or you've written a message in binary code, ask your parent or carer to share your work on Twitter tagging @OakNational and hashtag Learn with Oak. I really do hope you've enjoyed these lessons on different number systems and exploring different bases. I will see you again next time.

Unit Overview: Different number systems

Lessons:

Number systems: Writing numbers in different bases

Number systems: Rounding in different bases

Number systems: Operations in different bases

Number systems: Binary - the language of computers