Lesson video

Hi there, I'm Sara.

I'm your computing teacher for this unit, Representations.

And this is lesson four of six, where you'll been looking at numbers in binary.

You will need a pen and you will need paper also.

Pause the video, and get yourself ready, get a pen and paper.

Also, remember to turn off any notifications and remove distractions so you can begin.

I'll see you when you're ready.

In this lesson, you will explore how a sequence of binary digits can represent numbers, and convert between decimal, and binary numbers.

You've just arrived in Binary Land, and you'll be needing to spend your boins so let's look at what that means.

So to spend your boins you've got this strange coins we're not going to call them coins, we're going to call them boins 'cause you're in Binary Land.

You only have one of each.

You can only spend up to a maximum of 31 boins.

So, is there any amount that you won't be able to pay with these? Have a think for a moment, pause the video, unpause when you're ready to continue.

Yes quite rightly so, the answer is no, you can make up, up to 31 by including and excluding any boin.

So let's take a look.

Say you needed to spend 13 boins, what boins would you need? You would need eight, four, and one.

When you add up eight, plus four, plus one, that makes up the 13 boins that you need.

Let's look at another example, you have got 16 boins, eight boins, and two boins.

Adding those altogether, makes it 26 boins that you need altogether, for this transaction.

In this section, you will see more closely, how a sequence of digital digits are interpreted as numbers.

You've seen these before, these symbols, they're called digits, there are 10 of them, and a sequence of these digits, make a number, such as nine, one, six.

Let's take a closer look.

You've got before you a sequence of decimal digits, four and two.

And here are its place values or weights or multipliers.

So you've got one place value and tens place value.

How do you find out the number? You add the sum of the product, of the numbers, and that gives you the total number.

So you've got two, in the ones place value, so that two lots of ones.

So two times one, is two.

And you've got the tens place value.

You've got four lots of 10, so four times 10, is 40, 40 add two is 42.

When you see this number, you automatic call out 42 without thinking because you're used to it.

So this is how you interpret numbers or a sequence of decimal digits, as numbers rather.

Now let's take a look at another example.

You've got before you a sequence of three, one, four digits.

And what does that mean? It means you've got four in the ones place value, one in the 10th place value, and three in the hundreds place value.

So how'd you find out the number? Four times one is four, one times 10 is 10, 'cause you know, one lot of 10, and they're three parts or three hundreds, so three times hundred gives you three hundreds.

So when you add 300 plus 10 and four, you get 314.

Taking a look at another one.

You've got that number, which automatically will call out as 2,718.

So starting again you can see, you've added another place value.

So can you see a pattern there? Yes, there's a pattern in the multipliers.

So you multiply in powers of 10.

So why 10 digits? The common belief is, we're kind of used to 10 or we chose a 10 digit or the base-10 number system because we've got 10 fingers.

So digitus in Latin, is for fingers.

So, we commonly believe that we use the base-10 because it's more convenient for us, to use or count in base-10.

In this section, you'll be translating binary numbers to the familiar decimal numbers that you're well used to.

You've seen this one before, so have a pause to think, do not pause the video.

What do we call these symbols? And how many are there? That's correct.

We call the symbols binary digits, bits, and there are two of them.

And a sequence of binary digits make a number, like we saw in decimal.

So binary is base-2.

We've got the picture there of Leibniz.

And why have we got him there? He was a, notable figure, who wrote a lot about binary system.

He quite a bit about the binary system in early life, and he revisited the base-2 system and did a lot more writing in his career, over and over again.

According to Wikipedia actually, he's the first computer scientist and information theorist, but he did not invent binary, and he was well known for his work with mechanical calculators, such as the step reckoner.

So that's just more information about Leibniz.

So, going on it's the same reasoning you have in binary as you have in the decimal system.

So here, we know that there are two digits in the binary system, zero and one.

They're called binary digits and that has been shortened to bits.

Also, you saw earlier, in a one bit sequence, there are two possible numbers.

You also saw that in a two-bit sequence, there are four possible combination of numbers, and in a three bit sequence, you've got eight and each time you add a bit sequence the number doubles.

In binary, there are also multipliers or place value.

Can you remember, what this multiplier or place value is in base-10, in decimal? Yes, that's right, in powers of 10, 10.

So in binary then, can you guess or can you say, what the multiplier will be, in base-2? That's correct again, powers of two.

So each one is twice as much as the previous one.

Now let's look at an example.

So the multipliers are in powers of two, so the place value starting from the right hand side always, you start with one, and that doubles to two, and that doubles to four, and that doubles to eight and so on.

So for the digit sequence, one zero zero one, you would have to multiply the multiplier, by the digit, and that gives you the sum.

So one times one is one.

There's nothing in the second and the fourth place value, so we're not going to do that, it remains zero, and then eight times one is eight.

So the sum of the products, added up, would make nine in total.

So the decimal value for that binary number is nine.

So here is a summary of what we've just done.

Have a look for a moment, and just gather what you've just learnt.

So we saw earlier that in binary, we use two digits.

And that's very convenient for switches, because computers use switches to control the flow of electricity.

And that's why we use binary, to form the binary state zero, one, high or low voltage.

In a way you can compare binary digits to lamps.

When on, it's one, and when switched off is zero.

When switched on, you include the multiplier, in the sum.

So like the example we've got, eight and one switched on, both multipliers are included in the sum, you count that place value and that equals nine, for that particular number you can see on screen.

Right, let's look at another example.

You've got binary digit one zero one zero one.

You always start with writing the multipliers and you start with one from the right hand side and you go on writing so it's one, two, four, eight.

So what will the next multiplier be? Have a pause to think.

That's correct, 16.

So the multiplier is twice as big as the previous one.

In this particular example, can you see the decimal number? By all means pause the video and you can use your paper to work out this one.

That's correct.

Again, where you see one, include the multiplier in the sum, so for this particular number, or series of digits, you've got 16 and four and one.

So 16, add four, add one, makes 21 in decimal.

And that's how you do it.

Again, bits are like switches.

So where you've got the switch of the lamp turned on, you include that multiplier in the sum.

So you add the numbers together, and that gives you 21.

Right, task one, time for some practise.

Convert the binary number you see to decimal.

Pause the video and when you've completed, continue.

Right, well done.

Let's go for the solution.

So the first thing you do, is write down the multipliers from the right hand side starting from one and doubling as you go along, so you know what the multipliers correspond to.

That's the first thing you do.

Then look at all the bits set to one.

For all the bits set to one, you will circle them, and write the multipliers in the circle.

So that way they're selected.

The next step, is to add up, all the multipliers you've selected to get your final number.

In this case, you add 16 to eight to two and you get 26.

Right, this is like turning on the lamps 16, eight, and two multipliers.

So this is the lamp analogy where you've got the switches turned on.

And remember the boins? Yes, we've got the boins too.

And for another practise so you've got task two here, so convert the binary number to decimal again.

Pause the video and I'll see you when you've completed the task.

Right, well done for trying.

Again, write the multipliers over the decimal number so you know what you're working with.

So you start from one and you add you double each time as you go along so it's one, two, four, eight, and 16 and so on, and then the next step, you guess, that's correct.

Yes, so you circle, or write the multipliers underneath the digits with one.

So in this case you've got 16 and two, and what you do? Yes, you add them up, to get your final number in decimal.

This time you're going to learn another skill, converting decimal to binary digits.

So you're going to see how this is done.

It's exactly the opposite of what you've just seen, so the same principle applies, watch and see how to do this.

So, you're going to start with the decimal number 13, you're going to firstly before you begin, you write your multipliers or place value.

So you're going to start from the right most section you're going to write one.

In this example we've got one, two, four, eight, and 16.

We stopped at 16 because 16 is a multiplier that's just above the number one to convert, which is 13.

But when you're beginning to do your conversion, take a note, where actually you're going to be starting from the left hand side, the leftmost figure, which is 16.

So looking at 16, we're going to check now, what we're doing.

Now 16's quite big so 16's bigger than 13 so we're not going to work with that multiplier we're going to set that to zero, we're going to move on to the next one.

Right, what have we got there? We've got eight, right, yes eight goes into 13 quite nicely so we're going to set that to one, and then we're going to take eight away from 13 to give us five.

So 13 minus eight is five.

Now moving on to the next one, what do you think is going to happen? So, four goes into five nicely.

Are we going to use that? Set that to one? Will you set that to one? Yes, that will be set to one.

So four goes into five, five minus four is one.

So, you've got one to make up, now you've got the next column there so do we set this to one? Would you set this to one? Correct, the answer's no, because two's too big to go, two's too big for one so we're not going to set that to one so we're going to use the other multiplier the other place value column, because one fits in nicely, and then we can just set that to one and the answer's yes there.

Okay.

So, just take a moment to have a look before we go into the next screen.

Right, perfect.

So when you add up, just to check that you've got the correct answer.

You add up eight plus four plus one, it gives you 13.

Right, let's try another example.

Have a look at this.

We've got 22, again we write, first thing you write your multipliers starting from one so we've got the number 22 to consider, 16 goes nicely into 22 so you know, you're going to set that to one.

So the next thing you do is 22 minus 16, and that gives you six.

So you've got six, right, let's go on to the next, column there, and eight is too big for six so, you're going to set that to zero.

So moving on to four, what would you set that to? Would you set that to one? Yes.

Because four goes nicely into six, and you get a remainder so six takeaway four gives you two so you've got that to make up.

We go on to that column then what do we do? Do we set that column to one? Fantastic, and that's a yes.

So two minus two is zero so there's no more digits to consider and that gets set to zero.

Right.

So adding up all your multipliers where you've got one, gives you 22 in decimal, with 16 adding four, adding two being 22.

Right, so now you've seen how to do the reverse it's time to practise that skill.

Look at your task three, pause the video and when you've completed that task, you can unpause and we'll look at the solution together.

Right, well done for having a good go.

So the first thing you would do, would be to write down your place value or your multiplier so we've started, with one from the right hand side all the way to 16.

So remember we're converting 16 to binary.

So what we did is consider, the number 16 and that fits nicely in the first right most place value or multiplier, so that gets set to one and every other digit you replace that with a zero as a place holder that gives you your solution.

So note that down.

Fantastic.

That's just telling you how to get through that.

Right, okay.

So on to task four.

Pause the video and have another go, this time you've got 19 to convert to binary.

So decimal number 19 to convert to binary, pause the video and I'll see you when you get back.

Again, check your answer.

Well done.

And that should give you 19 in decimal.

If you're unsure any time just pause the video, go back and just check with your notes and then you should be able to work out how that's done in the correct way.

So well done.

Now this is a very interesting one to do, so you will enjoy this one.

So remember you're in Binary Land, you're in Binary World so your friend, the friend you've just made in Binary World has a birthday.

So your friend's birthday in binary is one one zero zero one, and one one zero zero.

So that's what your friend has written on a piece of paper to you about their birthday.

So can you work out their birthday in decimal, look through the worksheet, complete your task and when the task completed, continue the lesson.

Right, well done for trying.

Yes, this is a very interesting one.

So the first set of digits you saw, converts to 25.

So this must be the day, and the next set of digits you saw, converted there nicely to 12.

So what do we have there? It looks like your friend's birthday it's on Christmas Day, 25 of December.

So, her birthday's on Christmas Day.

Now the next one, how old is your friend? And the clue was in the candles.

So remember the lamp analogy where we had the lamps on, ones and zeros, ons and off.

So the first candle there's on, you've got the next two candles off, the next one's on and then the next one's off.

So that would give you in binary one zero zero one zero.

And when you convert that using all the skill you've learned, that's 18.

So your friend's 18.

So well done.

In this lesson you've acquired some really useful skills in binary, that would stand you in great stead in Binary World so you'll be at home with your friends in Binary Land.

So in this lesson, what have you learned? You have explored how numbers can be represented as sequences of decimal numbers, and binary digits.

You have also been able to convert between decimal and binary, and vice versa binary to decimal.

So in the next lesson, you're going to see how to count really, really, really long binary numbers.

So I hope you're looking forward to that as much as I am.

Right, before you go I'm going to leave you with a puzzle, and a joke.

So see if you get them.

Yes you're right, it's certainly not 10 so it's not for the bigger girl, it's one zero, and if you write your multiplier or your place value starting from the right most hand digit you have one and two, so that gives you two.

So that cake belongs to the two year old because we're in Binary Land.

So, well done for spotting that.

So the last one is a joke.

So there are one zero kinds of people in this world, those who understand binary and those who don't.

Have a pause to think.

Right, again I'm glad you did understand that because there are only two kinds of people in binary.

So well done, hope to see you next lesson.

But you've had such, you've acquired such good skills and it will be really nice to see what you've done for this lesson.

So if you'd like to share your work, please do ask your carer or your parent to share your work on Instagram, Facebook, or Twitter @OakNational with #learnwithOak.

Hope to see you next lesson.