Lesson video

Hello, my name's Mrs. Jones and I'm really pleased that you decided to join this lesson today.

In this lesson, we will look at binary and decimal in more detail and how to convert from one to the other.

So let's get started.

Welcome to today's lesson.

Today's lesson is called Numbers in Binary from the unit Data Representation, Text, and Numbers.

And by the end of this lesson, you'll be able to convert between decimal numbers and binary.

There are two key words to today's lesson.

Decimal.

Decimal is a number from the base-10 number system, also known as denary.

Binary.

Binary is a number system that has two digits, one and zero.

There are three sections to today's lesson.

The first is represent a number as a binary sequence.

Second, convert binary to decimal.

And third, convert decimal to binary.

So let's start with represent number as a binary sequence.

Take a look at these strange coins.

Let's call them boins.

You only have one of each.

Is there any amount that you won't be able to pay with these? So you can see that each has a number in it.

Is there an amount that we won't be able to pay with these? We won't be asked to pay for anything over 31 and that's because if you add all those together, you get 31, so the answer is no.

We can include or exclude each of these boins to form any sum up to 31.

In this example, we have 13 where we're using the eight, four, and one.

Add those together, we get the number 13.

Another example, 26.

If we add 16, 8, and 2, we get 26.

The four and the one are not used.

What do we call these symbols? Zero, one, two, three, four, five, six, seven, eight, nine.

How many of them are there? We call these symbols digits and there are 10 of them.

A sequence of decimal digits represents a number.

So let's think about this.

In decimal, here we have the numbers four and two.

We have two digits.

And these are called across the top multipliers or weights.

We have 1 and 10.

A digit multiplied by its multiplier is the product.

So we have 1 X 2 is 2 and 10 X 4 is 40.

40 + 2 is the sum of the products determines the value and we come out with 42.

The value of the two-digit number is therefore 42.

Let's have a look at this one.

Here we've got the multipliers across the top and we have an extra one.

1 X 4 is 4.

10 X 1 is 10.

We now have 100.

So we have our digits, our tens, and hundreds.

100 X 3 is 300 so we have 300 + 10 + 4 and we have the number 340.

Let's do a quick check.

Can you see a pattern in the multipliers? Pause the video to consider your answer and then we'll check it.

Let's check your answer.

Their multipliers are powers of 10.

You are multiplying by 10 each time we move across.

1, 10, 100, 1000.

Well done if you got that correct.

We use 10 digits and the decimal base-10 system for numbers, and this is probably because we have 10 fingers to count this.

You can see there we have 2,780.

You can see across the top multipliers 1, 10, 100, 1000.

What do we call these symbols? 0, 1.

How many of them are there? We call these symbols binary digits.

There are only two of them.

A sequence of binary digits represents a number.

Let's think about this.

In binary, there are also multipliers or weights.

You can see there one, two, four, eight and these multipliers are powers of two.

So now we're multiplying by two across the top.

1 X 2 is 2.

2 X 2 is 4.

4 X 2 is 8.

Let's have a quick check.

In binary, the multipliers are powers of A, 10? B, 2? C, 8? Pause the video to consider your answer and then we'll check it.

Let's check your answer.

The answer was B, 2.

Well done if you got that correct.

Let's look at this binary here again.

We have 1, 0, 0, 1 and the sum of the products is the number.

So here we have 1 X 1 is 1.

2 X 0 is 0.

4 X 0 is 0.

8 X 1 is 8.

We add 8 + 0 + 0 + 1, we have the number 9 and nine in decimal.

So now we've taken a binary number and converted it into a decimal number.

Let's do an activity, find the binary numbers corresponding to the decimal numbers.

Go through the multipliers from left to right.

If a multiplier needs to be included in the sum, set the corresponding binary digit to one and proceed with the number that remains.

Here is an example.

We have the decimal number 13.

So we need to look at eight will be used and what are we left with there? So 13 - 8 leaves us with 5 which is why there's a number five underneath and then we are moving along.

Four does go into that five so we can use that one.

So we're left with one.

So we can see there that we've got the eight, four, and the one being used.

Your activity has these numbers, 16, 19, 23, and 30.

Which multipliers need to be used to be able to convert this to the binary number? Pause the video to consider your answer, Use your worksheet, go back through the slides, and then we'll check it.

Let's check your answer.

So the first one 16 is we can see that we are using 16.

There is a multiplier 16.

So there's a one and the rest are all zeros because we've used everything.

So look at 19.

19 would be 16 + 2 + 1.

So once we've used that 16, we're left with 3.

So you can see there that we are using then the two and the one to make up three.

23, here we've got 16.

and 4, 2, and 1.

And 30, we've got the 16, 8, 4, and 2.

Well done if you've got those correct.

Let's move to the second part of today's lesson.

Convert binary to decimal.

In binary, we use two digits and the binary base-2 system for numbers.

This is convenient for systems using switches because the switch can only be on or off which represents a one or a zero.

And in this example here is the one we looked at, you have an eight plus the one, we get nine in decimal.

It's where those ones are.

In a sense, binary digits act like switches, flip one to on and the corresponding multiplier is included in the sum.

You see here that the one is under the eight and under the one.

Let's look at an example with more digits.

Now we have 1, 0, 1, 0, 1.

Let's have a quick check.

Always start with the multipliers.

What will the next one be? We have one, two, four, eight.

What would the next multiplier be? Pause the video to consider your answer and then we'll check it.

Let's check your answer.

The answer is 16 because here we're multiplying by two, remember? So it's 8 X 2 gives us 16.

Well done if you got that correct.

Each multiplier is twice as big as the one before.

Can you see the decimal number yet? When a binary digit equals one, its multiplier is included in the sum.

So here we have the 16 has a 1 underneath.

16.

The four has a one underneath, so we're using the four.

And the one has a one underneath, so we're using the one.

16 + 4 + 1.

It's 21 in decimal.

This is the decimal number for 1, 0, 1, 0, 1.

Again, bits are like switches.

A value of one means that the multiplier is included in the sum.

Instructions to convert binary to decimal.

Write the multipliers over the bits.

Start with one on the right and double as you go from right to left.

For each bit set to one, select its corresponding multiplier.

Here we've got the 16, the 8, and the 2 because there is a bit set to one.

Add up the selected multipliers.

The sum is the decimal number.

16 + 8 + 2 gives us 26 in decimal.

Remember the boins? Let's do a quick check.

What is the first step when converting binary to decimal? A, for each bit set to one, select its corresponding multiplier.

B, add up the selected multipliers.

C, write multipliers over the bits.

Pause the video to consider your answer and then we'll check it.

Let's check your answer.

The answer was C.

Write multipliers over the bits.

Well done if you've got that correct.

Let's do an activity and you'll need your worksheet for this.

Convert the following binary numbers to decimal.

So you have the binary numbers in the tables already for A, B, C, and D.

And in the first example, you do have the multipliers there.

Complete the activity using your worksheet and then we'll check your answers.

Let's check your answers.

The first one had eight, four, and one.

Add those together was 13.

B had a 1 under 16, 4, and 1 which equals 21.

C had ones under all of them and we add all those together, it was 31.

And D had a 1 under 16, 8, 4, and 1.

We add all those together, we get 29.

Well done if you've got those correct.

Let's move to the last part of today's lesson.

Convert decimal to binary.

Now we'll do the opposite.

Start with a decimal number and work out the corresponding binary number.

There are a few ways to do this.

We are only going to examine one of them.

Which multipliers do I select to assemble a sum of 13? Which binary digits do I set to one? So we're starting with the number, the decimal number 13.

So we start with the left most bit.

Go through the bits from left to right.

Do I need to select a multiplier 16 to assemble a sum of 13? Do I set this binary digit to one? So ask yourself, does 16 go into 13? And the answer is no.

It should be set to zero.

Setting it to one would include 16 in the sum, which would mean the sum would exceed 13.

So we've set that to zero.

Now move to the next one.

Do I need to select multiplier eight to assemble a sum of 13? Do I set this binary digit to one? And ask yourself, does 8 go into 13? Yes, it should be set to one.

Setting it to zero would exclude eight from the sum so the sum would never reach 13 because the rest of the multipliers only up to seven.

So we are now using that eight and we move to the next one.

But first, we need to work out what we've got left because we've used that eight.

We take that away from the 13.

So 13 - 8 leaves us with 5 and that's what we carry on looking at next.

So we are using that five.

Do I need to select a multiplier four to assemble a sum of five? Do I set this binary digit to one? Does four go into five? And the answer is yes, it should be set to one because setting it to zero would exclude four from the sum, so the sum would never reach five because the rest of the multipliers only add up to three.

So now we've got that one underneath.

And again, before moving on, we need to work out what we have left.

five minus that four that we've now used leaves us with one and we carry on with looking at that one.

Do I need to select multiplier two to assemble a sum of one? Do I set this binary digit to one? Ask yourself, does two go into one? And the answer is no.

It should be set to zero.

Setting it to one would include two in the sum, which would mean the sum would exceed one.

And we carry on to the last one.

Do I need to select multiplier one to assemble a sum of one? Do I set this binary digit to one? Does one go into one? Yes, it does.

So now we've turned that one into a one as well.

So we're left with that binary number 0, 1, 1, 0, 1 which we can check by adding eight, four, and one which gives us 13 in decimal.

Let's have a quick check.

Do I need to select multiplier 16 to assemble a sum of 22? Do I set this binary digit to one? Pause the video, consider your answer, and then we'll check it.

Let's check your answer.

Yes, it should be set to one.

Setting it to 0 would exclude 16 from the sum, so the sum would never reach 22 because the rest of the multipliers only add up to 15.

Well done if you got that correct.

You now need to work out what we had left because 22 minus the 16 we've already used leaves us with six.

What number would we carry into the next column? Is it, A, 22? B, 6? C, 16? Pause the video to consider your answer and then we'll check it.

Let's check your answer.

The answer is six.

Well done if you got that correct.

Let's do an activity.

Use the table to help you convert the following, decimal numbers to binary.

We have, A, 26.

B, 15.

C, 30.

D, 11.

Pause the video, use your worksheet, go back through the slides, and convert those decimal numbers to binary, and then we'll check your answers.

Let's check your answer.

For the first one, which was 26, the binary number was 1, 1, 0, 1, 0 because we are adding up 16, 8, and 2 which all had a one underneath.

Well done if you got that correct.

The second one was 15.

The binary number was 0, 1, 1, 1, 1.

We are adding up 8, 4, 2, and 1 to make 15.

Well done if you got that correct.

The third one was 30 and this one is 1, 1, 1, 1, 0.

You are adding there 16, 8, 4, and 2 gives us 30.

Well done if you got that one correct.

And the fourth one was 11 and the binary number was 0, 1, 0, 1, 1.

There if you add 8, 2, and 1, you get the decimal number 11.

Well done if you got that correct.

In summary, natural numbers like one, two, three can be represented in binary which uses only zeros and ones.

Each binary digit, a bit, represents a power of two.

Decimal digits represent powers of 10.

You can convert a decimal number to binary by finding which powers of two add up that number.

You can convert binary back to decimal by adding the values of all the ones in the binary number.

Well done for completing this lesson Numbers in Binary.