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Unit 1: Data representation�

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IGCSE Computer Science (0478)

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Representation

We use symbols to represent many things. �

Can you think of some examples?

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  • Letters of the alphabet
  • Digits
  • Mathematical operations, such as + for ‘add’
  • Musical notes
  • Sign language
  • Road signs, etc.

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Representation

Computers are good at storing numbers — we’ll find out why later

Programs, text, numbers, images, sound, etc. are all represented in a computer using numbers

We just need to agree a coding system, e.g. 65 means A

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Source: Pixabay

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Data and instructions

The invention of computers allowed us to perform calculations more quickly, accurately, and efficiently than people can.

What are the components of a calculation? e.g. 5×12

  • The 5 and 12 are the data
  • The × is a symbol that means ‘multiply’, and it is the operation or instruction

Source: Pixabay

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Data and instructions

How can we communicate our instructions to a computer?

By writing or running programs

How can we communicate data to a computer?

By entering or loading values

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Source: Pixabay

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Representation

How many states can a switch or bulb be in?

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1

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0

OR

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1

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0

OR

Two: on and off

These two states could also be called:

  • True/False
  • 1/0

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Representation

Think about how you could communicate a message to a friend in a different location using a two-state system.

You might like to use light, sound, electricity, or something else.

How could you communicate a short message to your friend?

Source: Pixabay

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Binary representation

If you have more bulbs, you can represent more things.

How many combinations are there for a group of two light bulbs?

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8

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0

1

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Binary representation: solution

If you have more bulbs, you can represent more things.

How many combinations are there for a group of two light bulbs?

Four combinations:

  • OFF-OFF 00
  • OFF-ON 01
  • ON-OFF 10
  • ON-ON 11

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9

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1

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Binary representation

In a computer, transistors are used instead of these bulbs.

Transistors are tiny electronic switches.

These 1s and 0s are actually the basis of a counting system called binary.

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10

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1

0

1

0

1

1

1

0

These switches could represent 10101110 in binary.

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Computers and electricity

Computers use electric circuits and switches to represent all data and instructions.

What are the tiny switches inside computers called?

Transistors. These are tiny at around 7 nanometers.

How many do you think would fit across a human hair?

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Source: Wikimedia commons, Human-hair-1

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14,285 transistors can fit across a single human hair

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Computers and electricity

These transistors allow electricity to be on or off in a circuit.

We combine lots of circuits to represent data.

Everything in a computer is represented with combinations of 1s and 0s.

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These switches could represent 10101110 in binary.

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Number bases

What is the value of this number?

9019

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14

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9

0

1

9

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Number bases

What is the value of this number?

9019

Nine thousand and nineteen

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15

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9

0

1

9

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Number bases

Why does the first 9 hold more value than the last 9?

16

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9

0

1

9

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Number bases

Why does the first 9 hold more value than the last 9?

Each number has a place value

17

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9

0

1

9

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Number bases

What are the place values in this number?

18

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9

0

1

9

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Number bases

What are the place values in this number?

19

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1

9

0

1

9

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Number bases

What are the place values in this number?

20

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10

1

9

0

1

9

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Number bases

What are the place values in this number?

21

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100

10

1

9

0

1

9

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Number bases

What are the place values in this number?

22

1000

100

10

1

9

0

1

9

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Number bases

You work out the next place value by multiplying by 10 as you move from right to left.

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100

10

1

9

0

1

9

x 10

x 10

x 10

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Digits

How many digits are there in our decimal number system?

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Source: Pixabay

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Digits

How many digits are there in our decimal number system?

We have 10 digits:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9

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Our number system is a base 10 number system because it has 10 digits.

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Source: Pixabay

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Binary

Binary is a base 2 number systems.

This is because it uses only 2 digits: 0 and 1.

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Make a prediction

What do you think the place value might be of each of the numbers in this table?

26

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1

1

0

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1

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Binary

Binary is a base 2 number systems.

This is because it uses only 2 digits: 0 and 1.

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Make a prediction

What do you think the place value might be of each of the numbers in this table?

27

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2

1

1

0

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1

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Binary

Binary is a base 2 number systems.

This is because it uses only 2 digits: 0 and 1.

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Make a prediction

What do you think the place value might be of each of the numbers in this table?

28

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4

2

1

1

0

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1

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Binary

Binary is a base 2 number systems.

This is because it uses only 2 digits: 0 and 1.

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Make a prediction

What do you think the place value might be of each of the numbers in this table?

29

8

4

2

1

1

0

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1

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Binary

You work out the next place value by multiplying by 2 as you move from right to left.

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8

4

2

1

1

0

0

1

x 2

x 2

x 2

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Binary

You work out the next place value by multiplying by 2 as you move from right to left.

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Each binary digit is called a bit.

31

8

4

2

1

1

0

0

1

x 2

x 2

x 2

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Number bases

To work out the value of a number, you need to know its place value.

You then multiple the digit by its place value.

32

1000

100

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9

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9

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Number bases

To work out the value of a number, you need to know its place value.

You then multiple the digit by its place value.

33

1000

100

10

1

9

0

1

9

9 x 1000

0 x 100

1 x 10

9 x 1

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Number bases

To work out the value of a number, you need to know its place value.

You then multiple the digit by its place value.

34

1000

100

10

1

9

0

1

9

9 x 1000

0 x 100

1 x 10

9 x 1

9000

0

10

9

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Number bases

Finally, you add all of those numbers together.

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1

9

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1

9

9 x 1000

0 x 100

1 x 10

9 x 1

9000

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10

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Number bases

Finally, you add all of those numbers together.

36

1000

100

10

1

9

0

1

9

9 x 1000

0 x 100

1 x 10

9 x 1

9000

0

10

9

Nine thousand and nineteen

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Number bases

You do exactly the same thing with binary numbers.

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8

4

2

1

1

0

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1

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Number bases

You do exactly the same thing with binary numbers.

You multiply the digit by its place value.

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8

4

2

1

1

0

0

1

1 x 8

0 x 4

0 x 2

1 x 1

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Number bases

You do exactly the same thing with binary numbers.

You multiply the digit by its place value.

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8

4

2

1

1

0

0

1

1 x 8

0 x 4

0 x 2

1 x 1

8

0

0

1

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Number bases

Then you add all of the values together.

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8

4

2

1

1

0

0

1

1 x 8

0 x 4

0 x 2

1 x 1

8

0

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1

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Number bases

Then you add all of the values together.

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8

4

2

1

1

0

0

1

1 x 8

0 x 4

0 x 2

1 x 1

8

0

0

1

Nine

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Number bases

Then you add all of the values together.

​

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In binary it is a little easier because you are always multiplying by either 1 or 0.

You don’t really need to multiply.

42

8

4

2

1

1

0

0

1

1 x 8

0 x 4

0 x 2

1 x 1

8

0

0

1

Nine

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Number bases

How can you tell the number base of a number that you are presented with?

e.g.

101

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Number bases

You can use a subscript at the end of the number to state the number base.

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1012

10110

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�A base of 2 means that it is a binary number.

A base of 10 means that it is a decimal number.

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Binary?

45

Base 10

Base 2

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Binary?

46

Base 10

Base 2

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How many digits does decimal have?

47

10

2

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48

10

2

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Converting from binary to decimal

In order to convert from binary to decimal you need to know the place value of each digit in the number.

When you are just starting to learn how to do this it is a good idea to always use a table like this one.

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Converting from binary to decimal

In order to convert from binary to decimal you need to know the place value of each digit in the number.

When you are just starting to learn how to do this it is a good idea to always use a table like this one.

Draw this table now.

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Converting from binary to decimal

If I give you the binary number 111 then you place the digits from right to left in the table.

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Converting from binary to decimal

If I give you the binary number 111 then you place the digits from right to left in the table.

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Converting from binary to decimal

If I give you the binary number 111 then you place the digits from right to left in the table.

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Converting from binary to decimal

If I give you the binary number 111 then you place the digits from right to left in the table.

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Converting from binary to decimal

Then you need to look at their place value and add those values together.

4 + 2 + 1 = 7

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Converting from binary to decimal

Then you need to look at their place value and add those values together.

4 + 2 + 1 = 7

111 in binary is 7 in decimal

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Converting from binary to decimal

Let’s try another.

Convert 1010 from binary to decimal.

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1

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Converting from binary to decimal

This time you have some 0s. These can be left out of your calculation.

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Converting from binary to decimal

This time you have some 0s. These can be left out of your calculation.

8 + 2 = 10

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Converting from binary to decimal

This time you have some 0s. These can be left out of your calculation.

8 + 2 = 10

1010 in binary is 10 in decmial

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Converting from binary to decimal

Let’s try a slightly larger binary number.

Convert 101111 from binary to decimal.

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Converting from binary to decimal

Let’s try a slightly larger binary number.

Convert 101111 from binary to decimal.

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Converting from binary to decimal

Let’s try a slightly larger binary number.

Convert 101111 from binary to decimal.

101111 in binary is 47 in decimal

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Converting from binary to decimal

Use your table to help you answer the following quick fire questions.

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Convert these numbers from binary to decimal

65

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101

66

5

6

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101

67

5

6

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1111

68

14

15

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1111

69

14

15

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10111

70

23

21

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10111

71

23

21

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1110

72

11

12

13

14

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1110

73

11

12

13

14

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10000011

74

128

131

129

132

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10000011

75

128

131

129

132

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Converting from decimal to binary

It is a little tricker to convert from decimal to binary.

You need to work from left to right.

And you need to subtract, not add.

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Converting from decimal to binary

Let’s start with a small decimal number.

5

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Converting from decimal to binary

Let’s start with a small decimal number.

5

You start by looking for the highest value that fits into the number 5.

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Converting from decimal to binary

Let’s start with a small decimal number.

5

You start by looking for the highest value that fits into the number 5.

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Converting from decimal to binary

Let’s start with a small decimal number.

5

You start by looking for the highest value that fits into the number 5.

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Converting from decimal to binary

Let’s start with a small decimal number.

5

You start by looking for the highest value that fits into the number 5.

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Converting from decimal to binary

Let’s start with a small decimal number.

5

You start by looking for the highest value that fits into the number 5.

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Converting from decimal to binary

Let’s start with a small decimal number.

5

You start by looking for the highest value that fits into the number 5.

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Converting from decimal to binary

Let’s start with a small decimal number.

5

You start by looking for the highest value that fits into the number 5.

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Converting from decimal to binary

Once you have found a value, you enter a 1 in that column.

You then take that value away from your current number (5).

5 - 4 = 1

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Converting from decimal to binary

Then you start looking for the highest value that fits into your remaining number.

5 - 4 = 1

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86

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Converting from decimal to binary

Then you start looking for the highest value that fits into your remaining number.

5 - 4 = 1

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Converting from decimal to binary

Then you start looking for the highest value that fits into your remaining number.

5 - 4 = 1

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Converting from decimal to binary

Found it!

5 - 4 = 1

1 - 1 = 0

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Converting from decimal to binary

Finally, you are left with your decimal to binary conversion.

5 in decimal is 101 in binary

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Converting from decimal to binary

You can double-check your maths by doing a quick conversion back to binary to just make sure.

101 in binary is 5 in decimal!

This becomes more essential with higher numbers, especially if you are in an exam!

5 in decimal is 101 in binary

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Converting from decimal to binary

Let’s try a higher number.

Convert the decimal number 60 into binary.

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Converting from decimal to binary

Let’s try a higher number.

Convert the decimal number 60 into binary.

What is the highest number in our table that will fit into 60?

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93

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Converting from decimal to binary

Let’s try a higher number.

Convert the decimal number 60 into binary.

What is the highest number in our table that will fit into 60?

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Converting from decimal to binary

Let’s try a higher number.

Convert the decimal number 60 into binary.

Take 32 away from 60, what are you left with?

60 - 32 = ?

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Converting from decimal to binary

Let’s try a higher number.

Convert the decimal number 60 into binary.

Take 32 away from 60, what are you left with?

60 - 32 = 28

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Converting from decimal to binary

Let’s try a higher number.

Convert the decimal number 60 into binary.

What is the highest value that fits into 28?

60 - 32 = 28

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Converting from decimal to binary

Let’s try a higher number.

Convert the decimal number 60 into binary.

Take 16 away from 28, what are you left with?

60 - 32 = 28

28 - 16 = ?

​

98

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1

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99 of 205

Converting from decimal to binary

Let’s try a higher number.

Convert the decimal number 60 into binary.

Take 16 away from 28, what are you left with?

60 - 32 = 28

28 - 16 = 12

​

99

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100 of 205

Converting from decimal to binary

Let’s try a higher number.

Convert the decimal number 60 into binary.

What is the highest value that fits into 12?

60 - 32 = 28

28 - 16 = 12

​

100

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101 of 205

Converting from decimal to binary

Let’s try a higher number.

Convert the decimal number 60 into binary.

What is the highest value that fits into 12?

60 - 32 = 28

28 - 16 = 12

​

101

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102 of 205

Converting from decimal to binary

Let’s try a higher number.

Convert the decimal number 60 into binary.

Take 8 away from 12, what are you left with?

60 - 32 = 28

28 - 16 = 12

12 - 8 = ?

​

102

128

64

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1

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www.thebiscomputing.com

103 of 205

Converting from decimal to binary

Let’s try a higher number.

Convert the decimal number 60 into binary.

Take 8 away from 12, what are you left with?

60 - 32 = 28

28 - 16 = 12

12 - 8 = 4

​

103

128

64

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8

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1

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www.thebiscomputing.com

104 of 205

Converting from decimal to binary

Let’s try a higher number.

Convert the decimal number 60 into binary.

What is the highest value that fits into 4?

60 - 32 = 28

28 - 16 = 12

12 - 8 = 4

​

104

128

64

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16

8

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2

1

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1

1

1

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105 of 205

Converting from decimal to binary

Let’s try a higher number.

Convert the decimal number 60 into binary.

What is the highest value that fits into 4?

60 - 32 = 28

28 - 16 = 12

12 - 8 = 4

​

105

128

64

32

16

8

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2

1

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1

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1

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​

www.thebiscomputing.com

106 of 205

Converting from decimal to binary

Let’s try a higher number.

Convert the decimal number 60 into binary.

Take 4 away from 4, what are you left with?

60 - 32 = 28

28 - 16 = 12

12 - 8 = 4

4 - 4 = ?

​

106

128

64

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16

8

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1

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1

1

1

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www.thebiscomputing.com

107 of 205

Converting from decimal to binary

Let’s try a higher number.

Convert the decimal number 60 into binary.

Take 4 away from 4, what are you left with?

60 - 32 = 28

28 - 16 = 12

12 - 8 = 4

4 - 4 = 0

​

107

128

64

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1

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www.thebiscomputing.com

108 of 205

Converting from decimal to binary

Let’s try a higher number.

Convert the decimal number 60 into binary.

There is nothing left so we enter 0s in the remaining columns.

60 - 32 = 28

28 - 16 = 12

12 - 8 = 4

4 - 4 = 0

​

108

128

64

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1

1

1

1

0

0

www.thebiscomputing.com

109 of 205

Converting from decimal to binary

Let’s try a higher number.

Convert the decimal number 60 into binary.

We have our answer!

60 in decimal is 111100 in binary

​

109

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110 of 205

Converting from decimal to binary

Let’s try a higher number.

Convert the decimal number 60 into binary.

Now we double-check that we were right.

32 + 16 + 8 + 4 = ?

​

110

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111 of 205

Converting from decimal to binary

Let’s try a higher number.

Convert the decimal number 60 into binary.

Now we double-check that we were right.

32 + 16 + 8 + 4 = 60

​

111

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112 of 205

Converting from decimal to binary

Try these conversions from decimal to binary.

  1. 128
  2. 120
  3. 80
  4. 200
  5. 190

​

​

112

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113 of 205

Converting from decimal to binary: solutions

Try these conversions from decimal to binary.

  1. 128
  2. 120
  3. 80
  4. 200
  5. 190

113

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128

64

32

16

8

4

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128

1

0

0

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120

​

1

1

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0

0

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80

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1

0

1

0

0

0

0

200

1

1

0

0

1

0

0

0

190

1

0

1

1

1

1

1

0

www.thebiscomputing.com

114 of 205

Converting from decimal to binary: solutions

Try these conversions from decimal to binary.

  1. 128
  2. 120
  3. 80
  4. 200
  5. 190

114

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128

64

32

16

8

4

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1

128

1

0

0

0

0

0

0

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120

​

1

1

1

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0

0

0

80

​

1

0

1

0

0

0

0

200

1

1

0

0

1

0

0

0

190

1

0

1

1

1

1

1

0

www.thebiscomputing.com

115 of 205

Converting from decimal to binary: solutions

Try these conversions from decimal to binary.

  1. 128
  2. 120
  3. 80
  4. 200
  5. 190

115

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128

64

32

16

8

4

2

1

128

1

0

0

0

0

0

0

0

120

​

1

1

1

1

0

0

0

80

​

1

0

1

0

0

0

0

200

1

1

0

0

1

0

0

0

190

1

0

1

1

1

1

1

0

www.thebiscomputing.com

116 of 205

Converting from decimal to binary: solutions

Try these conversions from decimal to binary.

  1. 128
  2. 120
  3. 80
  4. 200
  5. 190

116

​

128

64

32

16

8

4

2

1

128

1

0

0

0

0

0

0

0

120

​

1

1

1

1

0

0

0

80

​

1

0

1

0

0

0

0

200

1

1

0

0

1

0

0

0

190

1

0

1

1

1

1

1

0

www.thebiscomputing.com

117 of 205

Converting from decimal to binary: solutions

Try these conversions from decimal to binary.

  1. 128
  2. 120
  3. 80
  4. 200
  5. 190

117

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128

64

32

16

8

4

2

1

128

1

0

0

0

0

0

0

0

120

​

1

1

1

1

0

0

0

80

​

1

0

1

0

0

0

0

200

1

1

0

0

1

0

0

0

190

1

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1

1

1

1

1

0

www.thebiscomputing.com

118 of 205

Converting from decimal to binary

Try these conversions from decimal to binary.

  1. 65
  2. 129
  3. 160
  4. 210
  5. 176

​

​

118

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119 of 205

Converting from decimal to binary: solutions

Try these conversions from decimal to binary.

  1. 65
  2. 129
  3. 160
  4. 210
  5. 176

119

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128

64

32

16

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65

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1

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129

1

0

0

0

0

0

0

1

160

1

0

1

0

0

0

0

0

210

1

1

0

1

0

0

1

0

176

1

0

1

1

0

0

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0

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120 of 205

The four golden rules of binary addition

Rule 1

0 + 0 = 0

120

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In decimal, 0 + 0 = 0

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121 of 205

The four golden rules of binary addition

Rule 2

0 + 1 = 1

121

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In decimal, 0 + 1 = 1

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122 of 205

The four golden rules of binary addition

Rule 3

1 + 1 = 10

122

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In decimal, 1 + 1 = 2

www.thebiscomputing.com

123 of 205

The four golden rules of binary addition

Rule 4

1 + 1 + 1 = 11

123

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​

In decimal, 1 + 1 + 1 = 3

www.thebiscomputing.com

124 of 205

Binary addition

Rules

  • 0 + 0 = 0
  • 0 + 1 = 1
  • 1 + 1 = 10
  • 1 + 1 + 1 = 11

124

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1

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=

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​

​

www.thebiscomputing.com

125 of 205

Binary addition

Rules

  • 0 + 0 = 0
  • 0 + 1 = 1
  • 1 + 1 = 10
  • 1 + 1 + 1 = 11

125

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+

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1

0

=

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​

​

​

0

www.thebiscomputing.com

126 of 205

Binary addition

Rules

  • 0 + 0 = 0
  • 0 + 1 = 1
  • 1 + 1 = 10
  • 1 + 1 + 1 = 11

126

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+

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1

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=

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​

1

0

www.thebiscomputing.com

127 of 205

Binary addition

Rules

  • 0 + 0 = 0
  • 0 + 1 = 1
  • 1 + 1 = 10
  • 1 + 1 + 1 = 11

127

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1

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=

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1

1

0

www.thebiscomputing.com

128 of 205

Binary addition

Rules

  • 0 + 0 = 0
  • 0 + 1 = 1
  • 1 + 1 = 10
  • 1 + 1 + 1 = 11

100 + 10 = 110 in binary

4 + 2 = 6 in decimal

128

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1

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0

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129 of 205

Binary addition

Rules

  • 0 + 0 = 0
  • 0 + 1 = 1
  • 1 + 1 = 10
  • 1 + 1 + 1 = 11

129

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+

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=

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www.thebiscomputing.com

130 of 205

Binary addition

Rules

  • 0 + 0 = 0
  • 0 + 1 = 1
  • 1 + 1 = 10
  • 1 + 1 + 1 = 11

130

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=

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1

www.thebiscomputing.com

131 of 205

Binary addition

Rules

  • 0 + 0 = 0
  • 0 + 1 = 1
  • 1 + 1 = 10
  • 1 + 1 + 1 = 11

131

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=

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0

1

www.thebiscomputing.com

132 of 205

Binary addition

Rules

  • 0 + 0 = 0
  • 0 + 1 = 1
  • 1 + 1 = 10
  • 1 + 1 + 1 = 11

132

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+

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=

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1

0

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1

www.thebiscomputing.com

133 of 205

Binary addition

Rules

  • 0 + 0 = 0
  • 0 + 1 = 1
  • 1 + 1 = 10
  • 1 + 1 + 1 = 11

100 + 101 = 1001 in binary

4 + 5 = 9 in decimal

133

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=

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1

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1

www.thebiscomputing.com

134 of 205

Binary addition

Rules

  • 0 + 0 = 0
  • 0 + 1 = 1
  • 1 + 1 = 10
  • 1 + 1 + 1 = 11

134

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0

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+

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=

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www.thebiscomputing.com

135 of 205

Binary addition

Rules

  • 0 + 0 = 0
  • 0 + 1 = 1
  • 1 + 1 = 10
  • 1 + 1 + 1 = 11

135

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+

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=

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0

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1

​

www.thebiscomputing.com

136 of 205

Binary addition

Rules

  • 0 + 0 = 0
  • 0 + 1 = 1
  • 1 + 1 = 10
  • 1 + 1 + 1 = 11

136

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+

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1

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=

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0

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1

​

www.thebiscomputing.com

137 of 205

Binary addition

Rules

  • 0 + 0 = 0
  • 0 + 1 = 1
  • 1 + 1 = 10
  • 1 + 1 + 1 = 11

137

​

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1

0

1

+

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1

1

=

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0

0

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1

1

​

www.thebiscomputing.com

138 of 205

Binary addition

Rules

  • 0 + 0 = 0
  • 0 + 1 = 1
  • 1 + 1 = 10
  • 1 + 1 + 1 = 11

138

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1

0

1

+

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1

1

=

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​

0

0

​

​

​

1

1

​

www.thebiscomputing.com

139 of 205

Binary addition

Rules

  • 0 + 0 = 0
  • 0 + 1 = 1
  • 1 + 1 = 10
  • 1 + 1 + 1 = 11

139

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1

0

1

+

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1

1

=

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1

0

0

0

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​

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1

1

​

www.thebiscomputing.com

140 of 205

Binary addition

Rules

  • 0 + 0 = 0
  • 0 + 1 = 1
  • 1 + 1 = 10
  • 1 + 1 + 1 = 11

101 + 11 = 1000 in binary

5 + 3 = 8 in decimal

140

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1

1

​

www.thebiscomputing.com

141 of 205

Binary addition

Rules

  • 0 + 0 = 0
  • 0 + 1 = 1
  • 1 + 1 = 10
  • 1 + 1 + 1 = 11

141

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1

1

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+

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1

1

=

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www.thebiscomputing.com

142 of 205

Binary addition

Rules

  • 0 + 0 = 0
  • 0 + 1 = 1
  • 1 + 1 = 10
  • 1 + 1 + 1 = 11

142

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1

1

1

+

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1

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=

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0

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1

​

www.thebiscomputing.com

143 of 205

Binary addition

Rules

  • 0 + 0 = 0
  • 0 + 1 = 1
  • 1 + 1 = 10
  • 1 + 1 + 1 = 11

143

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1

1

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+

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1

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=

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0

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1

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www.thebiscomputing.com

144 of 205

Binary addition

Rules

  • 0 + 0 = 0
  • 0 + 1 = 1
  • 1 + 1 = 10
  • 1 + 1 + 1 = 11

144

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1

1

1

+

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1

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=

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1

0

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1

1

​

www.thebiscomputing.com

145 of 205

Binary addition

Rules

  • 0 + 0 = 0
  • 0 + 1 = 1
  • 1 + 1 = 10
  • 1 + 1 + 1 = 11

145

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1

1

1

+

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1

1

=

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1

0

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1

1

​

www.thebiscomputing.com

146 of 205

Binary addition

Rules

  • 0 + 0 = 0
  • 0 + 1 = 1
  • 1 + 1 = 10
  • 1 + 1 + 1 = 11

146

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1

1

1

+

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1

1

=

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1

0

1

0

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1

1

​

www.thebiscomputing.com

147 of 205

Binary addition

Rules

  • 0 + 0 = 0
  • 0 + 1 = 1
  • 1 + 1 = 10
  • 1 + 1 + 1 = 11

111 + 11 = 1010 in binary

7 + 3 = 10 in decimal

147

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=

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1

0

1

0

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1

1

​

www.thebiscomputing.com

148 of 205

Binary addition: try it yourself!

Rules

  • 0 + 0 = 0
  • 0 + 1 = 1
  • 1 + 1 = 10
  • 1 + 1 + 1 = 11
  1. 100 + 1 = ?
  2. 110 + 11 = ?
  3. 101 + 100 = ?
  4. 11 + 111 = ?
  5. 1010 + 1111 = ?

148

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www.thebiscomputing.com

149 of 205

Binary addition: solutions

Rules

  • 0 + 0 = 0
  • 0 + 1 = 1
  • 1 + 1 = 10
  • 1 + 1 + 1 = 11
  1. 100 + 1 = 101
  2. 110 + 11 = 1001
  3. 101 + 100 = 1001
  4. 11 + 111 = 1010
  5. 1010 + 1111 = 11001

149

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www.thebiscomputing.com

150 of 205

Binary shifting

​

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150

www.thebiscomputing.com

151 of 205

Binary shifting

Binary shifting is shifting the bits to the left or to the right.

If we shift to the left then we multiply.

If we shift to the right then we divide.

151

Source: Pixabay

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152 of 205

Binary shifting

Take a look at this example.

I want to multiply 1002 by 102

152

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153 of 205

Binary shifting

Take a look at this example.

I want to multiply 1002 by 102

I shift the bits to the left by one place

153

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154 of 205

Binary shifting

Take a look at this example.

I want to multiply 1002 by 102

I shift the bits to the left by one place

154

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155 of 205

Binary shifting

Take a look at this example.

I want to multiply 1002 by 102

I shift the bits to the left by one place

410 x 210 = 810

Or

1002 x 102 = 10002

155

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156 of 205

Binary shifting

You can use binary shifts to multiply by 2, 4, 8, 16 etc.

If you want to multiply by 4 then you shift left by 2 spaces.

156

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x2

x2

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157 of 205

Binary shifting

You can use binary shifts to multiply by 2, 4, 8, 16 etc.

If you want to multiply by 4 then you shift left by 2 spaces.

157

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x2

x2

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158 of 205

Binary shifting

410 x 410 = 1610

1002 x 1002 = 100002

158

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x2

x2

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Binary shifting

Let’s try one!

1112 x 10002 = ?

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Binary shifting

Let’s try one!

1112 x 10002 = ?

This is multiplying our number by 8.

How many shifts do you need to do to get from 1 to 8?

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Binary shifting

Let’s try one!

1112 x 10002 = ?

This is multiplying our number by 8.

How many shifts do you need to do to get from 1 to 8?

161

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x2

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Binary shifting

Let’s try one!

1112 x 10002 = ?

This is multiplying our number by 8.

This means that we need to shift our bits three spaces to the left.

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Binary shifting

Let’s try one!

1112 x 10002 = ?

This is multiplying our number by 8.

This means that we need to shift our bits three spaces to the left.

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Binary shifting

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1112 x 10002 = 1110002

Or

710 x 810 = 5610

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Which direction do you shift to multiply?

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RIGHT

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Which direction do you shift to multiply?

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Binary shifting

In order to divide, we do the same thing but step to the right!

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Binary shifting

Let’s try this one:

1002 / 102 = ?

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Binary shifting

Let’s try this one:

1002 / 102 = ?

We are dividing by 2 so we shift to the right by one place.

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Binary shifting

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1002 / 102 = 102

Or

410 / 210 = 210

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Binary shifting

Let’s try another one:

1012 / 1002 = ?

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Binary shifting

Let’s try another one:

1012 / 1002 = ?

Here we are dividing by 4 so we need to shift the values to the right by 2 places.

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Binary shifting

Let’s try another one:

1012 / 1002 = ?

Here we are dividing by 4 so we need to shift the values to the right by 2 places.

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Binary shifting

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1012 / 1002 = 12

In binary shifting, we can only work with whole numbers.

Any remainders are discarded.

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Binary shifting

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1012 / 1002 = 12

If we do the same division with our decimal numbers then we are left with a decimal value.

510 / 410 = 1.2510

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Which direction do you shift to divide?

176

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RIGHT

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Which direction do you shift to divide?

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Binary shifting: try it yourself!

MULTIPLY

  1. 11012 x 1002 = ?
  2. 1112 x 102 = ?
  3. 1111112 x 102 = ?

DIVIDE

  1. 11002 / 102 = ?
  2. 11100002 / 1002 = ?
  3. 1100002 / 10002 = ?

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Draw a table to help you!

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Binary shifting: solutions

MULTIPLY

  1. 11012 x 1002 = 1101002
  2. 1112 x 102 = 11102
  3. 1111112 x 102 = 11111102

DIVIDE

  1. 11002 / 102 = 01102
  2. 11100002 / 1002 = 00111002
  3. 1100002 / 10002 = 0001102

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Binary Subtraction

Can you subtract binary numbers? The answer is yes.

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Subtraction of binary numbers is an arithmetic operation similar to the subtraction of decimal numbers or base 10 numbers.

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Binary Subtraction

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Binary Subtraction

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2’s Complement - Positive Binary Numbers

Up until now, we have assumed all binary numbers are positive integers.

To allow the possibility of representing negative integers we make use of two’s

complement. In this section we will again assume 8-bit registers are being used.

Only one minor change to the binary headings needs to be introduced here:

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Example 1:

The following two examples show how we can write the following positive binary

numbers in the two’s complement format 19 and 4:

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2’s Complement

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-128

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Example 2:

Convert 38 to 8-bit binary numbers using the two’s complement format.

Since this number is positive, we must have a zero in the –128 column. It is then a

simple case of putting 1-values into their correct positions to make up the value

of 38:

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2’s Complement

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Example 3:

Convert 125 to 8-bit binary numbers using the two’s complement format.

Since this number is positive, we must have a zero in the –128 column. It is then a

simple case of putting 1-values into their correct positions to make up the value

of 125:

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2’s Complement

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Example 4:

Convert 125 to 8-bit binary numbers using the two’s complement format.

Since this number is positive, we must have a zero in the –128 column. It is then a

simple case of putting 1-values into their correct positions to make up the value

of 125:

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2’s Complement - Negative Binary Numbers

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Example 1:

Convert -109 to 8-bit binary numbers using the two’s complement format.

By following our normal rules, each time a 1 appears in a column, the column value is added to the total. So, we can see that in denary this is: −128 + 16 + 2 + 1 = −109.

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1 for negative numbers

-128 + 19 = -109

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2’s Complement - Negative Binary Numbers

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Example 2:

Convert -28 to 8-bit binary numbers using the two’s complement format.

By following our normal rules, each time a 1 appears in a column, the column value is added to the total. So, we can see that in denary this is: −128 + 64 + 32 + 4 = −28

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1 for negative numbers

-128 + 100 = -28

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2’s Complement - Negative Binary Numbers

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Example 3:

Convert -11 to 8-bit binary numbers using the two’s complement format.

By following our normal rules, each time a 1 appears in a column, the column value is added to the total. So, we can see that in denary this is: −128 + 64 + 32 + 16 + 4 + 1 = −11.

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1 for negative numbers

-128 + 117 = -11

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Converting negative denary numbers into binary numbers in two’s complement format

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Consider the number +67 in 8-bit (two’s complement) binary format:

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Method 1: Now let’s consider the number −67. One method of finding the binary equivalent to −67 is to simply put 1s in their correct places:

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−128 + 32 + 16 + 8

+ 4 + 1 = −67

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Method 2: Now let’s consider the number −67. One method of finding the binary equivalent to −67 is to simply put 1s in their correct places:

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first write the number as a positive binary value – in this case 67:

invert each binary value,which means swap the 1s and 0s around:

then add 1 to that number:

this gives us the binary for −67:

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1.2 Text, Sound and Images

1.2.1 Character sets – ASCII code

The ASCII code system (American Standard Code for Information Interchange) was set up in 1963 for use in communication systems and computer systems.

A newer version of the code was published in 1986.

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The standard ASCII code character set consists of 7-bit codes (0 to 127 in denary or 00 to 7F in hexadecimal) that represent the letters, numbers and characters found on a standard keyboard. The main disadvantage is that it

does not represent characters in non-Western languages.

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1.2.1 Character sets – Unicode

Unicode can represent all languages of the world, thus supporting many operating systems, search engines and internet browsers used globally.

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ASCII uses one byte to represent a character, whereas Unicode will support up to four bytes per character.

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1.2.2 Representation of sound

Sound waves are vibrations in the air. The human ear senses these vibrations and interprets them as sound.

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Each sound wave has a frequency, wavelength and amplitude.

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The amplitude specifies the loudness of the sound.

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Sound waves vary continuously. This means that sound is analogue. Computers cannot work with analogue data, so sound waves need to be sampled in order to be stored in a computer.

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Sampling means measuring the amplitude of the sound wave. This is done using an analogue to digital converter (ADC).

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Sampling rate is the number of sound samples taken per second. This is measured in hertz (Hz), where 1Hz means ‘one sample per second’.

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1.2.2 Representation of images

Bitmap images are made up of pixels; an image is made up of a two-dimensional matrix of pixels. Pixels can take different shapes such as:

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Each pixel can be represented as a binary number, and so a bitmap image is stored in a computer as a series of binary numbers.

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The number of bits used to represent each colour is called the colour depth.

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  • a black and white image only requires 1 bit per pixel – this means that each pixel can be one of two colours, corresponding to either 1 or 0.

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  • if each pixel is represented by 2 bits, then each pixel can be one of four colours

(22 = 4), corresponding to 00, 01, 10, or 11

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  • if each pixel is represented by 3 bits then each pixel can be one of eight colours

(23 = 8), corresponding to 000, 001, 010, 011, 100, 101, 110, 111.

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Modern computers have a 24 bit colour depth, which means over

16 million different colours can be represented

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1.3 Data storage and file compression

1.3.1 Measurement of data storage & 1.3.2 calculation for File size

A bit is the basic unit of all computing memory storage terms and is either 1 or 0.

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The file size of an image is calculated as:

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The size of a mono sound file is calculated as:

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image resolution (in pixels) × colour depth (in bits)

sample rate (in Hz) × sample resolution (in bits) × length of sample (in seconds)

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Question 1: To calculate the file size of this image, firstly although only two colours have been used, let's presume a colour depth of 8 bit for this image.

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Solution:

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Firstly; calculate the total amount of pixels used in the image

8 pixels wide by 8 pixels tall: 8 x 8 = 64 (A total of 64 pixels used to represent the entire image)

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Secondly; multiply the total pixels used by the colour depth (8 bits have been used to represent the content of each pixel) 64 pixels in total multiplied by 8(colour depth): 64 x 8 = 512

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Answer: Image has a file size of 512 bits

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Question 2: To calculate the file size of this image, firstly although only two colours have been used, let's presume a colour depth of 8 bit for this image.

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Solution:

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Firstly; calculate the total amount of pixels used in the image

8 pixels wide by 8 pixels tall: 16 x 16 = 256 (A total of 256 pixels used to represent the entire image)

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Secondly; multiply the total pixels used by the colour depth (8 bits have been used to represent the content of each pixel) 256 pixels in total multiplied by 8(colour depth): 256 x 8 = 2048

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Answer: Image has a file size of 2048 bits

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1.3 Data Compression

Sound and image files can be very large, so they are often compressed to reduce their size.Compression is necessary for several reasons:

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  • It saves storage space on devices such as hard drives or SSDs.�
  • It reduces the time needed to stream, upload, or download files.�
  • It uses less network bandwidth, since compressed files contain

fewer bits and therefore transfer faster.�

  • It lowers costs, for example, when using cloud storage or data plans that charge based on the amount of data stored or transferred.

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Feature

Lossless Compression

Lossy Compression

Definition

Reduces file size without losing any data. The file can be restored exactly to its original form.

Reduces file size by permanently removing some data that is less important or not easily noticed.

Data loss

❌ No data is lost

✅ Some data is lost permanently

File quality after compression

Quality stays the same as the original

Quality is slightly lower than the original

Can original file be restored?

✅ Yes, exactly the same

❌ No, some parts are lost forever

Common uses

Text files, software, PNG and GIF images

Photos, music, and videos

Examples

ZIP, PNG, GIF, FLAC

JPEG, MP3, MP4, AAC

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Exam Style Questions

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