Chapter 1

• New lesson editor

Denary

Hexadecimal

Explanation

Eg. "45A"

4

5

A

(4x256) + (5x16) + (10x1) = 1114 in denary

Note: A=10

162 =256

161 =16

160 =1

Denary

Hexadecimal

Explanation

Eg. "C8F"

C

8

F

(12x256) + (8x16) + (15x1) = 3215 in denary

Note: C=12, F=15

162 =256

161 =16

160 =1

DIY

What is the denary form of BF08?

Denary

Hexadecimal

DIY

B

F

0

8

162 =256

161 =16

160 =1

163 =4096

DIY

ANSWER

Denary

Hexadecimal

DIY

B

F

0

8

(11x4096) + (15x256) + (0x16) + (8x1) = 48904 in denary

162 =256

161 =16

160 =1

163 =4096

5

2

2

2

1

1

0

Denary

Binary

Convert 5 to binary:

5

RECAP

(Method 2)

remainder

remainder

2

0

remainder

1

Read the remainder

from bottom to top

Answer: 101

Denary

Hexadecimal

Explanation

Eg. "2004"

2004

16

16

125

remainder

2004/16 =

4

125/16 =

7

13

remainder

16

0

remainder

7

Answer: 7D4

Note: 13=D

125

remainder = 4

7

remainder = 13

DIY

What is the hexadecimal form of 3179?

Denary

Hexadecimal

DIY

3179

16

16

198

remainder

?

?

?

remainder

16

?

remainder

?

3179/16 = ?

DIY

16

6

What is the hexadecimal form of 3179?

Denary

Hexadecimal

DIY

3179

16

198

remainder

11

12

remainder

16

0

remainder

12

3179/16

198/16

Answer: C6B

= 198

remainder = 11

= 12

remainder = 6

PAST YEAR QUESTION

ANSWER

PAST YEAR QUESTION

ANSWER

Chapter 1.2

Use of hexadecimal system

Discussion Time

Use of hexadecimal system

Binary

Hexadecimal

110101111110100111001

1AFD39

Brainstorm time: Why is Hexadecimal used?

Explanation

Use of hexadecimal system

• One hex digit represents four binary digits

  • The hex number is far easier for humans to remember, copy and work with

  • Four uses of the hexadecimal system

Explanation

Usage 1: Error Code

• Error codes are often shown as hexadecimal values.

  • These numbers refer to the memory location of the error.

  • They are generated by the computer.

  • The programmer needs to know how to interpret the hexadecimal error codes.

Explanation

Usage 1: Error Code

Explanation

Usage 2: MAC address

• Media Access Control (MAC) address refers to

a number which uniquely identifies a device on a network.

• The MAC address refers to the network interface card (NIC) which is part of the device

  • The MAC address is rarely changed so that a particular device can always be identified no matter where it is.

Explanation

Usage 2: MAC address

00_1C_B3_4F_25_FE

00_1C_C3_4F_23_AE

Mac address uniquely identify a device on a Local Area Network

Message

Explanation

Usage 2: MAC address

00-1C-B3-4F-25-FE

NN-NN-NN-DD-DD-DD

00:1C:B3:4F:25:FE

NN:NN:NN:DD:DD:DD

Form 1

Form 2 2

Mac Address comes with 2 forms

Explanation

Usage 2: MAC address

00-1C-B3

4F-25-FE

Identity number of the manufacturer

Serial number of

a device

Eg. 00 – 14 – 22 which identifies devices made by Dell

00 – a0 – c9 which identifies devices made by Intel

Explanation

Usage 3: Internet Protocol Addresses

• Each device connected to a network is given an address known as the Internet Protocol address

  • An IPv4 address is a 32-bit number written in denary or hexadecimal form e.g. 109.108.158.1 (or 77.76.9e.01 in hex)

  • IPv4 has recently been improved upon by the adoption of IPv6. An IPv6 address is a 128-bit number broken down into 16-bit chunks, represented by a hexadecimal number.

Eg. a8f b:7a88:fff0:0fff:3d21:2085:66f b:f0fa

Explanation

Usage 4: HyperText Markup Language (HTML) colour code

• HyperText Mark-up Language (HTML) is used when writing and developing web pages.

  • It is not a programming language, but a markup language.

  • A mark-up language is used in the processing, definition and presentation of text.

Explanation

Overview

The 4 usages of Hexadecimals - EMIH

1. E - Error Codes

2. M - MAC Address

3. I - Internet Protocol Address

4. H - HTML Colour Code

PAST YEAR QUESTION

ANSWER

PAST YEAR QUESTION

ANSWER

Chapter 1.3

Addition of

binary number

Explanation

How do we perform add and carry in denary?

0 + 0 = 0

0 + 9 = 9

9 + 0 = 9

9 + 1 = 10

9

+1

0

1

1

Addition of binary number

Explanation

How do we perform add and carry in denary?

56

+79

6+9 = 15 (>9)

5

1

1+5+7 = 13 (>9)

3

1

1

Addition of binary number

Explanation

How do we perform add and carry in binary?

0 + 0 = 0

0 + 1 = 1

1 + 0 = 1

1 + 1 = 10

Addition of binary number

Explanation

How do we perform add and carry in binary?

00100111

+01001010

1

1

0

0

1

0

1

1

1

1

0

Addition of binary number

DIY

How do we perform add and carry in binary?

Perform

01111110 + 00111110

Addition of binary number

Explanation

The overflow condition

01101110

[phone]

1

1

0

0

1

0

1

1

0

1

1

1

1

0

1

Addition of binary number

Addition of binary number

Explanation

The overflow condition

01101110

[phone]

1

1

0

0

1

0

1

1

0

1

1

1

1

0

1

• The maximum denary of an 8-bit binary number (11111111) is (28 - 1 ) = 255

  • The generation of a 9th bit is a clear indication that the sum has exceeded this value.

  • This is known as an overflow error. The sum is too big to be stored using 8 bits.

Explanation

The overflow condition

01101110

[phone]

1

1

0

0

1

0

1

1

0

1

1

1

1

0

1

01101110 = 110

\11011110 = 222

110 + 222 = 322

322 > 255 (overflow)

The sum is too big to be stored in a 8 bit binary.

Addition of binary number

Lesson Objectives

Last lesson on the binary system

BINARY SHIFTING

Multiplication and division of binary numbers

TWO COMPLEMENTS

Represent negative number in binary

Chapter 1.4: Binary Shifting

BINARY SHIFTING

• Binary shift is a process that a CPU uses to perform multiplication and division.

BINARY SHIFTING - MULTIPLICATION

• For a CPU to multiply a binary number, the number needs to be shifted to the left and will fill the remaining gaps with zeros.

16

32

8

4

2

1

64

16

32

8

4

2

1

64

16

32

8

4

2

1

64

BINARY SHIFTING - MULTIPLICATION

Examples: 111 (Binary)

0

0

0

1

1

1

0

Examples: 1110 (Binary)

0

0

0

1

1

1

0

Examples: 11100 (Binary)

0

0

0

1

1

1

0

16

32

8

4

2

1

64

BINARY SHIFTING - MULTIPLICATION

Examples: 111 (Binary)

0

0

0

1

1

1

0

Multiply by 2, shift 1 place to the left

1110

Multiply by 4, shift 2 place to the left

Multiply by 8, shift 3 place to the left

11100

111000

Multiply by 2^n, shift n place to the left

BINARY SHIFTING - DIVISION

• For a CPU to multiply a binary number, the number needs to be shifted to the right.

16

32

8

4

2

1

16

32

8

4

2

1

16

32

8

4

2

1

BINARY SHIFTING - DIVISION

Examples: 101100 (Binary)

0

1

1

1

0

0

Examples: 10110 (Binary)

1

0

0

1

1

0

Examples: 1011 (Binary)

1

0

1

0

1

0

16

32

8

4

2

1

BINARY SHIFTING - DIVISION

Examples: 101100 (Binary)

0

1

1

1

0

0

Divide by 2, shift 1 place to the right

10110

Divide by 4, shift 2 place to the right

Divide by 8, shift 3 place to the right

1011

101

Divide by 2^n, shift n place to the right

BINARY SHIFTING WITH 8-BIT BINARY NUMBERS

• Registers contained within the CPU often have 8-bits limits on the amount of data they can hold at any one time.

  • The multiplying shifting process can cause bits to be lost at one end of the register, and zeros added at the opposite end.

  • This process is known as losing the most significant bit.

16

32

8

4

2

1

64

128

BINARY SHIFTING WITH 8-BIT BINARY NUMBERS

Examples: 10110101 (181 in denary)

1

1

0

1

0

1

0

1

10110101 -> 01101010

106 in denary

The bit lost is called the most significant bit, and when it is shifted beyond the furthest-column the binary data that is stored loses precision due to overflow.

16

32

8

4

2

1

64

128

16

32

8

4

2

1

64

128

BINARY SHIFTING WITH 8-BIT BINARY NUMBERS

The same process can happen when dividing an 8-bit binary number.

1

1

1

1

0

1

0

1

Example: 10111101 (189 in denary)

Divide this number by 32 (move 5 places to the right)

0

0

0

1

0

1

0

0

Least

Significant bit

The division shift produces the binary number 101 = 5, not 5.9 that arithmetic suggests.

11101

Lesson Objectives

Last lesson on the binary system

BINARY SHIFTING

Multiplication and division of binary numbers

TWO COMPLEMENTS

Represent negative number in binary

Chapter 1.5: Two Complements

TWO COMPLEMENTS

• A PROCESSOR CAN ALSO REPRESENT NEGATIVE NUMBERS.

  • ONE OF THE METHOD THAT A PROCESS REPRESENT NEGATIVE NUMBERS IS CALLED TWO'S COMPLEMENT.

TWO COMPLEMENTS

TWO COMPLEMENTS

• TO REPRESENT NEGATIVE NUMBERS, IT IS IMPORTANT TO THINK ABOUT THE PLACE VALUE OF THE FURTHEST-LEFT BIT IN A DIFFERENT WAY.

  • PROCESSOR CAN BE SET UP TO SEE THE BIT IN THE EIGHTH COLUMN AS A SIGN BIT.

0 = POSITIVE

1 = NEGATIVE

16

32

8

4

2

64

-128

CONVERT POSITIVE BINARY INTEGER TO A TWO'S COMPLEMENT 8-BIT INTEGER

Examples:13

0

0

1

1

0

0

0

1

1

Step 2: Put the number into the place value column

Step 3: Ensure that the the leftmost bit is 0 (+).

Step 1: Convert 13 into binary.

1101 in binary

DIY

Convert 19 into a Two's complement

8-bit Integer

16

32

8

4

2

64

-128

CONVERT POSITIVE BINARY INTEGER TO A TWO'S COMPLEMENT 8-BIT INTEGER

Examples:19

1

0

0

0

1

0

0

1

1

Step 2: Put the number into the place value column

Step 3: Ensure that the the leftmost bit is 0 (+).

Step 1: Convert 19 into binary.

10011 in binary

Answer: 00010011

16

32

8

4

2

64

-128

CONVERT TWO'S COMPLEMENT 8-BIT INTEGER TO A POSITIVE BINARY INTEGER

Examples: Convert 00010011 (two's complement) to denary

1

0

0

0

1

0

0

1

1

Step 1: Put the number into the place value column

Step 2: This shows that it is a positive number, we can just convert the binary into denary directly.

Step 3: Calculate the denary value.

(1x16) + (1x2) + (1x1) = 19

Convert 01010011 (two's complement)

to denary

DIY

16

32

8

4

2

64

-128

CONVERT TWO'S COMPLEMENT 8-BIT INTEGER TO A POSITIVE BINARY INTEGER

Examples: Convert 01010011 (two's complement) to denary

1

0

0

0

1

1

0

1

1

Step 1: Put the number into the place value column

Step 2: This shows that it is a positive number, we can just convert the binary into denary directly.

Step 3: Calculate the denary value.

(1x64) + (1x16) + (1x2) + (1x1) = 83

16

32

8

4

2

64

-128

CONVERT NEGATIVE BINARY NUMBERS IN TWO'S COMPLEMENT FORMAT AND CONVERT TO DENARY

Examples: 10010011

1

0

0

0

1

0

1

1

1

Step 1: Put the number into the place value column

Step 3: Compute the denary value as usual.

Step 2: The left-most bit is 1, this means that it is a negative number.

(1x -128) + (1x16) + (1x2) + (1x1)

= -128 + 16 + 2 + 1

= -109

Convert 10110011 (Two's Complement)

to denary

DIY

16

32

8

4

2

64

-128

CONVERT NEGATIVE BINARY NUMBERS IN TWO'S COMPLEMENT FORMAT AND CONVERT TO DENARY

Examples: 10110011

1

1

0

0

1

0

1

1

1

Step 1: Put the number into the place value column

Step 3: Compute the denary value as usual.

Step 2: The left-most bit is 1, this means that it is a negative number.

(1x -128) (1x32)+ (1x16) + (1x2) + (1x1)

= -128 + 32 + 16 + 2 + 1

= -77

CONVERTING NEGATIVE DENARY NUMBERS INTO BINARY NUMBERS IN TWO’S COMPLEMENT FORMAT

Examples: -67

Step 1: Convert the number to positive.

67

Step 2: Write the number in binary form (8 bits).

01000011

Step 3: Invert each binary value.

10111100

Step 4: Add 1 to the binary number.

1

10111101

+

10111100

Step 5: This gives us -67.

16

32

8

4

2

64

-128

1

1

1

1

0

0

1

1

1

-128 + 32 + 16 + 8 + 4 + 1 = -67

Convert -65 to 8 bit two's complement

binary number

DIY

CONVERTING NEGATIVE DENARY NUMBERS INTO BINARY NUMBERS IN TWO’S COMPLEMENT FORMAT

Examples: -65

Step 1: Convert the number to positive.

65

Step 2: Write the number in binary form (8 bits).

01000001

Step 3: Invert each binary value.

10111110

Step 4: Add 1 to the binary number.

1

10111111

+

Step 5: This gives us -65.

16

32

8

4

2

64

-128

1

1

1

1

1

0

1

1

1

-128 + 32 + 16 + 8 + 4 + 2 + 1 = -65

10111110

Summary: Convert negative denary to two's complement

Examples: -65

01000001

10111110

65

1

10111111

Convert to (+)

Convert to binary

Invert the digit

+1

Final result

Chapter 1.3

eight

Exercise for ASCII

• Using the ASCII table convert the word “BLUE” to binary.

• Using the ASCII table below, convert the following binary coded message into a word.

Activity

Using the ASCII table, try to decode this message:

01000011 01101111 01001101 01110000 01110101 01110100 01100101 01110010 01010011 01100011 01101001 01000101 01101110 01100011 01100101 01101001 01110011 01100110 01110101 01101110 00100001

each letter will be represented as an 8-bit binary number.

The ASCII character set represents characters using 8-bit binary numbers. This means that it can represent up to 256 characters (0 to 255).

A Unicode character set represents characters using 16-bit binary numbers.

This means it can represent a much greater number of characters, approximately 65 000.

1.1.3 Unicode

Sound is a vibration that propagates as an audible wave of pressure through the air. In human physiology ,sound is the reception of this waves and the perception by the brain.

The sound is generated by a sound source which is the vibrating diaphragm of a speaker and detected by microphone or human ears.

We can visualize sound as an analogue wave and to store the sound in binary we need to convert the analogue signal to a digital signal. This process is called analogue to digital conversion.

There are two main factors:

Sample rate :The sample rate is how often an audio analogue sound wave is sampled .

Sample resolution: Each of those sampling rates, what is the level of detail depth or fertility that analogue audio was recorded at .

OR

Number of bits that are used to represent each sample .

NOTE:

Amplitude is the distance between the wave's resting position and its maximum displacement.

 

Frequency is the number of waves that pass by a specific point per second. 2.

The meter is the SI unit of amplitude. The unit of frequency measurement is Hertz (Hz).

The number of bits used to represent sound amplitude in digital sound recording, as known as bit depth

• Image file types

  • BMP

  • JPG

  • GIF

  • PNG

1.2 How do computers represent images

• Creating an Image

• Each pixel is given a binary value

• Each value represents a different colour

• Using one bit per pixel allows only 2 values, 0 and 1

1 = Black, 0 = White

• Red, Green, Blue color system(RGB)

• The different intensity of the 3 primary colors make up the color you want

• The intensity will be represented by denary or hex numbers

For example: (255,0,0) / #FF0000 is red

RGB images

Measurement of Data Storage and Calculation of file size

Measurement of Data Storage

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

  • The byte is the smallest unit of memory in a computer.

8 bits = 1 byte

4 bits = 1 nibble

Memory

Size

System

Based on the SI (base 10) system of units where

1 kilo is equal to 1000.

SI-international System of Units

Memory

Size

System

Based on the IEC (base 2) system of units where

1 kilo is equal to 1024 (2^10).

As memory size is actually measured in terms of powers of 2...

Memory

Size

System

Converting Bytes into KiB, MiB and GiB

68719476736 Bytes

68719476736 Bytes / 1024

=

= 67108864 KiB

=

67108864 KiB / 1024

= 65536 MiB

=

65536 MiB / 1024

= 64 GiB

• Sound file formats

• .WAV – uncompressed files(Wave Form Audio)

• .FLAC or .M4A lossless compression, slightly smaller files-Free Lossless Audio Codec,MPEG-Audio 4

• .MP3 – Lossy compression, much smaller files

• MP-4 /MPEG-4 file format –Motion picters Experts Group

• MP4 is a digital multimedia format most commonly used to store video and audio

  • It can also be used to store subtitles and still images

  • It allows different
    multimedia streams (video, audio, text) to be combined into one file

• MIDI files

• MIDI stands for Musical Instrument Digital Interface

• A MIDI file:

  • is not a recording of a live
    music source

  • is a set of instructions for digital
    instruments to play synthesised sounds

  • can be used to synchronise an orchestra
    of digital instruments to play simultaneously

  • uses up to 1000 times less disk space than a
    conventional recording

  • is commonly used for mobile phone ringtones

Conversion between bits and bytes

Memory

Size

System

Converting Gib, Mib, Kib into bytes

= 68719476736 Bytes

64 x 1024

=

= 65536 MiB

=

65536 x 1024

= 67108864 KiB

=

67108864 x 1024

64 GiB

Calculation

of file

size

Image

Audio

Calculation

of file

size - Image

Image Resolution - The number of pixels that make up an image.

The higher the image resolution, the higher the quality of the image.

Calculation

of file

size - Image

Formula

image resolution (pixels) x colour depths (bits)

Calculation

of file

size - Image

Example 1

00

01

10

11

2px

2px

Total pixels = 2 x 2 = 4

Colour depth = 2

Calculation = (2x2) x 2

= 8 bits = 1 byte

Calculation

of file

size - Image

Example 2

Formula : image resolution (pixels) x colour depths (bits)

Question:

Image Resolution = 1024 x 1080

Colour depth = 32

Calculate the size of this image in Bytes.

Workings:

1024 x 1080 = 1105920 pixels

1105920 x 32 = 35389440 bits

Answer in byte: 35389440/8 = 4423680 bytes

Calculation

of file

size - Image

Example 2

Question:

Image Resolution = 1024 x 1080

Colour depth = 32

Calculate the size of this image in Bytes. How many photograph of this size would fit onto a memory stick of 64Gib.

Each image = 4423680 bytes

First convert 64 Gib into bytes:

64 x 1024 = 65536 MiB

65536 x 1024 = 67108864 KiB

67108864 x 1024 = 68719476736 bytes

Calculation

of file

size - Image

Example 2

Question:

Image Resolution = 1024 x 1080

Colour depth = 32

Calculate the size of this image in Bytes. How many photograph of this size would fit onto a memory stick of 64Gib.

Each image = 4423680 bytes

First convert 64 Gib into bytes = 68719476736 bytes

68719476736/4423680 = 15534 photos.

DIY

Question:

Image Resolution = 2048 x 2048

Colour depth = 16

Calculate the size of this image in Bytes.

DANSWER

Question:

Image Resolution = 2048 x 2048

Colour depth = 16

Calculate the size of this image in Bytes.

Answer:

2048 x 2048 x 16 = 67108864 bits

= 67108864/8

= 8388608 bytes

DIY

Question:

Image Resolution = 2048 x 2048

Colour depth = 16

Calculate the size of this image in Bytes (Answer: 8388608 bytes).

What is the size of the image in MiB.

DANSWER

Question:

Image Resolution = 2048 x 2048

Colour depth = 16

Calculate the size of this image in Bytes (Answer: 8388608 bytes).

What is the size of the image in MiB.

8388608 / 1024 = 8192 KiB

8192 / 1024 = 8 MiB

Calculation

of file

size - Sound

Formula

Sample Rate (in Hz) x Sample Resolution (in bits) x length of sample (in seconds)

Calculation

of file

size - Sound

Mono Sound vs Stereo Sound

Comparison

The difference between monophonic (mono) and stereophonic (stereo) sound is the number of channels used to record and playback audio.

Mono signals are recorded and played back using a single audio channel, while stereo sounds are recorded and played back using two audio channels.

Calculation

of file

size - Sound

Example 1 - Mono Sound

Question:

Sample Rate: 44100

Sample Resolution: 8 bits

Length of the music: 20 seconds

Calculate the size of the audio in KiB.

44100 x 8 x 20 = 7056000 bits

7056000/8 = 882000 bytes

882000 / 1024 = 861.328 KiB

Calculation

of file

size - Sound

Example 1 - Stereo Sound

An audio CD has a sample rate of 44100 and a sample resolution of 16 bits. The music being sampled uses two channels to allow for stereo recording. Calculate the file size for a 60-minute recording.

44100 x 16 x 3600 = 2540160000 bits

2540160000 x 2 = 5080320000 bits

5080320000 / 8 = 635040000 bytes

635040000 / 1024 = 620156.25 KiB

620156.25 / 1024 = 605.62 MiB

DIY

An audio CD has a sample rate of 44100 and a sample resolution of 8 bits. The music being sampled uses two channels to allow for stereo recording. Calculate the file size for a 25-minute recording.

DIY

An audio CD has a sample rate of 44100 and a sample resolution of 8 bits. The music being sampled uses two channels to allow for stereo recording. Calculate the file size for a 25-minute recording in MiB.

44100 x 8 x 1500 = 529200000 bits

529200000 x 2 = 1058400000 bits

1058400000 / 8 = 132300000 bytes

132300000 / 1024 = 129199.218 KiB

129199.218 / 1024 = 126.17 MiB

PAST YEAR QUESTION

ANSWER

DATA

COMPRESSION

DATA COMPRESSION

DATA COMPRESSION

• FILE SIZE OF IMAGES AND SOUND CAN BE VERY LARGE.

  • THEREFORE, DATA COMPRESSION IS NEEDED TO REDUCE THE SIZE OF A FILE.

DATA

COMPRESSION

WHAT ARE SOME BENEFITS OF REDUCING THE FILE SIZE?

DATA

COMPRESSION

Benefits of Data Compression

SAVE STORAGE SPACE

REDUCE

STREAMING TIME

REDUCE

TIME TAKEN TO UPLOAD AND DOWNLOAD MEDIA

REDUCE

COST

LOSSY

FILE

COMPRESSION

LOSSLESS

FILE

COMPRESSION

DATA

COMPRESSION

• Two types of image compression

  • Lossy Compression (JPG)

  • Lossless Compression (PNG)

• What is compression?

A method that uses an algorithm to reduce the size of a file.

3. Data Compression

• Removes data permanently

• Much smaller file sizes but some loss of quality

e.g. reducing the resolution:

Note: Mainly used in image or a sound file .

3.1 Lossy Compression

128KiB

21KiB

Note:

1. The size of the sound file can be reduced by reducing the sample rate and the sample resolution

2. The lossy compression algorithm for an image file could reduce the size of the file by reducing the colur depth .

• Lossless compression reduces the file size without permanent loss of data, e.g. run length encoding (RLE)

• This algorithm will group together repeating pixels and store how many times they occur.

3.2 Lossless Compression

Run Length Encoding (RLE)

If W is white, Y is yellow, R is red and G is green, RLE could compress the image into the following data:

12W, 3Y, 5W, 2Y, 1R, 2Y, 3W, 2Y, 3R, 2Y, 3W, 2Y, 1R, 2Y, 5W, 3Y, 7W, 1G, 5W, 2G,

1W, 1G, 1W, 2G, 4W, 3G, 7W, 1G, 4W.

3.2 Lossless Compression

DIY

DIY

REDUCE COLOUR

DEPTH

REDUCE

IMAGE

RESOLUTION

PAST YEAR QUESTION

ANSWER

PAST YEAR QUESTION

ANSWER