CHAPTER 1: INFORMATION REPRESENTATION

1.1 DATA REPRESENTATION

1.1.1 Fundamental Characteristics of Number Systems

Every number system has two fundamental characteristics:

1. Base (Radix): The number of different digits that a system can use to represent numbers
2. Place Value: The specific value of a digit based on its position within a number

1.1.2 Denary (Decimal) System - Base 10

• Uses digits 0-9
• Each position represents powers of 10 (10⁰, 10¹, 10², etc.)
• Example: 3,567 = (3 × 10³) + (5 × 10²) + (6 × 10¹) + (7 × 10⁰)

1.1.3 Binary System - Base 2

Key Points:

• Uses only two digits: 0 and 1
• Each bit (binary digit) represents a power of 2
• All data and characters in computers are represented in binary

Binary Place Values:

Example - Converting Denary to Binary:

• Denary 65 in binary: 01000001
• Calculation: 64 + 1 = 65

Example - Converting Binary to Denary:

• Binary 01000001 = 64 + 1 = 65

1.1.4 Binary Prefixes vs Decimal Prefixes

It is crucial to understand the difference between binary prefixes (based on powers of 2) and decimal prefixes (based on powers of 10):

Important: Always use the correct prefix:

• Computer storage uses binary prefixes (KiB, MiB, GiB, TiB)
• Data transfer rates often use decimal prefixes (kbps, Mbps, Gbps)

1.1.5 Binary Coded Decimal (BCD)

Definition: Binary representation where each individual denary digit is represented by a sequence of 4 bits (nibble).

Characteristics:

• Each nibble can represent denary digits 0-9
• Uses only specific 4-bit patterns (0000 to 1001)
• The patterns 1010 to 1111 are not used in BCD

Example - Converting 429 to BCD:

Practical Applications:

• Electronic devices displaying numbers (calculators)
• Accurately measuring decimal fractions
• Electronically coding denary numbers

1.1.6 Two's Complement Representation

Two's complement is used to represent negative numbers in binary.

Converting Negative Denary to Binary (Example: -42):

Step 1: Find binary equivalent (ignoring sign)

Step 2: Convert to one's complement (flip all bits)

Step 3: Add 1 to get two's complement

Converting Binary Two's Complement to Denary (Example: 11010110):

Step 1: Flip all bits

Step 2: Add 1

Step 3: Convert to denary and apply negative sign

Range in 8-bit Two's Complement:

• Maximum positive: +127 (01111111)
• Maximum negative: -128 (10000000)

Overflow:

• Occurs when the result of an arithmetic operation is too large/small to fit in the allocated bits
• Example: Adding 127 + 1 in 8-bit gives -128 (overflow)

1.1.7 Hexadecimal System - Base 16

Characteristics:

• Uses digits 0-9 and letters A-F
• A=10, B=11, C=12, D=13, E=14, F=15

Converting Denary to Hexadecimal: Example: 165 to Hex

Converting Hexadecimal to Denary: Example: A5 to Denary

Practical Applications:

• Defining colours in HTML (#FF0000 = red)
• Defining MAC addresses
• Assembly languages and machine code
• Debugging via memory dumps

1.1.8 Character Sets and Encoding

Definition: A character set is a collection of characters that can be represented using binary codes. It typically includes upper and lower case letters, number digits, punctuation marks, and other characters.

Character Encoding Standards:

ASCII:

• Only supports English alphabet
• 7 bits = 128 possible characters
• Includes control characters (0-31), printable characters (32-126)

Extended ASCII:

• 8 bits = 256 possible characters
• Includes most European languages' alphabets
• Still limited for global languages

Unicode:

• Modern international standard
• Supports all global languages
• UTF-8 uses 1-4 bytes per character
• Backward compatible with ASCII

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1.2 MULTIMEDIA - GRAPHICS AND SOUND

1.2.1 Bitmap Images

Definition: Bitmap images are created by assigning a solid colour to each pixel using bit patterns. The image is represented as a grid of pixels, where each pixel's colour is encoded using binary values.

Key Terms:

• Pixel: The smallest picture element whose colour can be accurately represented by binary code
• File Header: Contains metadata including image size, number of colours, etc.

Image Resolution:

• Definition: The number of pixels that make up an image
• Example: 4096 × 3192 pixels
• Effect: Higher resolution results in sharper, more detailed images

Screen Resolution:

• Definition: The number of pixels that can be viewed horizontally and vertically on a device's screen
• Example: 1680 × 1080 pixels

Colour Depth:

• Definition: The number of bits used to represent the colour of a single pixel
• Formula: If n bits are used, there are 2ⁿ colours per pixel
• Example: 16-colour bitmap = 4 bits per pixel (2⁴ = 16)
• Effect: Increasing colour depth improves colour quality but increases file size

File Size Calculation:

Example Calculation:

Applications:

• Scanned images
• Digital photographs
• Computer screen displays
• Small file sizes and easy manipulation when needed

1.2.2 Vector Graphics

Definition: Made up of drawing objects (mathematically defined constructs like rectangles, lines, circles, curves).

Components:

• Drawing List: A set of commands defining the vector
• Properties: Basic geometric data determining shape and appearance
• Encoding: Data is encoded using mathematical formulas

Advantages over Bitmap:

• Objects can be resized without losing quality
• Scalability is the key benefit
• Smaller file sizes for simple images
• Can be enlarged infinitely without pixelation

Disadvantages:

• Cannot represent complex images like photographs
• More complex to create

Applications:

• Company logos
• Architectural drawings
• Icons and symbols
• Fonts (TrueType, PostScript)

1.2.3 Sound Representation

Analogue vs Digital:

Sound as Analogue Data:

• Sound consists of vibrations through a medium
• Inherently analogue due to infinite detail variation

Conversion Process (Analogue to Digital):

1. Sampling: The sound wave's amplitude is measured at set time intervals
2. Quantization: Each sample is assigned a binary value
3. Encoding: Binary values are stored

Key Terms:

• Sampling Rate: Number of samples taken per unit of time (measured in Hz)

  • Effect: Increasing sampling rate improves accuracy but increases file size
  • CD quality: 44,100 Hz

• Sampling Resolution: Number of bits used to encode each sample

  • Effect: Increasing resolution improves accuracy but increases file size
  • CD quality: 16 bits

• Bit Rate: Number of bits used to store 1 second of sound

  • Formula: Bit Rate = Sampling Rate × Sampling Resolution
  • Example: 44,100 × 16 = 705,600 bps (approximately 706 Kbps)

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1.3 COMPRESSION

1.3.1 Need for Compression

Definition: Compression is the process of reducing file size without significant loss in quality.

Benefits:

• Reduced storage requirements
• Faster data transfer (uses less bandwidth)
• Reduced time needed to search for data

1.3.2 Lossless Compression

Definition: A type of compression that allows original data to be perfectly reconstructed from the compressed file.

Key Feature:

• Uses some form of replacement (substitution)
• No data is permanently deleted

Examples:

• PNG images (for graphics with sharp edges)
• ZIP files
• Text file compression
• Database records
• Run-Length Encoding (RLE)

Run-Length Encoding (RLE):

Definition: A form of lossless compression used for compressing text files and bitmap images.

Mechanism:

• Reduces file size by encoding sequences of adjacent, identical elements
• Encodes as two values: run count and run value

Example: Original: AAAAAAABBBBBCCCCCC Compressed: 7A5B6C

Example - Bitmap: Original row: White White White White White Black Black Compressed: 5W2B

Applications:

• Simple graphics with large areas of same colour
• Database records with repeated values

1.3.3 Lossy Compression

Definition: A type of compression that irreversibly eliminates unnecessary data.

Characteristics:

• File accuracy/quality is lower than lossless
• File size is significantly reduced (often to about 10% of lossless size)
• Some original data is permanently lost

Examples:

• MP3 (sound files)
• JPEG (images)
• MP4 (video files)

Mechanism in Sound Files (MP3):

• Perceptual Coding: Removes parts of the sound that are less audible or discernible to human hearing
• Removes frequencies outside human hearing range
• Removes subtle volume differences

Mechanism in Images (JPEG):

• Removes high-frequency details
• Uses mathematical approximations
• Reduces colour precision in less important areas

When to Use Lossy vs Lossless:

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