CHAPTER 1: INFORMATION REPRESENTATION

1.1 DATA REPRESENTATION

1.1.1 Fundamental Characteristics of Number Systems

Every number system has two fundamental characteristics:

Base (Radix): The number of different digits that a system can use to represent numbers
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:

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128 | 64 | 32 | 16 | 8 | 4 | 2 | 1
 2⁷ 2⁶ 2⁵ 2⁴ 2³ 2² 2¹ 2⁰

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):

Denary Prefix	Factor	Value	Binary Prefix	Factor	Value
kilo- (k)	×10³	1,000	kibi- (Ki)	×2¹⁰	1,024
mega- (M)	×10⁶	1,000,000	mebi- (Mi)	×2²⁰	1,048,576
giga- (G)	×10⁹	1,000,000,000	gibi- (Gi)	×2³⁰	1,073,741,824
tera- (T)	×10¹²	1,000,000,000,000	tebi- (Ti)	×2⁴⁰	1,099,511,627,776

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:

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4 = 0100
2 = 0010
9 = 1001
Therefore, 429 in BCD = 0100 0010 1001

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)

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42 = 00101010 (8-bit representation)

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

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00101010 → 11010101

Step 3: Add 1 to get two's complement

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11010101 + 1 = 11010110

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

Step 1: Flip all bits

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11010110 → 00101001

Step 2: Add 1

<TEXT>

00101001 + 1 = 00101010

Step 3: Convert to denary and apply negative sign

<TEXT>

00101010 = 42
Therefore: -42

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

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165 ÷ 16 = 10 remainder 5
10 = A
Therefore: 165 = A5 (hex)

Converting Hexadecimal to Denary: Example: A5 to Denary

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A5 = (10 × 16) + (5 × 1) = 160 + 5 = 165

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:

Standard	Description	Bits per Character	Characters
ASCII	American Standard Code for Information Interchange	7 bits	128
Extended ASCII	Extension of ASCII	8 bits	256
Unicode	Superset of ASCII and extended ASCII	16 or 32 bits	65,536+

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

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:

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File Size = Number of Pixels × Colour Depth

Example Calculation:

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Image: 1024 × 768 pixels, 24-bit colour
Number of Pixels = 1024 × 768 = 786,432
Colour Depth = 24 bits
File Size = 786,432 × 24 = 18,874,368 bits
 = 18,874,368 ÷ 8 = 2,359,296 bytes
 ≈ 2.36 MB

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:

Analogue	Digital
Continuous electrical signals	Discrete electrical signals
Infinite detail	Finite representation
Cannot be stored directly	Can be stored in binary

Sound as Analogue Data:

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

Conversion Process (Analogue to Digital):

Sampling: The sound wave's amplitude is measured at set time intervals
Quantization: Each sample is assigned a binary value
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)

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:

Lossless	Lossy
Text documents	Photography
Database files	Video streaming
Program files	Music (streaming)
Spreadsheets	Web graphics (where size matters)

Revision #2
Created 2026-03-16 06:50:51 UTC by Samuel Lee
Updated 2026-03-17 01:14:04 UTC by Samuel Lee