# 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:**
<TEXT>
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:**
<TEXT>
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)
<TEXT>
42 = 00101010 (8-bit representation)
**Step 2:** Convert to one's complement (flip all bits)
<TEXT>
00101010 → 11010101
**Step 3:** Add 1 to get two's complement
<TEXT>
11010101 + 1 = 11010110
**Converting Binary Two's Complement to Denary (Example: 11010110):**
**Step 1:** Flip all bits
<TEXT>
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
<TEXT>
165 ÷ 16 = 10 remainder 5
10 = A
Therefore: 165 = A5 (hex)
**Converting Hexadecimal to Denary:** Example: A5 to Denary
<TEXT>
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:**
<TEXT>
File Size = Number of Pixels × Colour Depth
**Example Calculation:**
<TEXT>
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):**
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)
---
## 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:**