Data Representation & Number Systems: Binary, Hex, ASCII, and Memory

TAKEAWAYS:

• Computers use binary (base-2) to represent all data, including text, images, and numbers.
• Hexadecimal (base-16) simplifies binary representation for humans and software.
• ASCII and Unicode encode characters (e.g., letters, symbols) into binary for storage and transmission.
• Memory hierarchy (registers → RAM → SSD → HDD) balances speed and cost in computer systems.
• Data conversion (binary ↔ decimal ↔ hex) is essential for debugging and software development.
• Real-world examples like eSewa’s transaction codes and Daraz’s inventory tracking rely on these concepts.

1. Introduction to Data Representation

Computers cannot process human-readable data (e.g., text, images) directly. Instead, they convert everything into binary (0s and 1s). This section covers how data is encoded and represented internally.

Why Binary?

• Simplicity: Electronic circuits (transistors) naturally switch between two states: ON (1) and OFF (0).
• Universality: All data—numbers, text, images—is stored as binary.
• Efficiency: Binary operations (AND, OR, NOT) are fast and hardware-friendly.

2. Number Systems

Computers use multiple number systems to represent data. The key systems are:

• Decimal (Base-10): Human default (digits 0–9).
• Binary (Base-2): Computer default (digits 0–1).
• Hexadecimal (Base-16): Shorthand for binary (digits 0–9, A–F).
• Octal (Base-8): Rarely used today, but historically important.

Conversion Between Number Systems

Example: Convert 13 (decimal) to binary and hex.

• Decimal to Binary:

  13 ÷ 2 = 6  (remainder 1)
   6 ÷ 2 = 3  (remainder 0)
   3 ÷ 2 = 1  (remainder 1)
   1 ÷ 2 = 0  (remainder 1)
  

  Reading remainders bottom-up: 1101 (binary).
• Decimal to Hex: 13 ÷ 16 = 0 (remainder D). So, 13 (decimal) = D (hex).

Mermaid Diagram: Conversion Flowchart

flowchart TD
    A["Decimal"] -->|"Divide by 2"| B["Binary"]
    A -->|"Divide by 16"| C["Hex"]
    B --> D["Check remainders"]
    C --> E["Use A-F for 10-15"]

3. Binary Arithmetic

Computers perform arithmetic using binary logic gates. Key operations:

• Addition: Follows rules like 0+0=0, 1+1=10 (carry over).
• Subtraction: Uses 2’s complement for negative numbers.
• Multiplication/Division: Extends binary addition/subtraction.

Example: Binary Addition (1011 + 1101)

  1011
+ 1101
-------
 10100  (Result: `20` in decimal)

4. Data Representation: ASCII and Unicode

Text is stored as binary using character encoding schemes:

• ASCII (American Standard Code for Information Interchange): 7-bit (128 characters), covers basic English letters, symbols, and numbers.
• Unicode (UTF-8/UTF-16): Extends ASCII to support global languages (e.g., Nepali, Chinese, Arabic).

Mermaid Diagram: ASCII Table Snippet

Why Unicode?

• eSewa/Khalti: Use Unicode to display Nepali characters in transaction confirmations.
• WhatsApp/YouTube: Unicode ensures messages/videos render correctly across languages.

5. Memory Hierarchy and Data Storage

Computers use a memory hierarchy to balance speed and cost:

1. Registers (Fastest, smallest, e.g., CPU registers like AX, BX).
2. Cache Memory (L1, L2, L3: ~MBs, faster than RAM).
3. RAM (Random Access Memory) (GBs, volatile, holds running programs).
4. SSD (Solid State Drive) (GBs–TB, non-volatile, faster than HDD).
5. HDD (Hard Disk Drive) (TB, slowest, cheapest).

Layers from registers to HDD with speed/cost trade-offs (Image: ComputerMemoryHierarchy.png: User:Danlash at en.wikipedia.or, Public domain, via Wikimedia Commons)

Example: Daraz’s Inventory System

• RAM: Stores real-time stock levels of products (e.g., 100 units of "Nepalgunj Rice").
• SSD: Logs historical sales data for analytics.
• HDD: Archives old inventory reports.

6. Real-World Applications

## In the Real World

1. eSewa/Khalti:

   • Idea: Hexadecimal encoding of transaction IDs (e.g., A1B2C3D4) for compact storage in databases.
   • How: Transaction IDs are often stored as hex to save space and enable quick binary processing.

2. Daraz Order Processing:

   • Idea: Binary flags to mark order status (e.g., 101 = "Processing," 110 = "Shipped").
   • How: Logical operations on binary flags automate workflows (e.g., AND to check if an order is paid and shipped).

3. NTC/Ncell Network Routing:

   • Idea: IP addresses (e.g., 192.168.1.1) are stored in binary internally but represented in decimal for humans.
   • How: Routers use binary logic to forward data packets based on IP prefixes.

Worked Example: Calculating Loan Interest (Banking) A bank uses binary arithmetic to compute interest:

• Suppose a loan amount is ₹50,000 (stored as binary: 01100011 00101000 00000000 00000000).
• Interest rate: 5% (stored as 00000101 in binary).
• Calculation:
  1. Convert 50,000 to binary (as above).
  2. Multiply by 5% using binary multiplication.
  3. Result: ₹2,500 (interest), stored back in binary.

7. Advantages and Limitations

Exam Tip

• Focus on conversions: Practice decimal ↔ binary ↔ hex (e.g., 255 → FF → 11111111).
• ASCII/Unicode: Know key codes (e.g., 65 = A, 97 = a) and why Unicode is needed for multilingual apps.
• Memory hierarchy: Compare speed/cost trade-offs (e.g., "Why does a CPU use cache?").
• Real-world links: Relate to apps like eSewa (hex IDs), Daraz (binary flags), or NTC (IP routing).
• Common mistakes:
  • Confusing ASCII (7-bit) and Unicode (variable-bit).
  • Forgetting 2’s complement for negative binary numbers.
  • Misaligning bits during conversions (e.g., 101 vs 0101).

Sample Exam Question: "Explain how a computer stores the character ‘ने’ in binary using Unicode. Why is ASCII insufficient for Nepali text?" Answer Structure:

1. Unicode uses UTF-8 (multi-byte encoding).
2. ‘ने’ is encoded as 01001110 01101001 01101110 (UTF-8 bytes).
3. ASCII lacks Nepali characters (only covers 128 symbols).

Mermaid Diagram: ASCII vs Unicode

mindmap
  root((Character Encoding))
    ASCII
      only 128 characters
      limited to English
    Unicode
      UTF-8/UTF-16
      supports global languages
      multi-byte per character