1. Data Representation

Number Systems

Using Binary to Represent Data in a Computer System

• Why Computers Use Binary:
  • Data must be converted to binary for processing.
  • Logic gates, switches, transistors process data in two states: high-low / on-off / 1 and 0.
  • Data is stored in registers.
• How Binary Represents Data:
  • Binary value assigned to each character/pixel.

Binary Number System

• Base: 2
• Values: 0, 1
• Units Increase: Powers of 2 (e.g., 2^0, 2^1)
• Related Systems:
  • Denary (Base 10): Values 0-9
  • Hexadecimal (Base 16): Values 0-9, A-F

Conversions

• Denary to Binary:
  • Use powers of 2, write 1 if power is ≤ number.
  • Check by adding powers with 1.
• Binary to Denary:
  • Add up power values with binary digit 1.
• Binary to Hexadecimal:
  • Group binary digits into nibbles, convert to hex.
• Hexadecimal to Binary:
  • Convert each hex digit to binary.
• Denary to Hexadecimal: Convert via binary.
• Hexadecimal to Denary: Convert via binary.

Binary Addition

• Addition rules:
  • 0+0=0,
  • 0+1=1,
  • 1+1=10 (carry 1),
  • 1+1+1=11 (carry 1)
• Overflow:
  • Occurs when calculation exceeds register limit.

Binary Shifts

• Left Shift: Multiplies by power of 2.
• Right Shift: Divides by power of 2.
• Effect: Bits lost at one end, zeros added at the other.

Two's Complement

• Method for representing signed integers.
• Conversion for Negatives:
  • Binary of positive equivalent,
  • Invert bits,
  • Add 1.

Benefits of Hexadecimal

• Easier/quicker to read and debug than binary.
• Uses fewer digits.
• Applications: IP, MAC addresses, error codes, HTML colors.

Text, Sound, Images

Representing Text

• Text converted to binary for computer processing.
• Character Sets:
  • ASCII: 7-bit binary code.
  • Unicode: Variable-length, more characters and languages.

Representing Sound

• Sound is an analog signal sampled for binary conversion.
• Quality Factors:
  • Sample rate (samples per second).
  • Sample resolution (bits per sample).

Representing Images

• Image made of pixels, each with a binary code.
• Quality Factors:
  • Resolution (pixels dimensions).
  • Color depth (bits per color).

Data Storage

• Units: Byte, Kibibyte, Mebibyte, etc.
• File Size Calculation:
  • Images: resolution x color depth.
  • Sound: sample rate x resolution x time.

Data Compression

Purpose

• Reduce file size, bandwidth, transmission time.

Lossy Compression

• Removes data permanently.
• Reduces file size.

Lossless Compression

• No data loss.
• Original file can be restored.

Applications

• Lossy: Media files.
• Lossless: Text, code files.