Number Systems

Basic Definitions

Number Bases

• Binary (Base 2): Uses digits 0, 1
• Octal (Base 8): Uses digits 0-7
• Decimal (Base 10): Uses digits 0-9
• Hexadecimal (Base 16): Uses digits 0-9, A-F

Key Concepts

• Positional Number System: Each digit’s value depends on its position
• Multiple Representations: Same binary pattern can represent different values depending on the number system used
  • Example: could be 10, -2, -5, or -6

Base Conversions

Binary to Decimal

Use positional notation:

Binary to Octal

Group binary digits in sets of 3 (from right):

Binary to Hexadecimal

Group binary digits in sets of 4 (from right):

Other Examples

Negative Binary Numbers

Three Main Approaches

1. Sign-and-Magnitude
2. Ones’-Complement
3. Two’s-Complement

Common Features

• MSB (Most Significant Bit) is the sign bit:
  • 0 = Positive
  • 1 = Negative

Two’s-Complement (Preferred Method)

• Advantages:
  • Widely used in modern systems
  • Simplifies arithmetic operations
  • Only one representation for zero
• Process: To get negative of a number:
  1. Invert all bits (ones’-complement)
  2. Add 1 to the result

Binary Numbers with Decimals

Fractional Binary Representation

• Uses negative powers of 2 for digits after the binary point
• Example:

Converting Decimal Fractions to Binary

Method: Multiply by 2, take integer part, repeat with fractional part

• Example: Convert to binary
  • → 0
  • → 1
  • → 1
  • Result:

Limitations

• Some decimal fractions cannot be exactly represented in binary
• Example: (repeating)
• This leads to precision issues in computer calculations

Floating Point Numbers

IEEE 754 Standard

The most common floating point representation:

Single Precision (32-bit)

Sign	Exponent	Mantissa/Significand
1	8	23

Double Precision (64-bit)

Sign	Exponent	Mantissa/Significand
1	11	52

Components

1. Sign Bit: 0 for positive, 1 for negative
2. Exponent: Biased representation (actual exponent + bias)
   • Single: bias = 127
   • Double: bias = 1023
3. Mantissa: Fractional part (normalized, implicit leading 1)

Formula

Special Values

• Zero: Exponent = 0, Mantissa = 0
• Infinity: Exponent = all 1s, Mantissa = 0
• NaN (Not a Number): Exponent = all 1s, Mantissa ≠ 0
• Denormalized: Exponent = 0, Mantissa ≠ 0

Example

Single precision representation of :

1. Convert to binary:
2. Normalize:
3. Sign: 0 (positive)
4. Exponent:
5. Mantissa:

A/D and D/A Conversion

• Real-world signals: Continuous/analog form
• Digital systems: Model these as 0s and 1s
• Conversion needed: Between analog and digital representations