Positional and Non-Positional Number Systems,

 In computer science and mathematics, number systems are fundamental for representing numerical values. They are broadly categorized into Positional and Non-Positional Number Systems, each with unique characteristics and applications.

1. Positional Number System

In this system, the position of each digit relative to others determines its value. The value of a number is computed based on a base or radix, and each position represents a power of the base.

Key Characteristics:

  • Base/ Radix: The number of distinct digits used, including 0. For example:
    • Binary (Base 2): Uses digits 0 and 1.
    • Decimal (Base 10): Uses digits 0 through 9.
    • Hexadecimal (Base 16): Uses digits 0-9 and letters A-F.
  • Positional Weight: Each digit’s value is multiplied by the base raised to the power of its position. For instance, in decimal 432=4×102+3×101+2×100432 = 4 \times 10^2 + 3 \times 10^1 + 2 \times 10^0432=4×102+3×101+2×100.
  • Examples in Computing:
    • Binary is used in computer architecture.
    • Octal and hexadecimal simplify binary representation for humans.

Advantages:

  • Efficient for performing arithmetic operations.
  • Compact and scalable for large numbers.

Examples:

  1. Binary: 10112=1×23+0×22+1×21+1×20=11101011_2 = 1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 = 11_{10}10112​=1×23+0×22+1×21+1×20=1110​
  2. Decimal: 342.5=3×102+4×101+2×100+5×10−1342.5 = 3 \times 10^2 + 4 \times 10^1 + 2 \times 10^0 + 5 \times 10^{-1}342.5=3×102+4×101+2×100+5×10−1

2. Non-Positional Number System

In this system, the position of a digit does not affect its value. Each symbol represents a fixed value, and the overall value is determined by simple addition.

Key Characteristics:

  • No base or positional weight.
  • Limited arithmetic utility; primarily used in historical or specific contexts.

Examples:

  • Roman Numerals: XII=10+1+1=12XII = 10 + 1 + 1 = 12XII=10+1+1=12
  • Egyptian Numerals: Based on symbols for 1, 10, 100, etc., where repetition indicates addition.

Applications:

  • Historical and cultural significance.
  • Limited modern use except in clocks, book chapters, etc.

Advantages:

  • Simple representation for small numbers.
  • Easy to learn and use for basic counting.

Comparison:

Feature

Positional System

Non-Positional System

Base/ Radix

Required (e.g., 2, 10, 16)

Not required

Positional Weight

Present

Absent

Arithmetic Utility

Highly efficient

Limited

Examples

Binary, Decimal, Hexadecimal

Roman Numerals, Tally Marks

Application in Computers

Universally used

Rarely used

Importance in Computers:

  • Positional Number Systems are essential for digital systems:
    • Binary underpins machine-level operations.
    • Hexadecimal simplifies memory addressing and debugging.
  • Non-Positional systems have no practical computational application but offer educational insights into the evolution of numerical representation.

 

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