Unit III — Number Systems: Positional & Non-Positional

A number system is a mathematical notation for expressing numbers. Computers internally use the binary number system. Understanding number systems and their inter-conversions is fundamental to computer science.

---

What is a Number System?

A number system defines:

• A set of symbols (digits) used to represent values.
• A base (radix) — the total count of distinct digits.

---

Types of Number Systems

1. Non-Positional Number Systems

In a non-positional system, the value of a digit does not depend on its position in the number.

• Example: Roman Numerals — I, V, X, L, C, D, M
• I = 1, V = 5, X = 10, L = 50, C = 100, D = 500, M = 1000
• XIV = 14 (X + IV = 10 + 4)
• The value is determined by the digit itself, not by where it appears.
• Disadvantage: Very difficult to perform arithmetic operations.

2. Positional Number Systems

In a positional system, the value of each digit depends on its position (place value) in the number.

Formula: Value = Σ (digit × base^position)

The base (radix) determines the number of unique digits:

---

Place Value Concept

In Decimal (Base-10): The number 3425 means:

• 3 × 10³ = 3000
• 4 × 10² = 400
• 2 × 10¹ = 20
• 5 × 10⁰ = 5
• Total = 3425

In Binary (Base-2): The number 1101 means:

• 1 × 2³ = 8
• 1 × 2² = 4
• 0 × 2¹ = 0
• 1 × 2⁰ = 1
• Total = 13 (decimal)

---

Why Different Number Systems?

---

Subscript Notation

To avoid ambiguity, a subscript denotes the base:

• (1101)₂ = binary number 1101
• (25)₈ = octal number 25
• (FF)₁₆ = hexadecimal number FF
• (255)₁₀ = decimal number 255

Key Takeaway: Positional number systems assign place values to digits based on their position and the base. All four systems used in computing — binary, octal, decimal, and hexadecimal — are positional. Binary is the native language of computers; hex and octal are convenient shorthand for binary.