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

Position Value = Digit × Baseⁿ

Used in computer arithmetic, coding, memory and digital systems.

What is a Number System?

A Number System is a method of representing and expressing numbers using a set of symbols called digits. Every number system is defined by its Base (Radix), which specifies the total number of unique digits available.

Examples:
Decimal → Base 10 → Digits 0-9
Binary → Base 2 → Digits 0,1
Octal → Base 8 → Digits 0-7
Hexadecimal → Base 16 → Digits 0-9,A-F

Base or Radix

The base of a number system indicates the number of distinct symbols used to represent numbers.

Base 2 → Binary
Base 8 → Octal
Base 10 → Decimal
Base 16 → Hexadecimal

Positional Number System

Most modern number systems are positional. The value of a digit depends on: 1. The digit itself
2. Its position in the number

Value = Digit × Baseⁿ

Example: (345)₁₀ = 3×10² + 4×10¹ + 5×10⁰ = 300 + 40 + 5 = 345

Binary Number System

Binary is the fundamental language of computers. Digits Used: 0 and 1 Example: (1011)₂ = 1×2³ + 0×2² + 1×2¹ + 1×2⁰ = 8 + 0 + 2 + 1 = 11₁₀

Advantages: • Simple electronic implementation
• Reliable storage and transmission
• Directly compatible with digital circuits

Octal Number System

Base = 8 Digits = 0 to 7 Each Octal Digit represents exactly 3 Binary Bits. Example: 563₈ = 101110011₂

Used to shorten long binary numbers.

Decimal Number System

Base = 10 Digits = 0 to 9 Most commonly used number system in daily life. Example: 528₁₀ = 5×10² + 2×10¹ + 8×10⁰

Hexadecimal Number System

Base = 16 Digits: 0 1 2 3 4 5 6 7 8 9 A B C D E F Where: A = 10
B = 11
C = 12
D = 13
E = 14
F = 15

Each Hex Digit represents exactly 4 Binary Bits. Example: FA₁₆ = 11111010₂

Widely used in: • Memory Addressing
• Programming
• Machine Language
• Color Codes (#FFFFFF)

Why Computers Use Binary?

Electronic circuits naturally have two stable states: ON / OFF
HIGH / LOW
TRUE / FALSE These are represented as: 1 / 0 Hence computers process all information using binary digits.

Number System Conversions

Common conversions include: • Binary ↔ Decimal
• Binary ↔ Octal
• Binary ↔ Hexadecimal
• Decimal ↔ Octal
• Decimal ↔ Hexadecimal

Complements

Complements simplify subtraction operations. Examples: 1's Complement
2's Complement
7's Complement
8's Complement
9's Complement
10's Complement

Digital Codes

Special coding schemes used in digital electronics: • BCD (Binary Coded Decimal)
• Excess-3 Code
• Gray Code

Signed Number Representation

Negative numbers can be represented using: • Sign Magnitude
• 1's Complement
• 2's Complement

Floating Point Representation

Real numbers are represented using: • Fixed Point Format
• Floating Point Format
• IEEE-754 Standard

Important Exam Facts

• Binary uses 2 symbols.
• Octal uses 8 symbols.
• Decimal uses 10 symbols.
• Hexadecimal uses 16 symbols.
• 1 Octal Digit = 3 Bits.
• 1 Hex Digit = 4 Bits.
• Base is also called Radix.
• 2's Complement is most commonly used for signed integers.
• IEEE-754 is the standard floating-point representation.
• Gray Code changes only one bit between consecutive values.

Applications of Number Systems

• Computer Architecture
• Digital Electronics
• Programming Languages
• Networking
• Data Communication
• Embedded Systems
• Microprocessors
• Memory Management
• Operating Systems
• Scientific Computing