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What is a Number System?

A number system is a mathematical way of representing numbers using a specific base. Computers use different number systems to process, store, and transfer data efficiently. The most commonly used number systems in computing include:

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Types of Number Systems

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1. Decimal (Base-10)

  • The standard number system used in everyday life.
  • Uses digits 0-9.
  • Example: 472 in decimal represents (4 × 10²) + (7 × 10¹) + (2 × 10⁰) = 472.

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2. Binary (Base-2)

  • The fundamental number system for computers.
  • Uses only 0 and 1.
  • Example: 1011 in binary represents (1 × 2³) + (0 × 2²) + (1 × 2¹) + (1 × 2⁰) = 11 in decimal.

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3. Octal (Base-8)

  • Uses digits 0-7.
  • Commonly used as a shorthand for binary.
  • Example: 57 in octal represents (5 × 8¹) + (7 × 8⁰) = 47 in decimal.

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4. Hexadecimal (Base-16)

  • Uses digits 0-9 and letters A-F (where A=10, B=11, …, F=15).
  • Used in memory addressing, networking (MAC addresses), and color codes in web design.
  • Example: 2F in hexadecimal represents (2 × 16¹) + (F × 16⁰) = 47 in decimal.

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Number System Conversions

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Decimal to Binary

  1. Divide the decimal number by 2.
  2. Record the remainder.
  3. Repeat until the quotient is 0.
  4. Read the remainders in reverse order.

Example: Convert 13 to binary:

13 ÷ 2 = 6, remainder = 1
6 ÷ 2 = 3, remainder = 0
3 ÷ 2 = 1, remainder = 1
1 ÷ 2 = 0, remainder = 1
Binary: **1101**

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Binary to Decimal

Multiply each bit by 2^n (where n is its position from right, starting at 0), then sum the results.

Example: Convert 1101 to decimal:

(1 × 2³) + (1 × 2²) + (0 × 2¹) + (1 × 2⁰)
= 8 + 4 + 0 + 1 = **13**

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Hexadecimal to Binary

Replace each hex digit with its 4-bit binary equivalent.

Example: Convert 2F to binary:

2  →  0010
F  →  1111
Binary: **00101111**

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Octal to Binary

Replace each octal digit with its 3-bit binary equivalent.

Example: Convert 57 to binary:

5 → 101
7 → 111
Binary: **101111**

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Importance of Number Systems in Computing

  • Binary is the language of computers (1s and 0s).
  • Octal & Hexadecimal simplify binary representation.
  • Decimal is used in high-level programming and user interactions.
  • Efficient Data Processing: Different systems help optimize memory storage and computation.

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Real-World Applications

  • Computer Memory & Storage: Data is stored in binary.
  • Networking: IP addresses and MAC addresses use hexadecimal.
  • Programming: Low-level languages (e.g., Assembly) work with different number systems.
  • Digital Electronics: Logic gates operate using binary.

Understanding number systems is fundamental for computer science, programming, and electronics. Mastering conversions and operations across these systems is essential for working with low-level data structures and memory management!