Number Systems

Overview

Number systems are different methods of representing numbers, each with its own base and symbols. In digital electronics, we primarily work with binary (base-2), octal (base-8), decimal (base-10), and hexadecimal (base-16) systems. Understanding these systems and conversions between them is fundamental to digital design, computer architecture, and programming.

Detailed Explanation

Common Number Systems Table

Base Name     | Base | Digits Used
--------------|------|-------------
Binary        |   2  | 0,1
Octal         |   8  | 0-7
Decimal       |  10  | 0-9
Hexadecimal   |  16  | 0-9,A-F

Position Values

System    Position Values (Right to Left)
Decimal:  10⁰, 10¹, 10², 10³, ...
Binary:   2⁰,  2¹,  2²,  2³,  ...
Octal:    8⁰,  8¹,  8²,  8³,  ...
Hex:      16⁰, 16¹, 16², 16³, ...

Quick Reference Table

Decimal | Binary  | Octal | Hex
--------|---------|-------|----
   0    |  0000   |   0   |  0
   1    |  0001   |   1   |  1
   2    |  0010   |   2   |  2
   3    |  0011   |   3   |  3
   4    |  0100   |   4   |  4
   5    |  0101   |   5   |  5
   6    |  0110   |   6   |  6
   7    |  0111   |   7   |  7
   8    |  1000   |  10   |  8
   9    |  1001   |  11   |  9
  10    |  1010   |  12   |  A
  11    |  1011   |  13   |  B
  12    |  1100   |  14   |  C
  13    |  1101   |  15   |  D
  14    |  1110   |  16   |  E
  15    |  1111   |  17   |  F

Conversion Methods

1. To Decimal

   Binary:  1101₂
   = 1×2³ + 1×2² + 0×2¹ + 1×2⁰
   = 8 + 4 + 0 + 1
   = 13₁₀

2. From Decimal

   Division Method:
   13÷2 = 6 rem 1
   6÷2  = 3 rem 0
   3÷2  = 1 rem 1
   1÷2  = 0 rem 1
   Result: 1101₂ (read remainders bottom-up)

Practice Problems

1. Convert these numbers to all other bases:

   • Decimal: 25
   • Binary: 1010
   • Octal: 37
   • Hex: 2F

2. Arrange in ascending order:

   • 1010₂
   • 12₈
   • 0A₁₆
   • 11₁₀

References

• Digital Design by Morris Mano
• Computer System Architecture by M. Morris Mano
• Number Systems Tutorial

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