Notes on Number Systems

Overview

• Different number systems (or bases) to understand: Binary, Decimal, Hexadecimal

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Decimal Number System (Base 10)

• Most familiar number system
• Contains 10 unique digits: 0-9
• Evolved possibly due to having 10 fingers
• Representation of numbers:
  • No unique digit for 10 or higher; uses combination of digits
  • Example: 10 is represented as 10 (1 in the tens column and 0 in the ones column)
  • Larger number example: 4273
    • Breakdown:
      • 4 (thousands) + 2 (hundreds) + 7 (tens) + 3 (ones)
      • 4273 = 4,000 + 200 + 70 + 3
• Column headings increase by a factor of 10

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Binary Number System (Base 2)

• Contains 2 unique digits: 0 and 1
• Representation of numbers:
  • Column headings are weighted as: 1, 2, 4, 8, etc. (doubling each time)
  • Example: 111 (binary) = 3 (decimal)
    • Breakdown: 1 (8s) + 0 (4s) + 1 (2s) + 1 (1s)
  • Example: 1011 (binary) = 11 (decimal)
    • Breakdown: 1 (8) + 0 (4) + 1 (2) + 1 (1)
• Maximum representable number in 16 bits: 65535
• Smallest representable number in 16 bits: 0

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Hexadecimal Number System (Base 16)

• Contains 16 unique digits: 0-9 and A-F
  • A = 10, B = 11, C = 12, D = 13, E = 14, F = 15
• Follows similar principles to decimal and binary
• Example: 1A (hex) = 26 (decimal)
  • Breakdown: 1 (16s) + A (10)

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Summary of Base Number Systems

• Base 10 (Decimal), Base 2 (Binary), Base 16 (Hexadecimal)
• Each system can represent the number 0 as a single digit
• Unique digits available are limited in binary, requiring combinations for larger numbers
• Hexadecimal condenses binary representation, allowing for more compact expressions

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Importance of Number Systems in Computing

• Binary: Represents two states (on/off, high/low voltage)
• Decimal: Familiar to most, used in everyday counting
• Hexadecimal: More compact representation of binary, offers advantages in computing

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Additional Resources

• Further exploration on advantages of hexadecimal over binary in future discussions.