Tags: #◇Computer Science #◇CSARCH1 #◇Computer Architecture #◇Data Representation #◇Positional Number System

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Positional Number System

◇Overview

This module introduces the basic concepts of positional (or place-value) number systems and how to convert any number system to decimal.

◇Key Definitions

• Positional Number System: A number system where the value of a symbol depends on its position within the number.
• Radix (Base): The total number of unique symbols available in that system.
  • The largest-valued symbol always has a magnitude of one less than the radix.
• Digit Value: The digit multiplied by its place value.
• Place Value: The base raised to the power of its position index.
  • Position starts at 0 from the left side of the radix point and increments by 1 moving left (0, 1, 2...).
  • Position starts at -1 from the right side of the radix point and decrements by 1 moving right (-1, -2...).

◇Common Positional Number Systems

1. Denary / Decimal (Base 10)
   • Radix: 10
   • Symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
   • Notation: Subscript 10, prefix 0d or suffix d (e.g., $124.5_{10}$).
2. Binary (Base 2)
   • Radix: 2
   • Symbols: 0, 1
   • Notation: Subscript 2, prefix 0b, suffix b (e.g., $0101.1_2$, 0b0101.1). The radix point is called the binary point.
3. Octal (Base 8)
   • Radix: 8
   • Symbols: 0, 1, 2, 3, 4, 5, 6, 7
   • Notation: Subscript 8, prefix 0o or 0q (e.g., $67.4_8$, 0q67.4). The radix point is called the octal point.
4. Hexadecimal (Base 16)
   • Radix: 16
   • Symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F
   • Notation: Subscript 16, prefix 0x or 0h, suffix h (e.g., $0A2.8_{16}$, 0xA2.8). The radix point is called the hexadecimal point.

◇Converting Any Base to Decimal

The decimal value is the sum of all digit values (digit $\times$ place value).

Example: Binary to Decimal ($101.1_2$)

• $1 \times 2^2 = 4$
• $0 \times 2^1 = 0$
• $1 \times 2^0 = 1$
• $1 \times 2^{-1} = 0.5$
• Sum = 5.5

Example: Hexadecimal to Decimal ($2A7.8_{16}$)

• $2 \times 16^2 = 512$
• $A (10) \times 16^1 = 160$
• $7 \times 16^0 = 7$
• $8 \times 16^{-1} = 0.5$
• Sum = 679.5

Example: Quaternary (Base-4) to Decimal ($213.2_4$)

• $2 \times 4^2 = 32$
• $1 \times 4^1 = 4$
• $3 \times 4^0 = 3$
• $2 \times 4^{-1} = 0.5$
• Sum = 39.5

◇Other Number Systems

• Base 3: Ternary
• Base 4: Quaternary
• Base 5: Quinary
• Base 12: Duodecimal
• Base 20: Vigesimal
• Base 60: Sexagesimal

◇References & Navigation

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