Number System Base Conversion

Tags: #◇Computer Science #◇CSARCH1 #◇Problem Solving #◇M2

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◇Overview

◇Important Formulas & Algorithms

1. Decimal to Target Base

To convert a decimal number to any target base (e.g., Binary, Octal, Hexadecimal):

• Integer Part (Whole Number):
  Perform successive ("reverse") division by the target base. Write down the remainders at each step. The final result is read from the last quotient/remainder bottom-up.
• Fractional Part:
  Perform successive multiplication of the fraction by the target base. Extract the integer part generated at each step. Continue until the fractional part becomes zero (or until 4 decimal places for non-terminating fractions). The result is read top-down.

Example: $6.625_{10}$ to Binary

• Integer (6):
  • $6 \div 2 = 3$ remainder 0
  • $3 \div 2 = 1$ remainder 1
  • $1 \div 2 = 0$ remainder 1
  • Read bottom-up: 110
• Fraction (0.625):
  • $0.625 \times 2 = \mathbf{1}.250$ (Extract 1, keep 0.250)
  • $0.250 \times 2 = \mathbf{0}.500$ (Extract 0, keep 0.500)
  • $0.500 \times 2 = \mathbf{1}.000$ (Extract 1, fraction is now 0)
  • Read top-down: .101
• Result: $110.101_2$

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◇Power-of-Two Conversions (Binary, Octal, Hex)

Because $8 = 2^3$ and $16 = 2^4$, conversions between these bases and binary can be done directly through grouping, without routing through decimal.

Binary $\leftrightarrow$ Octal (Base 8)

• Rule: 1 Octal digit = 3 Binary bits.
• Binary to Octal: Starting from the radix point (decimal point), group the binary bits into clusters of 3. Pad with leading/trailing zeros if necessary. Convert each 3-bit cluster into its octal equivalent.
  • Example: $011\ 101\ 001.111_2 \rightarrow 351.7_8$
• Octal to Binary: Convert each octal digit directly into a 3-bit binary sequence.
  • Example: $726.5_8 \rightarrow 111\ 010\ 110.101_2$

Binary $\leftrightarrow$ Hexadecimal (Base 16)

• Rule: 1 Hexadecimal digit = 4 Binary bits (a nibble).
• Binary to Hex: Starting from the radix point, group bits into clusters of 4. Convert each nibble to its hex equivalent (0-9, A-F).
  • Example: $0111\ 0101\ 1001.1011_2 \rightarrow 759.B_{16}$
• Hex to Binary: Convert each hex digit directly into a 4-bit binary sequence.
  • Example: $829.A_{16} \rightarrow 1000\ 0010\ 1001.1010_2$

Hexadecimal $\leftrightarrow$ Octal

You cannot convert directly between Hex and Octal. You must use Binary as an intermediate bridge.

• Hex to Octal: Hex $\rightarrow$ Expand to 4-bit Binary $\rightarrow$ Regroup into 3-bit Octal.
  • Example ($829.A_{16}$):

1. Hex to Bin: $1000\ 0010\ 1001.1010_2$
2. Regroup to 3s: $100\ 000\ 101\ 001.101\ 000_2$ (added trailing zero)
3. Bin to Oct: $4051.50_8$

• Octal to Hex: Octal $\rightarrow$ Expand to 3-bit Binary $\rightarrow$ Regroup into 4-bit Hex.

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◇General Base Conversion ($Base\ X \leftrightarrow Base\ Y$)

To convert between two non-power-of-two bases (e.g., Base-7 to Base-5), you must use Decimal (Base 10) as the intermediate bridge.

1. Step 1 ($Base\ X \rightarrow Decimal$): Use positional expansion (summing digits multiplied by their positional weight $X^i$).
2. Step 2 ($Decimal \rightarrow Base\ Y$): Use the successive division/multiplication algorithm on the resulting decimal number to reach Base Y.

◇References & Navigation

• Previous Topic: ◇Positional Number System
• Next Topic: ◇Binary Data Organization
• Related Notes: ◇Introduction to Number Systems