Base Conversions: Decimal, Binary, Octal, Hex

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Base Conversions: Decimal, Binary, Octal, Hex

This page explains how to convert numbers between the most important bases: decimal (10), binary (2), octal (8), and hexadecimal (16).

General method: from any base to decimal

To convert from another base into decimal, expand the number into positional values.

Example: 725₈ = 7 × 8² + 2 × 8¹ + 5 × 8⁰ = 7 × 64 + 2 × 8 + 5 × 1 = 448 + 16 + 5 = 469₁₀.

General method: from decimal to another base

Use repeated division by the target base:

  1. Divide the number by the base.
  2. Record the remainder.
  3. Continue dividing the quotient until 0.
  4. Read the remainders backwards.
Example: Convert 232622₁₀ to octal  
232622 ÷ 8  = 29077   remainder 6  
29077 ÷ 8   = 3634    remainder 5  
3634 ÷ 8    = 454     remainder 2  
454 ÷ 8     = 56      remainder 6  
56 ÷ 8      = 7       remainder 0  
7 ÷ 8       = 0       remainder 7  
Result: 706256₈.

Binary and octal

Binary and octal map directly:

  • Each octal digit = 3 binary digits.
  • Group binary digits into groups of 3 to convert to octal.
  • Expand octal digits into 3-bit groups to convert to binary.

Example: 110110₂ → group 110 110 → 66₈.

Binary and hexadecimal

Binary and hexadecimal map directly:

  • Each hex digit = 4 binary digits.
  • Group binary digits into groups of 4 to convert to hex.
  • Expand hex digits into 4-bit groups to convert to binary.

Example: 111100111101₂ → group 1111 0011 1101 → F3D₁₆.

Octal and hexadecimal

Convert through binary:

  • Octal → binary (3 bits per digit).
  • Regroup binary into 4 bits.
  • Binary → hex.

Example: 157₈ → 1 = 001, 5 = 101, 7 = 111 → 001101111₂. Group as 0001 1011 11 → 1B7₁₆.

Practice examples

Decimal to binary: 37
37 ÷ 2      = 18      remainder 1  
18 ÷ 2      = 9       remainder 0  
9 ÷ 2       = 4       remainder 1  
4 ÷ 2       = 2       remainder 0  
2 ÷ 2       = 1       remainder 0  
1 ÷ 2       = 0       remainder 1  
Read backwards: 100101₂.

Decimal to hex: 988664  
988664 ÷ 16 = 61791   remainder 8  
61791 ÷ 16  = 3861    remainder 15 (F)  
3861 ÷ 16   = 241     remainder 5  
241 ÷ 16    = 15      remainder 1  
15 ÷ 16     = 0       remainder 15 (F)  
Result: F15F8₁₆ = 0xF15F8.

Why conversions matter

  • Binary is closest to hardware but unreadable in long form.
  • Octal and hex are compact and map cleanly to binary.
  • Decimal is human-friendly.
  • Conversions are essential for reading memory dumps, debugging, and low-level programming.
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