Bidirectional conversion between quaternary (base 4) and decimal with detailed steps
Input Value
Result
Conversion Result
Step-by-Step Derivation
Base Conversion Principle
Quaternary → Decimal: Place value expansion (each digit × 4ⁿ) and sum
Decimal → Quaternary: Repeated division by 4 (reverse remainders)
Digits: 0, 1, 2, 3
Quaternary (base 4) is a positional numeral system with four as its base. It uses four digits: 0, 1, 2, and 3. Base 4 is a power-of-2 base and is very convenient for converting to/from binary (base 2).
⚠Quaternary numbers use digits 0-3 only. Any digit beyond 3 is invalid in base 4.
What Is Quaternary (Base 4)?
Quaternary is a base-4 number system. It uses four digits: 0, 1, 2, and 3. Since 4 = 2², base 4 is very convenient for converting to/from binary (base 2), as each quaternary digit corresponds to exactly two binary digits.
Binary Compatibility
Each quaternary digit corresponds to exactly two binary digits, making conversion between base 4 and base 2 very simple and efficient.
Place Value Expansion
Quaternary to decimal: multiply each digit by 4^position. For example, 123₄ = 1×16+2×4+3×1 = 27₁₀.
Repeated Division by 4
Decimal to quaternary: repeatedly divide by 4, collect remainders (0-3), and read in reverse.
Genetic Research
Base 4 relates to DNA structure (4 nucleotide bases: A, T, C, G), making it relevant in bioinformatics and genetic research.
💡 Teaching Example: Convert quaternary 123₄ to decimal. Place value expansion: 1×4²+2×4¹+3×4⁰ = 1×16+2×4+3×1 = 16+8+3 = 27₁₀. Conversely, 27₁₀: 27÷4=6 r3, 6÷4=1 r2, 1÷4=0 r1 → 123₄.
Quaternary is a base-4 number system using digits 0-3. It is a power-of-2 base (2²) and is very convenient for converting to/from binary (base 2). Each quaternary digit corresponds to exactly two binary digits.
How do you convert quaternary to decimal?▼
Place value expansion: multiply each quaternary digit by its power of 4, then sum. For example, 123₄ = 1×4²+2×4¹+3×4⁰ = 1×16+2×4+3×1 = 16+8+3 = 27₁₀.
How do you convert decimal to quaternary?▼
Repeated division by 4: repeatedly divide by 4, collecting remainders (0-3). Read remainders in reverse. For example, 27÷4=6 r3, 6÷4=1 r2, 1÷4=0 r1 → 123₄.
What is special about base 4?▼
4 is 2 squared, making base 4 very convenient for conversion to/from binary. Each quaternary digit corresponds to exactly two binary digits. Base 4 is also used in some genetic research (DNA has 4 bases), digital communication, and error-correcting codes.
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