Bidirectional conversion between senary (base 6) and decimal with detailed steps
Input Value
Result
Conversion Result
Step-by-Step Derivation
Base Conversion Principle
Senary → Decimal: Place value expansion (each digit × 6ⁿ) and sum
Decimal → Senary: Repeated division by 6 (reverse remainders)
Digits: 0, 1, 2, 3, 4, 5
Senary (base 6) is a positional numeral system with six as its base. It uses six digits: 0, 1, 2, 3, 4, and 5. Base 6 is convenient for divisibility by 2 and 3, and it is compact for binary encoding.
⚠Senary numbers use digits 0-5 only. Any digit beyond 5 is invalid in base 6.
What Is Senary (Base 6)?
Senary is a base-6 number system. It uses six digits: 0, 1, 2, 3, 4, and 5. Base 6 is historically tied to counting on fingers (5 fingers) and has nice divisibility properties, being divisible by both 2 and 3.
Divisibility Advantage
6 is divisible by 1, 2, 3, and 6. Fractions like 1/2 and 1/3 terminate in base 6, making it convenient for calculations.
Place Value Expansion
Senary to decimal: multiply each digit by 6^position. For example, 123₆ = 1×36+2×6+3×1 = 51₁₀.
Repeated Division by 6
Decimal to senary: repeatedly divide by 6, collect remainders (0-5), and read in reverse.
Binary Encoding
Each senary digit represents ~2.58 bits, making it a compact way to encode binary data in some applications.
💡 Teaching Example: Convert senary 123₆ to decimal. Place value expansion: 1×6²+2×6¹+3×6⁰ = 1×36+2×6+3×1 = 36+12+3 = 51₁₀. Conversely, 51₁₀: 51÷6=8 r3, 8÷6=1 r2, 1÷6=0 r1 → 123₆.
Applications
Traditional CountingBinary EncodingFraction CalculationMath EducationNumber Theory
Frequently Asked Questions
What is senary (base 6)?▼
Senary is a base-6 number system using digits 0-5. It is historically significant as it relates to counting on one hand (5 fingers) and is convenient for divisibility by 2 and 3. Base 6 is compact for binary encoding (each digit = ~2.58 bits).
How do you convert senary to decimal?▼
Place value expansion: multiply each senary digit by its power of 6, then sum. For example, 123₆ = 1×6²+2×6¹+3×6⁰ = 1×36+2×6+3×1 = 36+12+3 = 51₁₀.
How do you convert decimal to senary?▼
Repeated division by 6: repeatedly divide by 6, collecting remainders (0-5). Read remainders in reverse. For example, 51÷6=8 r3, 8÷6=1 r2, 1÷6=0 r1 → 123₆.
Why is base 6 interesting?▼
Base 6 has factors 1, 2, 3, and 6. Fractions like 1/2 and 1/3 terminate in base 6, and it is compact for binary encoding. Senary also has interesting number theory properties and is used in some traditional counting systems.
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