Bidirectional conversion between nonary (base 9) and decimal with detailed steps
Input Value
Result
Conversion Result
Step-by-Step Derivation
Base Conversion Principle
Nonary → Decimal: Place value expansion (each digit × 9ⁿ) and sum
Decimal → Nonary: Repeated division by 9 (reverse remainders)
Digits: 0, 1, 2, 3, 4, 5, 6, 7, 8
Nonary (base 9) is a positional numeral system with nine as its base. It uses nine digits: 0, 1, 2, 3, 4, 5, 6, 7, and 8. Base 9 is a perfect square base and is convenient for conversion to/from ternary (base 3).
⚠Nonary numbers use digits 0-8 only. Any digit beyond 8 is invalid in base 9.
What Is Nonary (Base 9)?
Nonary is a base-9 number system. It uses nine digits: 0, 1, 2, 3, 4, 5, 6, 7, and 8. Since 9 = 3², base 9 is very convenient for converting to/from ternary (base 3), as each nonary digit corresponds to exactly two ternary digits.
Ternary Compatibility
Each nonary digit corresponds to exactly two ternary digits, making conversion between base 9 and base 3 very simple and efficient.
Place Value Expansion
Nonary to decimal: multiply each digit by 9^position. For example, 123₉ = 1×81+2×9+3×1 = 102₁₀.
Repeated Division by 9
Decimal to nonary: repeatedly divide by 9, collect remainders (0-8), and read in reverse.
Perfect Square Base
9 is 3 squared, giving base 9 unique mathematical properties related to squares and quadratic residues in number theory.
💡 Teaching Example: Convert nonary 123₉ to decimal. Place value expansion: 1×9²+2×9¹+3×9⁰ = 1×81+2×9+3×1 = 81+18+3 = 102₁₀. Conversely, 102₁₀: 102÷9=11 r3, 11÷9=1 r2, 1÷9=0 r1 → 123₉.
Applications
Ternary ConversionNumber TheoryTraditional CountingMath EducationRecreational Math
Frequently Asked Questions
What is nonary (base 9)?▼
Nonary is a base-9 number system using digits 0-8. It is a perfect square base (3²) and is convenient for converting to/from ternary (base 3). Base 9 is used in some traditional counting systems and has interesting number theory properties.
How do you convert nonary to decimal?▼
Place value expansion: multiply each nonary digit by its power of 9, then sum. For example, 123₉ = 1×9²+2×9¹+3×9⁰ = 1×81+2×9+3×1 = 81+18+3 = 102₁₀.
How do you convert decimal to nonary?▼
Repeated division by 9: repeatedly divide by 9, collecting remainders (0-8). Read remainders in reverse. For example, 102÷9=11 r3, 11÷9=1 r2, 1÷9=0 r1 → 123₉.
What is special about base 9?▼
9 is 3 squared, making base 9 very convenient for conversion to/from ternary (base 3). Each nonary digit corresponds to two ternary digits. Base 9 also has nice divisibility properties and is used in some traditional counting systems.
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