Convert between balanced ternary and decimal with detailed step-by-step derivation
Input Value (use T for -1)
Result
Conversion Result
Step-by-Step Derivation
Conversion Principle
Balanced Ternary → Decimal: digit × 3ⁿ and sum
Decimal → Balanced Ternary: Divide by 3, adjust remainder
Digits: T (-1), 0, 1 (balanced base 3)
Balanced ternary is a base-3 numeral system using digits -1, 0, and +1 (often written as T, 0, 1). It's elegant and can represent negative numbers without a minus sign.
⚠For balanced ternary, use T for -1. When converting decimal to balanced ternary: remainder 0 → 0, remainder 1 → 1, remainder 2 → T (-1) and add 1 to quotient.
What Is Balanced Ternary?
Balanced ternary is a base-3 number system using digits -1, 0, and +1 (often written as T, 0, 1). It has elegant mathematical properties and can represent negative numbers without a separate minus sign.
Elegant Symmetry
Balanced ternary has beautiful symmetry. The negative of a number is just its digits flipped (T ↔ 1).
Place Value Expansion
Balanced ternary to decimal: digit × 3^position. For example, 1T1 = 1×3²+(-1)×3¹+1×3⁰ = 7.
Adjusted Division
Decimal to balanced ternary: divide by 3, remainder 2 → T (-1) and add 1 to quotient.
Historical Interest
Balanced ternary was studied in early computing for its elegance, though binary ultimately dominated.
💡 Teaching Example: Convert balanced ternary 1T1 to decimal. Place value expansion: 1×3²+(-1)×3¹+1×3⁰ = 9 - 3 + 1 = 7. Conversely, 7: 7÷3=2 remainder 1 → digit 1. 2÷3=0 remainder 2 → digit T and quotient +1 → new quotient 1. 1÷3=0 remainder 1 → digit 1. Read in reverse: 1 T 1.
Balanced ternary is a base-3 number system using digits -1, 0, and +1 (often written as T, 0, 1). It can represent negative numbers without a minus sign and has elegant mathematical properties.
How do you convert balanced ternary to decimal?▼
Place value expansion: each digit × 3^position, then sum. For example, 1T1 = 1×3²+(-1)×3¹+1×3⁰ = 9 - 3 + 1 = 7.
How do you convert decimal to balanced ternary?▼
Repeated division by 3: if remainder 2 → use T (-1) and add 1 to quotient; remainder 1 → use 1; remainder 0 → use 0. Collect digits, read in reverse.
What are the advantages of balanced ternary?▼
Balanced ternary is elegant: negative numbers without a minus sign, efficient representation, symmetric, and has interesting theoretical properties in computing.
Free online calculators and tools covering mathematics, unit conversion, text processing, and daily life. Accurate, fast, mobile-friendly, and completely free to use.