What is balanced ternary?

Balanced ternary is a base-3 numeral system whose three digits are −1, 0 and +1 instead of 0, 1 and 2. The negative digit is usually written T. Each position represents a power of three, just like decimal positions represent powers of ten.

Why does balanced ternary not need a minus sign?

Because the digit set is symmetric around zero, the negation of a number is found by swapping every 1 with a T and every T with a 1. The leading digit's sign tells you whether the whole number is positive or negative, so no separate sign bit or minus symbol is required.

How does the conversion algorithm work?

You repeatedly divide by 3 and look at the remainder. A remainder of 0 or 1 becomes that digit. A remainder of 2 is rewritten as −1 (digit T) with a carry of +1 added to the quotient, since 2 equals 3 minus 1. The digits are read from last to first.

Where is balanced ternary actually used?

The Soviet Setun computer of the 1950s used balanced ternary, and the system appears in puzzles like weighing with a balance scale, where each weight can sit on the left pan, the right pan, or neither. It also simplifies rounding because truncation rounds to the nearest value.

What does the digit T mean?

T is the conventional symbol for the digit −1 (sometimes written as an underlined or inverted 1). For example the balanced ternary 1T means 1 times 3 plus −1 times 1, which equals 2 in decimal.

What is the Balanced Ternary Converter?

Free balanced ternary converter. Convert any integer to and from balanced ternary, the base-3 system that uses digits −1, 0 and +1 (written T, 0, 1). Handles negative numbers without a sign bit. Shows the correct carry-on-2 algorithm. Runs entirely in your browser. It runs free in your browser on Gera Tools, with nothing uploaded.

Balanced Ternary Converter

Name: Balanced Ternary Converter
Creator: Gera Tools
License: https://creativecommons.org/licenses/by/4.0/

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Convert numbers to and from balanced ternary

The Balanced Ternary Converter translates ordinary decimal integers into balanced ternary — the elegant base-3 system whose digits are −1, 0 and +1 — and back again. Negative numbers are handled naturally, with no sign bit and no minus symbol, because the digit set is symmetric around zero.

How it works

To convert a positive integer to balanced ternary, repeatedly divide by 3 and inspect the remainder:

remainder 0 -> digit 0
remainder 1 -> digit 1
remainder 2 -> digit T (-1), and add 1 carry to the quotient

The carry on a remainder of 2 is the key trick: since 2 = 3 − 1, you write −1 in this position and bump the next position up by one. Reading the recorded digits from last to first gives the balanced ternary value. To convert back, evaluate the string left to right with total = total × 3 + digit, where T is −1.

Negation is free: swap every 1 with T and every T with 1. That is why the converter handles negative inputs by converting the absolute value and flipping the digits.

Worked examples

Decimal 5 → balanced ternary

5 ÷ 3 = 1 remainder 2  →  digit T, carry 1 → quotient becomes 1+1 = 2
2 ÷ 3 = 0 remainder 2  →  digit T, carry 1 → quotient becomes 0+1 = 1
1 ÷ 3 = 0 remainder 1  →  digit 1
read digits last-to-first: 1 T T
check: 1×9 + (−1)×3 + (−1)×1 = 9 − 3 − 1 = 5  ✓

Decimal −5 → balanced ternary

Negate 1TT by swapping every 1 with T and every T with 1 → T11

Check: (−1)×9 + 1×3 + 1×1 = −9 + 3 + 1 = −5 ✓

Why balanced ternary is interesting

Most numeral systems handle negative numbers with a separate sign bit (binary) or a minus symbol. Balanced ternary encodes the sign implicitly: if the leading digit is 1 the number is positive; if it is T the number is negative. This means:

Negation is trivially cheap — flip all digits, no subtraction circuit needed.
Rounding is built in — truncating a balanced ternary fraction always rounds to the nearest value, not always downward as binary truncation does.
Symmetric range — an n-trit balanced ternary number can represent the range −(3^n − 1)/2 to +(3^n − 1)/2, centred perfectly on zero.

These properties inspired the Soviet Setun (1958) and Setun 70 computers, the only balanced-ternary electronic computers ever built commercially. They also make balanced ternary the natural language for the classic balance-scale puzzle, where each weight can go on the left pan, the right pan, or stay off the scale — exactly three states per weight.

Reading the output

The converter displays the result using T for −1, 0 for zero, and 1 for positive one. To verify by hand, multiply each digit by 3 raised to its position (rightmost = position 0) and sum: 1TT is 1×9 + (−1)×3 + (−1)×1 = 5. Inputs are capped at JavaScript’s safe integer range so the repeated-division algorithm stays exact.