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.

Balanced Ternary Converter

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.

Example

Decimal 2 has remainder 2 when divided by 3, so the digit is T and a carry of 1 is added, making the next quotient 1. The result is 1T, which evaluates as 1×3 + (−1)×1 = 2. Decimal −2 is simply T1, the digit-flipped form.

Notes

Balanced ternary truncation rounds to the nearest value, a property that makes it appealing for certain arithmetic and the reason it powered the experimental Setun computer. The converter limits input to safe integers so the repeated division stays exact.