Balanced Ternary Calculator

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Balanced Ternary Arithmetic

Conversion Table (−13 to +13)

What Balanced Ternary Is, and Why It Behaves So Well

Balanced ternary is base 3 written with the digit set −1, 0, +1 instead of the usual 0, 1, 2. Each position is still a power of three — 1, 3, 9, 27, 81 — but a digit may now subtract its place value as well as add it. The digits are called trits, and −1 is conventionally written T so that a number stays one character per position.

Every integer has exactly one balanced ternary representation, and it needs no sign bit to say which side of zero it is on. Take 42. The largest place value that helps is 81, so start there and subtract your way down: 81 − 27 − 9 − 3 = 42, which is 1 T T T 0 — one 81, minus one 27, minus one 9, minus one 3, and no units. Enter 42 in the converter above and that is the answer it produces.

Negation is free

Now negate it. In balanced ternary you flip the sign of every trit and stop: 1 T T T 0 becomes T 1 1 1 0, which is −81 + 27 + 9 + 3 = −42. There is no two's complement, no carry, no add-one step, and no asymmetry between the most negative and most positive value a fixed width can hold. In hardware that is one inverter per trit rather than an adder, which is why subtraction costs the same as addition here and rounding to the nearest integer is simply truncation.

Why three and not two

Representing a range of numbers costs roughly (number of digits) × (digits per position), and that product is minimised at base e ≈ 2.718. Three is the nearest whole number to it, so base 3 carries more information per position than base 2 does per bit — one trit is log₂3 ≈ 1.585 bits. That has been known since Setun, the balanced ternary computer built at Moscow State University in 1958. What was missing then was a device that is naturally three-state rather than a three-state code faked on top of two-state parts.

Where this connects to the rest of the lab

maniTLab's premise is that a carbon nanotube gives you that device: current one way, no current, current the other way — a physical −1, 0, +1 with no encoding overhead. How it works walks through the stack from that device upward; Thatte1 is the device patent itself, and balanced ternary vs binary puts the two number systems side by side, including where binary still wins. If you want to see trits doing something structural rather than arithmetic, the trit-trie visualizer builds a three-way prefix tree a trit at a time.