About

Ternary logic operates on base 3. Unlike binary systems that rely on two states, ternary systems utilize three distinct values. This offers higher information density and specific arithmetic advantages. Standard ternary uses digits 0, 1, and 2. Balanced ternary, a variation often cited in historical computing architectures like the Setun computer, uses digits -1, 0, and 1 (often notated as T, 0, 1). This distinction allows for the representation of negative numbers without a separate sign bit. Errors in base conversion or arithmetic in these systems lead to significant logic failures in algorithm design or low-level data structure implementation.

This tool performs addition, subtraction, multiplication, and integer division directly within the ternary ecosystem. It handles large integer inputs and visualizes the underlying decimal values to verify accuracy. The implementation strictly differentiates between unsigned standard ternary and signed balanced ternary logic. It serves engineers working on non-binary logic gates, students studying radix economy, or developers exploring efficient data compression techniques.

Formulas

The value of a standard ternary number is calculated by summing the coefficients multiplied by powers of 3. For a string of digits d:

N = n−1∑i=0 d_i ⋅ 3ⁱ

Where d_i ∈ {

0, 1, 2 (Standard)−1, 0, 1 (Balanced)

Reference Data

Decimal (Base 10)	Binary (Base 2)	Standard Ternary (Base 3)	Balanced Ternary (T, 0, 1)	Power of 3 Reference
-5	-101	N/A (Signed)	T11	-
-4	-100	N/A (Signed)	TT	-
-3	-11	N/A (Signed)	T0	-3¹
-2	-10	N/A (Signed)	T1	-
-1	-1	N/A (Signed)	T	-
0	0	0	0	0
1	1	1	1	3⁰
2	10	2	1T	-
3	11	10	10	3¹
4	100	11	11	-
5	101	12	1TT	-
6	110	20	1T0	-
7	111	21	1T1	-
8	1000	22	10T	-
9	1001	100	100	3²
10	1010	101	101	-
13	1101	111	111	-
27	11011	1000	1000	3³

Frequently Asked Questions

Balanced ternary reduces the need for a separate sign bit because negative numbers are inherent to the system (using the "T" or -1 digit). This simplifies addition and subtraction hardware logic, as the operation is identical regardless of the sign of the number, reducing the complexity of the arithmetic logic unit (ALU).

In balanced ternary notation, "T" typically represents the value -1 (negative one). Sometimes it is also written as a "1" with a bar over it. This tool uses "T" for ease of typing. So, a number like "1T" means 1 times 3 minus 1, which equals 2.

This tool focuses on standard arithmetic (+, -, *, /). Modulo in balanced ternary is mathematically complex because the remainder can be positive or negative depending on the specific definition of Euclidean division used. For standard integer division results, use the Division operator; the remainder is not explicitly displayed but can be derived from the decimal result.

Yes. While the main calculator expects ternary inputs to perform math, you can use the result visualization to verify conversions. Simply enter a value (like '100') in one field, add "0", and the result section will show the decimal equivalent alongside the ternary representation.

The calculator uses arbitrary-precision integers (BigInt) in the backend. Practically, you can input very long strings of ternary digits. However, extremely large numbers may cause UI text wrapping issues, though the mathematical calculation remains precise.