About

Radix conversion between base-2 and base-3 is a non-trivial operation because 3 is not a power of 2. You cannot simply regroup digits. Each conversion requires full positional arithmetic: the source string must be evaluated as N = k∑i=0 d_i ⋅ bⁱ, then decomposed by repeated division in the target base. A single misread bit in binary propagates unpredictably through every ternary digit. This tool uses native BigInt arithmetic to handle inputs of thousands of digits without the precision ceiling of IEEE 754 double-precision floats, which silently corrupt integers beyond 2⁵³. It approximates nothing. Every digit is exact.

Formulas

A number N represented in base b₁ as digits d_kd_k−1…d₁d₀ has the decimal value:

N = k∑i=0 d_i × b₁ⁱ

To convert N into base b₂, perform repeated Euclidean division:

N = b₂ × q + r, 0 ≤ r < b₂

The remainders r, read in reverse order of computation, form the digits in base b₂. For binary (b₁ = 2) to ternary (b₂ = 3), the intermediate value N is computed via Horner's method to avoid explicit exponentiation:

N = d_k × 2^k + … + d₁ × 2 + d₀

Where d_i ∈ {0, 1} for binary and r ∈ {0, 1, 2} for ternary output. The digit count in the target base is floor(log_b₂(N)) + 1. This tool uses BigInt internally, so precision is unlimited.

Reference Data

Decimal	Binary (Base-2)	Ternary (Base-3)	Octal (Base-8)	Hex (Base-16)
0	0	0	0	0
1	1	1	1	1
2	10	2	2	2
3	11	10	3	3
4	100	11	4	4
5	101	12	5	5
6	110	20	6	6
7	111	21	7	7
8	1000	22	10	8
9	1001	100	11	9
10	1010	101	12	A
15	1111	120	17	F
16	10000	121	20	10
20	10100	202	24	14
27	11011	1000	33	1B
32	100000	1012	40	20
42	101010	1120	52	2A
64	1000000	2101	100	40
81	1010001	10000	121	51
100	1100100	10201	144	64
128	10000000	11202	200	80
243	11110011	100000	363	F3
255	11111111	100110	377	FF
256	100000000	100111	400	100
512	1000000000	200222	1000	200
729	1011011001	1000000	1331	2D9
1000	1111101000	1101001	1750	3E8
1024	10000000000	1102011	2000	400

Frequently Asked Questions

Binary to octal works by grouping 3 binary digits because 8 = 2³. Ternary's base (3) is not a power of 2, so no fixed-width grouping exists. Every binary digit influences every ternary digit through carry propagation. Full positional evaluation is required.

The converter accepts up to 10,000 digits. It uses JavaScript's native BigInt type, which handles arbitrary-precision integers. A 10,000-bit binary number is approximately 3,010 decimal digits - well within BigInt's capability on modern browsers.

The ratio of digit counts is log(2)/log(3) ≈ 0.6309. A 100-digit binary number produces roughly 63 ternary digits. Conversely, a 100-digit ternary number requires approximately 159 binary digits. The ternary representation is always shorter.

Leading zeros are stripped during conversion. The input "00101" is treated identically to "101". The output never contains leading zeros unless the value is zero itself, which is represented as a single "0".

Ternary appears in balanced ternary computing (Soviet Setun computer, 1958), in information-theoretic optimal radix analysis (e ≈ 2.718, closest integer is 3), in redundant number systems for fast carry-free addition, and in some error-correcting codes. It is also used in combinatorial puzzles like the balance scale problem.

The converter validates every character against the expected alphabet: only "0" and "1" for binary mode, or "0", "1", and "2" for ternary mode. Any invalid character triggers an immediate error notification, and no conversion is attempted.