Convert Base 3 to Base 6

Convert Base 3 (base 3) to Base 6 (base 6) instantly and accurately.

Selected Base 3 (base 3) - available characters: 0, 1, 2

Conversion Formula

Step-by-step example using the value 42 (decimal):

Convert Base 3 (base 3) → Base 6 (base 6)

Step 1: Expand each digit of 1120 (base 3) by position:

        1 × 3^3 = 27
        1 × 3^2 =  9
        2 × 3^1 =  6
        0 × 3^0 =  0
        ────────────
        Sum = 42 (decimal)

Step 2: Divide 42 by 6 repeatedly (read remainders upward):

        42 ÷ 6 = 7  r 0
         7 ÷ 6 = 1  r 1
         1 ÷ 6 = 0  r 1
        Read remainders upward: 110

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Result:  1120 (base 3) = 110 (base 6)

About Base 3

Ternary (base 3) uses digits 0, 1, 2. Each position is a power of 3: 3⁰=1, 3¹=3, 3²=9, 3³=27. Balanced ternary (−1, 0, +1) represents negatives without a sign and minimises rounding error. The Soviet Setun computer (1958) was the only practical balanced-ternary machine. Ternary underlies the Cantor set and Toom-Cook multiplication. Conversion: decimal 42 = 1×27+1×9+2×3+0 = 1120₃.

About Base 6

Senary (base 6) uses digits 0-5. Each position is a power of 6: 6⁰=1, 6¹=6, 6²=36. Six is divisible by 1, 2, 3, and 6, making fractions clean: 1/2 = 0.3, 1/3 = 0.2 (both terminating). Native to the Ndom language of Papua New Guinea. Standard dice produce one uniform senary digit per roll. Donald Knuth noted senary's compact multiplication table. Conversion: decimal 42 = 1×36+1×6+0 = 110₆.

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