Convert Base 9 to Base 3

Convert Base 9 (base 9) to Base 3 (base 3) instantly and accurately.

Selected Base 9 (base 9) - available characters: 0, 1, 2, 3, 4, 5, 6, 7, 8

Conversion Formula

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

Convert Base 9 (base 9) → Base 3 (base 3)

Step 1: Expand each digit of 46 (base 9) by position:

        4 × 9^1 = 36
        6 × 9^0 =  6
        ────────────
        Sum = 42 (decimal)

Step 2: Divide 42 by 3 repeatedly (read remainders upward):

        42 ÷ 3 = 14  r 0
        14 ÷ 3 =  4  r 2
         4 ÷ 3 =  1  r 1
         1 ÷ 3 =  0  r 1
        Read remainders upward: 1120

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Result:  46 (base 9) = 1120 (base 3)

About Base 9

Nonary (base 9) uses digits 0-8. Each position is a power of 9: 9⁰=1, 9¹=9, 9²=81. Because 9 = 3², each nonary digit maps to exactly 2 ternary digits (0=00 … 8=22). The finite field GF(9) = GF(3²) is used in certain error-correcting codes and algebraic geometry. The 3×3 magic square sums to 15 in every row/column/diagonal and is deeply tied to mod-9 arithmetic. Conversion: decimal 42 = 4×9+6 = 46₉.

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.

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