Convert Base 9 to Base 7

Convert Base 9 (base 9) to Base 7 (base 7) 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 7 (base 7)

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 7 repeatedly (read remainders upward):

        42 ÷ 7 = 6  r 0
         6 ÷ 7 = 0  r 6
        Read remainders upward: 60

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Result:  46 (base 9) = 60 (base 7)

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 7

Septenary (base 7) uses digits 0-6. Each position is a power of 7: 7⁰=1, 7¹=7, 7²=49. Seven is prime, so every fraction p/q (q not divisible by 7) produces a purely repeating expansion. Seven days of the week derive from Babylonian astronomy (the 7 naked-eye celestial objects). Reed-Solomon codes used in QR codes and Blu-ray draw on prime-field properties of GF(7). Conversion: decimal 42 = 6×7+0 = 60₇.

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