Convert Base 9 to Binary

Convert Base 9 (base 9) to Binary (bin) 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) → Binary (base 2)

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 2 repeatedly (read remainders upward):

        42 ÷ 2 = 21  r 0
        21 ÷ 2 = 10  r 1
        10 ÷ 2 =  5  r 0
         5 ÷ 2 =  2  r 1
         2 ÷ 2 =  1  r 0
         1 ÷ 2 =  0  r 1
        Read remainders upward: 101010

────────────────────────────
Result:  46 (base 9) = 101010 (bin)

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 Binary

Binary (base 2) is the foundational numeral system of all digital electronics, using only 0 and 1. Each position represents a power of 2: 2⁰=1, 2¹=2, 2²=4, 2³=8. Leibniz formalised it in 1679; Shannon applied Boolean logic to circuits in 1948. 8 bits = 1 byte (256 values) - the fundamental unit in every CPU from the 8080 to ARM Cortex-X. Every integer, float, character, pixel, and instruction in any digital device is ultimately binary. Conversion: decimal 42 = 32+8+2 = 2⁵+2³+2¹ = 101010₂.

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