Convert Base 4 to Base 7

Convert Base 4 (base 4) to Base 7 (base 7) instantly and accurately.

Selected Base 4 (base 4) - available characters: 0, 1, 2, 3

Conversion Formula

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

Convert Base 4 (base 4) → Base 7 (base 7)

Step 1: Expand each digit of 222 (base 4) by position:

        2 × 4^2 = 32
        2 × 4^1 =  8
        2 × 4^0 =  2
        ────────────
        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

────────────────────────────
Result:  222 (base 4) = 60 (base 7)

About Base 4

Quaternary (base 4) uses digits 0-3. Each position is a power of 4: 4⁰=1, 4¹=4, 4²=16. Because 4 = 2², each quaternary digit maps to exactly 2 bits (0=00, 1=01, 2=10, 3=11). Used in bioinformatics to encode DNA nucleotides (A, C, G, T = 0-3), storing the human genome (3.2 Gbp) in ≈750 MB. 4-QAM encodes exactly 2 bits per symbol. Conversion: decimal 42 = 2×16+2×4+2 = 222₄.

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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