Convert Base 3 to Base 4

Convert Base 3 (base 3) to Base 4 (base 4) 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 4 (base 4)

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 4 repeatedly (read remainders upward):

        42 ÷ 4 = 10  r 2
        10 ÷ 4 =  2  r 2
         2 ÷ 4 =  0  r 2
        Read remainders upward: 222

────────────────────────────
Result:  1120 (base 3) = 222 (base 4)

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 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.

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