Convert Base 3 to Decimal

Convert Base 3 (base 3) to Decimal (dec) 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) → Decimal (base 10)

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: Target is decimal - Step 1 result is the final answer.

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Result:  1120 (base 3) = 42 (dec)

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 Decimal

Decimal (base 10) is the universal positional system using digits 0-9. Each position is a power of 10: 10⁰=1, 10¹=10, 10²=100. Attributed to human anatomy (ten fingers). The Hindu-Arabic system - including zero - was formalised by Brahmagupta (628 CE) and spread via al-Khwarizmi (c. 820 CE). SI and metric systems are base-10. IEEE 754-2008 added decimal64/128 formats for exact decimal arithmetic in financial applications. Conversion: decimal 42 is the reference value used throughout this converter.

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