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How Does Base Conversion Work?

Any base-b number system uses only the digits 0 through b−1, with each position worth a power of b — the same idea as decimal, just with a different number of symbols. Converting from decimal into another base uses the division-remainder method: repeatedly divide the number by the target base and record each remainder, then read the remainders in reverse order. Converting decimal 100 to base 7: 100÷7 = 14 remainder 2, 14÷7 = 2 remainder 0, 2÷7 = 0 remainder 2 — reading the remainders bottom-up gives 202 in base 7.

Converting from any base into decimal uses the reverse idea, the positional-value-sum method: multiply each digit by the base raised to its position's power (counting from 0 on the right) and add the results. So base-7 202 converts back with 2×7² + 0×7¹ + 2×7⁰ = 98 + 0 + 2 = 100. This tool runs both directions simultaneously between any two bases from 2 to 36, using big-integer arithmetic so results stay exact even for very long numbers — no 15-digit rounding like a pocket calculator.

The same two methods generalize to any pair of non-decimal bases too: converting directly from, say, base 5 to base 12 is usually done by first landing on decimal as an intermediate step (base 5 → decimal via positional-value-sum, then decimal → base 12 via division-remainder), which is exactly what this tool automates behind the scenes for any of the 35 possible base combinations from 2 to 36.

What to Know

• Letters extend the digit set past 10: once a base needs more than 10 symbols, it borrows letters where digits run out — A=10, B=11, up to Z=35 for base 36, the largest base this tool supports. Input is case-insensitive.
• Hex is popular because it maps cleanly to binary: each hex digit corresponds to exactly 4 bits, so converting binary↔hex is a simple lookup per group of 4 digits, unlike binary↔decimal, which requires full positional arithmetic.
• Octal (base 8) groups binary by 3 bits: it was common in early computing (Unix file permissions still use it, e.g. 755) because 3 binary digits map to exactly one octal digit.
• Leading zeros and sign: this converter works with unsigned magnitudes — a negative number should be entered with an explicit minus sign, since base systems above 10 have no reserved character of their own for it.
• Bases below 2 or above 36 do not exist here on purpose: base 1 (unary) has no meaningful positional digits, and beyond 36 there are no more case-insensitive single-character symbols left in the Latin alphabet plus digits to represent higher digit values unambiguously.