About

Positional numeral systems encode integer values using a fixed set of digits weighted by powers of a radix r. Converting between bases requires decomposing the source representation into its canonical value, then re-encoding under the target radix. Errors in manual conversion compound quickly: a single misidentified digit in base-16 can shift the result by 16^k, where k is the digit position. This tool performs exact arbitrary-precision conversion for any integer across bases 2 through 36, using the standard alphanumeric digit set (0 - 9, A - Z). It handles negative values and numbers of arbitrary length without floating-point truncation.

Note: this tool operates on integers only. Fractional base conversion introduces repeating-digit edge cases that require separate treatment. Results are exact; no rounding or approximation is applied.

Formulas

An integer N represented in base r with digits d_k has the canonical value:

N = n−1∑k=0 d_k ⋅ r^k

To convert from base r₁ to base r₂, first evaluate the sum above to obtain the canonical integer, then extract digits in base r₂ via repeated division:

d_k = N mod r₂ , N ← Nr₂

Digits are collected least-significant first and reversed to produce the final representation.

Where: N = integer value, r = radix (base), d_k = digit at position k, n = total number of digits.

Reference Data

Base	Name	Digits Used	Common Use	Example: 255₁₀
2	Binary	0-1	Digital logic, CPU instructions	11111111
3	Ternary	0-2	Balanced ternary computing	100110
4	Quaternary	0-3	DNA nucleotide encoding	3333
5	Quinary	0-4	Tally systems	2010
6	Senary	0-5	Dice arithmetic	1103
7	Septenary	0-6	Week-day calculations	513
8	Octal	0-7	Unix file permissions, PDP-11	377
10	Decimal	0-9	Everyday arithmetic	255
12	Duodecimal	0-9, A - B	Time (12 hours), imperial units	193
16	Hexadecimal	0-9, A - F	Memory addresses, CSS colors	FF
20	Vigesimal	0-9, A - J	Maya and Aztec numeral systems	CF
32	Duotrigesimal	0-9, A - V	Crockford Base32 encoding	7V
36	Hexatrigesimal	0-9, A - Z	URL shorteners, compact IDs	73

Frequently Asked Questions

JavaScript Number uses IEEE-754 double-precision floats, which can only represent integers exactly up to 2⁵³ − 1 (about 9 × 10¹⁵). Beyond that, digits are silently lost. BigInt has no upper bound, so a 256-bit hexadecimal value converts without any precision loss.

The standard convention uses Latin letters: A = 10, B = 11, … Z = 35. This covers every base up to 36. Input is case-insensitive; the tool normalises output to uppercase.

The sign is stripped before conversion and reattached to the output. This tool does not use two's complement or sign-magnitude encoding. If you need a two's complement representation, convert the absolute value and apply the complement manually for your target bit width.

No. Fractional base conversion can produce infinitely repeating digit sequences (analogous to 1/3 = 0.333… in decimal). This tool is restricted to integers to guarantee exact, finite results.

Because 16 = 2⁴, each hex digit maps to exactly 4 binary digits, allowing a simple grouping shortcut. Internally, however, this tool always goes through the canonical BigInt intermediate for correctness across all base pairs, and the operation is O(n) on digit count, so it completes in microseconds regardless.

There is no hardcoded limit. The practical limit is your browser's BigInt implementation. Modern browsers handle integers with thousands of digits without issue. If you paste a value exceeding ~50,000 digits, you may notice a brief delay.