About

Converting between binary and Roman numeral systems requires a two-stage transformation: base-2 to base-10, then base-10 to additive-subtractive Roman notation. Errors in either stage cascade silently. A misread binary digit doubles or halves the decimal intermediate, producing a completely wrong Roman output. This tool computes d = parseInt(b, 2) then decomposes d using the greedy algorithm against the standard 13-symbol Roman table extended with vinculum notation for values above 3,999. The conversion handles subtractive pairs (IV, IX, XL, XC, CD, CM) per modern convention, not the archaic additive form.

Standard Roman numerals cap at 3,999 (MMMCMXCIX). This tool extends to 3,999,999 using vinculum: an overline multiplies a symbol by 1,000. For example, 5,000 is represented as V with an overline. Note: zero has no Roman representation. Binary input 0 returns an explicit error rather than a misleading empty string. Pro tip: verify your binary source uses unsigned integers. Signed binary (two's complement) produces negative decimals, which Roman numerals cannot express.

Formulas

The conversion proceeds in two deterministic stages. Stage 1 converts the binary string b of length n into its decimal equivalent d:

d = n−1∑i=0 b_i ⋅ 2ⁱ

where b_i is the i-th bit from the right (LSB at position 0).

Stage 2 applies the greedy decomposition algorithm. Given the Roman value-symbol table R sorted in descending order of value:

R = {(1000, M), (900, CM), (500, D), (400, CD), (100, C), (90, XC), (50, L), (40, XL), (10, X), (9, IX), (5, V), (4, IV), (1, I)}

For each pair (v, s) in R: while d ≥ v, append s to output and subtract d ← d − v. The algorithm terminates when d = 0. The vinculum extension multiplies each base symbol by 1,000 and renders it with an overline (e.g., V̅ = 5,000).

Reference Data

Binary	Decimal	Roman	Notes
1	1	I	Unit
10	2	II	Additive
11	3	III	Max repeat of I
100	4	IV	Subtractive pair
101	5	V	Quinary symbol
1001	9	IX	Subtractive pair
1010	10	X	Decimal symbol
101000	40	XL	Subtractive pair
110010	50	L	Half-centum
1011010	90	XC	Subtractive pair
1100100	100	C	Centum
110000100	388	CCCLXXXVIII	Longest standard < 1000
110010000	400	CD	Subtractive pair
111110100	500	D	Half-mille
1110000100	900	CM	Subtractive pair
1111101000	1000	M	Mille
11000011010	1562	MDLXII	Year example
11111010000	2000	MM	Two thousand
11111011001	2009	MMIX	Year example
11111100100	2020	MMXX	Year example
11111100101	2025	MMXXV	Current year
110000110101	3125	MMMCXXV	5⁵
111110011111	3999	MMMCMXCIX	Max standard Roman
1001110001000	5000	V̅	Vinculum (×1000)
10011100010000	10000	X̅	Vinculum
11110100001001000000	1000000	M̅	One million vinculum

Frequently Asked Questions

The Roman numeral system has no symbol for zero. It was developed before the concept of zero was adopted in Western mathematics. The decimal value 0 is outside the valid domain of the Roman numeral function, so the converter explicitly flags it rather than returning a misleading empty string.

Standard Roman numerals represent values from 1 to 3,999. Vinculum notation places an overline above a symbol to multiply its value by 1,000. For example, V̅ = 5,000 and X̅ = 10,000. This extends the representable range to 3,999,999. The converter applies vinculum automatically when the decimal intermediate exceeds 3,999.

Leading zeros do not affect the decimal value. Binary 00001010 and 1010 both equal decimal 10, which converts to X. The converter strips leading zeros during display of the decimal intermediate but accepts them in input without error.

The converter supports binary values that produce decimals up to 3,999,999 (the vinculum limit). In binary, that is 1111010000100100111111 (22 bits). Inputs producing values above this threshold trigger an explicit range error because no standard Roman representation exists beyond this point.

The subtractive notation used here follows the modern conventional form codified in most reference works and ISO recommendations. Six subtractive pairs are recognized: IV (4), IX (9), XL (40), XC (90), CD (400), and CM (900). Historical inscriptions sometimes used additive forms (IIII for 4), but this tool follows the modern convention exclusively.

No. Roman numerals have no concept of negative values. This converter treats all input as unsigned binary. If you input a two's complement negative representation (e.g., a 32-bit signed integer), the converter will interpret it as a large unsigned positive number, which may exceed the 3,999,999 limit. Verify your binary source uses unsigned encoding.