Roman Numeral Converter

Formula and variables

n₀ = n; v_j = max{v ∈ V ∶ v ≤ n_{j−1}}; n_j = n_{j−1} − v_j (j = 1,…,m); n_m = 0; R = s₁s₂⋯s_m; N = Σ_{k=1}^{ℓ} q_k v_k; q_k = {−1, v_k < v_{k+1}; 1, v_k ≥ v_{k+1}}; v_{ℓ+1} = 0

n

Decimal integer 1–3999

V

Table value set {1,4,5,9,10,40,50,90,100,400,500,900,1000}

j

Encoding step, 1 ≤ j ≤ m

v_j

Largest value in V with v_j ≤ n_{j−1}

s_j

Table symbol paired with v_j (I, IV, V, …, M)

n_j

Remainder after step j, n_j = n_{j−1} − v_j

m

Number of symbols in R; n_m = 0

R

Roman numeral: concatenation s₁s₂⋯s_m

k

Position in the input numeral, 1 ≤ k ≤ ℓ

ℓ

Number of input symbols

v_k

Value of the kth input symbol I,V,X,L,C,D,M

q_k

−1 before a larger value, otherwise +1

v_{ℓ+1}

Sentinel 0, so the last symbol has q_ℓ = 1

N

Decoded number

Canonical Roman numerals only, from I to MMMCMXCIX (1–3999); invalid forms such as IIII and IC are rejected.