How to convert decimal numbers to Roman numerals

Roman numerals, a number system originating from ancient Rome, are still widely used today in various contexts, such as clock faces, book chapters, and historical references. Converting decimal numbers to Roman numerals involves translating a base-10 number into its equivalent in this classical numeral system. This conversion process is not only a fascinating exercise in understanding ancient numeral systems but also a valuable skill in applications where Roman numerals are preferred for their aesthetic or traditional value.

Problem statement

Given a decimal number, the task is to convert it to Roman numerals using a systematic algorithmSystematic algorithm refers to a well-defined and structured set of rules or steps for converting a decimal number to Roman numerals. It implies a methodical and organized approach, ensuring consistency and clarity in the conversion process..

Representation

Symbols in Roman numerals are formed by combining specific base values. For example, “M” indicates 1000, whereas “CM” is a combination of “C” (100) and “M,” resulting in 900. Similarly, “XC” reflects 90, with “X” (10) subtracted from “C.”

Note: To explore further details about Roman numerals, please refer to the Roman to integer problem.

The following table shows the important Roman numeral symbols for the suitable values:

Algorithm

Here’s the algorithm for converting decimal values to Roman numerals:

Begin with the provided decimal number.
Determine the largest base value that is less than or equal to the current number.
Divide this number by the determined base value.
Repeat the corresponding Roman numeral symbol in the result quotient times.
For subsequent division and repetitions, replace the number with the remainder.
Continue this process until the number reaches zero.

Decimal to Roman numeral conversion process

The tables below demonstrate the step-by-step conversion of decimal numbers to their Roman numeral equivalents. Each row represents a key step in the algorithm, such as identifying the largest base value, computing the quotient and remainder, determining the Roman numeral symbol, and incrementally updating the result list.

Note: The "Number" column represents the value utilized for further division in each iteration, providing a more concise and informative representation of the conversion process.

Example 1: Convert 789 to Roman numerals

#include <iostream>
using namespace std;
string convertToRoman(int decimal) {
    string symbols[] = {"M", "CM", "D", "CD", "C", "XC", "L", "XL", "X", "IX", "V", "IV", "I"};
    int values[] = {1000, 900, 500, 400, 100, 90, 50, 40, 10, 9, 5, 4, 1};
    int remaining = decimal;
    string result = "";
    for (size_t i = 0; i < sizeof(symbols) / sizeof(symbols[0]); ++i) {
        while (remaining >= values[i]) {
            result += symbols[i]; // Append the current symbol to the result
            remaining -= values[i]; // Subtract the current value from the remaining
        }
    }
    return result;
}
int main() {
    int decimalNumber = 789;
    // Display the original decimal number
    cout << "C++ code" << endl;
    cout << "Decimal Number: " << decimalNumber << endl;
    // Convert the decimal number to Roman numerals
    string romanNumeral = convertToRoman(decimalNumber);
    // Display the Roman numeral representation
    cout << "Roman Numeral: " << romanNumeral << endl;
    return 0;
}

C++ code: Convert decimal to Roman numerals

Code explanation

This code implements a common approach for converting decimal values to Roman numerals across several programming languages:

Function definition: The convertToRoman function accepts an integer decimal as input and returns a string with the matching Roman numeral.
Symbol and value definitions: The function initializes two lists: symbols which contain Roman numeral symbols and values which have the matching decimal values.
Initialization: Variables such as remaining (initialized with the input decimal) and result (initialized as an empty string) are set up.
Iteration through symbols and values: A loop traverses the symbols and values, identifying the largest base value that is less than or equal to the remaining decimal.
Symbol repetition: Within the loop, the code repeats the current symbol while the remaining value is greater or equal to the current value. The symbol is appended to the result string.
Value subtraction: The code subtracts the current value from the remaining, updating it for future iterations.
Result string building: The loop continues until the remaining value becomes zero. The result string accumulates the repeated symbols.
Return result: The function returns the final result string, representing the Roman numeral equivalent of the input decimal.
Main function: The main function demonstrates the usage of the conversion function by providing an example decimal number (e.g., 789).

Complexity

Time complexity: The time complexity of the algorithm is $O(log N)$ , where $N$ is the input decimal number. The loop iterates through the base values, reducing the decimal number with each iteration.
Space complexity: The space complexity is $O(1)$ because the algorithm uses only a constant amount of extra space for variables, making it space-efficient regardless of the input size.

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Iteration	Number	Largest Base Value	Quotient	Remainder	Roman Numeral	Updated Result List
1	789	500	1	289	D	D
2	289	100	2	89	C	DC
3	89	50	1	39	L	DCL
4	39	10	3	9	X	DCLXXX
5	9	9	1	0	IX	DCLXXXIX

Iteration	Number	Largest Base Value	Quotient	Remainder	Roman Numeral	Updated Result List
1	1492	1000	1	492	M	M
2	492	400	1	92	CD	MCD
3	92	90	1	2	XC	MCDXC
4	2	1	2	0	II	MCDXCII