Conversion from Decimal to roman number

In number systems, Roman numerals are a traditional way of representing numbers using a combination of letters from the Latin alphabet. Conversion from decimal numbers to Roman numerals is a common problem in mathematics and computer science. Roman numerals are often used in various contexts, including clock faces, numbering of book chapters, and naming of monarchs.

Problem Statement and Description

The problem is to convert a given decimal number into its Roman numeral representation. Roman numerals have specific symbols for certain values, such as "I" for 1, "V" for 5, "X" for 10, and so on. The algorithm should identify the largest Roman numeral value that is less than or equal to the given decimal number, append the corresponding Roman numeral symbol to the result, and continue this process until the entire number is represented in Roman numerals.

Example

Let's consider an example where we want to convert the decimal number 604 to a Roman numeral. The algorithm should identify the largest Roman numeral values that fit into 604 (i.e., D for 500 and C for 100), and combine them to form the Roman numeral DCIV.

Idea to Solve the Problem

To convert a decimal number to a Roman numeral, we need to identify the largest Roman numeral values that fit into the given decimal number and construct the Roman numeral representation step by step. Here's a more detailed explanation of the approach:

1. Create a Lookup for Roman Numerals: Define a lookup that associates Roman numeral symbols with their corresponding decimal values. This lookup will help us identify the appropriate Roman numeral symbols for various values.

2. Iterative Conversion: Start by considering the largest Roman numeral value (e.g., 1000, represented as "M") and check if the given decimal number is greater than or equal to that value. If it is, subtract the value from the decimal number and append the corresponding Roman numeral symbol to the result.

3. Continue with Smaller Values: Repeat this process for the remaining Roman numeral values in decreasing order (e.g., 900, 500, 400, and so on). At each step, subtract the value from the decimal number and append the corresponding symbol to the result.

4. Repeat for Smaller Symbols: After handling the larger symbols, move on to the smaller symbols (e.g., 100, 90, 50, 40, and so on). Follow the same process: check if the decimal number is greater than or equal to the value, subtract the value, and append the symbol.

5. Finish with Single-Digit Symbols: Finally, handle the single-digit Roman numeral symbols (e.g., 10, 9, 5, 4, 1) in the same way. Iterate through each value, subtract it from the decimal number, and append the corresponding symbol to the result.

6. Constructing the Roman Numeral: As you iterate through the values and subtract them from the decimal number, the result will be constructed by appending the corresponding symbols. Once the decimal number reaches 0, the conversion is complete.

Pseudocode

Here's the pseudocode for the algorithm:

function result(n):
    Display the corresponding Roman numeral symbol for n

function select(number, collection, size):
    Initialize n to 1
    For i from 0 to size-1:
        If number >= collection[i]:
            n = collection[i]
        Else:
            Break the loop
    Call result(n)
    Return number - n

function romanNo(number):
    If number <= 0:
        Return
    Define an array collection with common Roman numeral values
    Get the size of the collection
    Display the original number
    While number > 0:
        Set number to select(number, collection, size)
    Display a new line

main:
    Call romanNo for test cases

Algorithm Explanation

1. Define the result function to display the corresponding Roman numeral symbol for a given value.
2. Define the select function to identify the largest Roman numeral value that fits into the given number.
3. Define the romanNo function that takes a decimal number as input.
4. Inside the romanNo function, initialize the collection array with common Roman numeral values and get its size.
5. Display the original number.
6. Iterate through the collection using the select function to convert the number into Roman numerals.
7. Display a new line.

Code Solution

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Time Complexity

The time complexity of this algorithm depends on the number of iterations needed to convert the decimal number to Roman numerals. The number of iterations is proportional to the number of digits in the decimal number. The conversion process involves mainly arithmetic operations, comparisons, and lookups in the collection array, all of which are typically constant time operations. Therefore, the overall time complexity is reasonable for practical scenarios.