Question

Roman numbers. Write a program that converts a positive integer into the Roman number system. The Roman number system has digits \(\begin{array}{ll}\text { I } & 1 \\ \text { V } & 5 \\ \text { X } & 10 \\\ \text { L. } & 50 \\ \text { C } & 100 \\ \text { D } & 500 \\ \text { M } & 1,000\end{array}\) Numbers are formed according to the following rules: a. Only numbers up to 3,999 are represented. b. As in the decimal system, the thousands, hundreds, tens, and ones are expressed separately. c. The numbers 1 to 9 are expressed \(\begin{array}{ll}\text { I } & \text { 1 } \\ \text { II } & 2 \\ \text { III } & 3 \\ \text { IV } & 4 \\ \text { V } & 5 \\ \text { VI } & 6 \\\ \text { VII } & 7 \\ \text { VIII } & 8 \\ \text { IX } & 9\end{array}\) As you can see, an I preceding a \(V\) or \(X\) is subtracted from the valuc, and you can never have more than three I's in a row. d. Tens and hundreds are done the same way, except that the letters \(\mathrm{X}, \mathrm{L}, \mathrm{C}\) and \(\mathrm{C}\), \(\mathrm{D}, \mathrm{M}\) are used instead of \(\mathrm{I}, \mathrm{V}, \mathrm{X}\), respectively. Your program should take an input, such as 1978 , and convert it to Roman numerals, MCMLXXVIII.

Short answer

Convert 1978 to MCMLXXVIII using descending value pairs.

Step-by-step solution

Understand the Roman Numerals Mapping

Each Roman numeral has a corresponding value: I=1, V=5, X=10, L=50, C=100, D=500, M=1000. Additionally, certain combinations subtract: IV=4, IX=9, XL=40, XC=90, CD=400, CM=900.

Set Up Value Pairs

Create a list of tuples to hold Roman numeral symbols in descending value pairs: [(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')]. This helps to determine which numerals to use when converting.

Start Conversion Process

Initialize an empty string to build the Roman numeral result. Use a loop to iterate through the list of value pairs from highest to lowest.

Check and Append Roman Numerals

For each value in the list, check how many times it can fit into the current integer. While the integer is greater than or equal to the current value, subtract the value from the integer and append the corresponding numeral character to the result string.

Compile the Result

Once the entire integer has been converted, exit the loop. The result string should contain the correct Roman numeral representation.

Key concepts

Integer Algorithms

When tackling a problem like converting integers to Roman numerals, understanding integer algorithms is essential. These algorithms provide step-by-step instructions on how to perform calculations with whole numbers. In this case, the goal is to repeatedly subtract values from a given integer and use Roman numeral symbols to reflect those deductions.

• First, identify the repeated subtraction process, which ensures we appropriately use each Roman numeral value.
• By arranging Roman numeral values from largest to smallest, follow an algorithm that deducts these values sequentially.
• This creates a loop within the algorithm, which checks if the current highest Roman numeral can be subtracted from the integer.
• The program then appends the corresponding numeral to a result string and continues the conversion process with the new integer value.

Utilizing integer algorithms allows for efficient conversions and helps automate what could otherwise be a cumbersome manual task.

Historical Number Systems

Roman numerals are a fascinating aspect of historical number systems, rooted in ancient Roman civilization. Unlike the base-ten Arabic numbers that most of us are familiar with today, Roman numerals use combinations of Latin letters to form numbers.

• The Romans typically combined these letters to indicate numbers, using addition and sometimes subtraction for these purposes.
• For example, the numeral "IX" uses subtraction to represent 9 (10 - 1), while "XII" uses addition to denote 12 (10 + 1 + 1).
• This system imposes certain limitations, such as only extending to 3,999 and using specific rules to prevent more than three consecutive identical letters.

Despite its limitations, the Roman numeral system is still widely used in specific contexts, such as for marking hours on a clock face or numbering chapters in books, due to its historical significance and aesthetic appeal.

Basic Programming Logic

Basic programming logic is crucial for implementing a solution to convert integers to Roman numerals. The process uses logical thinking to break down the conversion tasks systematically.

• First, understand the sequence of operations necessary to transform a decimal number into Roman numerals based on pre-defined rules.
• A loop constructs a sequence of logical decisions, where the program repeatedly performs actions based on specific conditions.
• If-else statements are used extensively to decide when a particular Roman numeral will be appended to the output string.
• Finally, knowing when to stop these iterations and compile the final result is equally important, ensuring the program outputs the correct Roman numeral.

This logical framework, combining loops, conditionals, and string manipulations, allows for a functional and efficient program capable of converting any integer up to 3,999 into Roman numerals accurately.