Question

Write the following Roman numerals as Arabic numbers. \(\mathrm{LXXV}\) ______

Short answer

The Roman numeral LXXV is 75.

Step-by-step solution

Understand Roman Numerals

Roman numerals are made up of the letters I, V, X, L, C, D, and M, where each letter represents a different value. The letter L represents 50, X represents 10, and V represents 5.

Break Down the Roman Numeral

The Roman numeral LXXV consists of L, X, X, and V. Each has the value of L=50, X=10, X=10, and V=5.

Add the Values Together

Since Roman numerals are generally additive (except in some cases), you simply add up the values: 50 for L, 10 for the first X, 10 for the second X, and 5 for V. The sum is calculated as: \[50 + 10 + 10 + 5\]

Calculate the Total

Perform the addition based on the previous step: \[50 + 10 + 10 + 5 = 75\]

Write the Arabic Number

Thus, the Roman numeral \(\mathrm{LXXV}\) corresponds to the Arabic number 75.

Key concepts

Arabic numbers

Arabic numbers are the most common numeral system in use today. They are the familiar digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. These numbers are known as 'Arabic' because they were introduced to Europe in the 10th century through scholars who translated Arabic mathematics texts. Their simplicity and usability make them ideal for performing mathematical operations like addition, subtraction, multiplication, and division. In contrast to Roman numerals, Arabic numbers utilize a positional system, meaning the position of a digit affects its value. This system provides a base-10 framework, where each position in a number represents a power of ten.

Additive Roman numerals

The Roman numeral system is primarily additive, meaning that when reading or writing Roman numerals, each symbol is added together to form the total value. This feature is apparent in the example of \(\mathrm{LXXV}\). Roman numerals are constituted by combining the values of each character:

• L = 50
• X = 10
• X = 10
• V = 5

In additive situations, like in our example, you directly sum the values. For \(\mathrm{LXXV}\), this is \[50 + 10 + 10 + 5 = 75\]. However, note there are exceptions where Roman numerals can also be subtractive; when a smaller numeral appears before a larger one, it is subtracted (e.g., IV for 4). The additive nature of most Roman numeral sequences simplifies their conversion to Arabic numbers, as it typically involves straightforward addition.

Roman numeral values

Each Roman numeral consists of specific letters that symbolize fixed values. Understanding these helps in the conversion to Arabic numbers. Here are the most commonly used Roman numerals:

• I = 1
• V = 5
• X = 10
• L = 50
• C = 100
• D = 500
• M = 1000

Roman numerals don’t use a symbol for zero and are less intuitive for complex arithmetic operations compared to Arabic numbers.In our exercise, the Roman numeral \(\mathrm{LXXV}\) features L (50), X (10), and V (5). When converting Roman numerals, recognizing these key values and their arrangements enable easy calculations. Keep in mind that Roman numerals are structured in descending order, so the values simply add up unless smaller numerals precede larger ones, which would indicate a subtractive situation.

Mathematical conversion

Converting Roman numerals to Arabic numbers involves a few simple mathematical steps. Recognizing the system of Roman numerals, their values, and the patterns whether additive or subtractive, is crucial. **Basic Steps of Conversion:***

• Identify each numeral and its value.
• Check the order for possible subtractive pairs (e.g., IX for 9).
• Add or subtract based on identified values.

In our specific example with \(\mathrm{LXXV}\):1. Break the numeral into individual components: L, X, X, and V.2. Translate each into numerical values: L=50, X=10, X=10, V=5.3. Sum these values since they are in descending order: \[50 + 10 + 10 + 5 = 75\]By following these steps, converting any Roman numeral to its Arabic equivalent becomes manageable, ensuring accuracy in results.