Question

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

Short answer

The Arabic number for DXIV is 514.

Step-by-step solution

Understand Roman Numerals

Roman numerals are a numeral system originating from ancient Rome, utilizing combinations of letters from the Latin alphabet: I, V, X, L, C, D, and M. Each has a specific value. D = 500, X = 10, I = 1, and V = 5.

Break Down the Numeral

Break down the numeral \(\mathrm{DXIV}\) into its individual components: \(D, X, I, ext{ and } V\) . Each part corresponds to a numerical value: \(\mathrm{D} = 500, \mathrm{X} = 10, \mathrm{I} = 1, \text{ and } \mathrm{V} = 5\).

Determine the Order of Calculation

Roman numerals are calculated from left to right. Larger numbers precede smaller ones in the numeral. If a smaller numeral precedes a larger numeral, it is subtracted from the larger number. Otherwise, the values are added from left to right.

Apply Addition and Subtraction Rules

Starting with D (500), add X (10). The next numeral is I, which precedes V, so it should be subtracted from V. Hence, 1 is subtracted from 5, resulting in 4 (IV). Thus, the numeral becomes: 500 (D) + 10 (X) + 4 (IV).

Calculate the Total

Add the calculated values to convert thoroughly:\(500 \text{ (D)} + 10 \text{ (X)} + 4 \text{ (IV)} = 514\).

Key concepts

Arabic numbers

Arabic numbers, also referred to as Hindu-Arabic numerals, are the ten digits that we use universally today. These are 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. They form the basis for the decimal numeral system, which is a positional notation system. Each digit has a value depending on its position in a number. For example, in the number 514, the 5 is in the hundreds place, indicating five hundred, the 1 is in the tens place, indicating ten, and the 4 is in the ones place. This numeral system is widely used for its simplicity and efficiency in arithmetic calculations. It allows for easy operations like addition, subtraction, multiplication, and division. Compare this to Roman numerals, which do not have a zero and use different principles for combining numbers.

numeral system

A numeral system is a method or convention to represent numbers in a consistent manner. Ancient cultures developed numeral systems that suited their needs for counting and calculations. Roman numerals, for example, stem from ancient Rome. It uses combinations of Latin letters: I, V, X, L, C, D, and M, each assigned a specific value. Meanwhile, the Hindu-Arabic numeral system, used internationally today, is a base-10 or decimal system. This is due to its easy usability in performing mathematical operations.
Every numeral system has its own rules for interpreting and using symbols. With Roman numerals, understanding the value each letter represents, such as D for 500, X for 10, is key. By contrast, the Arabic numeral system is a placeholder-based system, meaning each digit has a value based on its position in the number.

addition and subtraction rules

The addition and subtraction rules are crucial when it comes to converting from Roman numerals to Arabic numbers. These rules dictate how values are combined to form the number represented by a Roman numeral.
1. **Addition**: If a Roman numeral has letters arranged from largest to smallest (from left to right), you add their values. For example, in DX (500 + 10), both values are simply added together resulting in 510.
2. **Subtraction**: If a smaller value precedes a larger one, you should subtract the smaller number. Take, for instance, IV. Here, I (1) precedes V (5), so you subtract 1 from 5, giving an outcome of 4.
These fundamental rules help in decoding Roman numerals accurately and efficiently. Remember, without these rules, confusion can arise since not all sequences of letters represent straightforward additions.

step-by-step conversion

Step-by-step conversion from Roman numerals to Arabic numbers involves breaking down the numeral and applying the addition and subtraction rules methodically. Let's take the Roman numeral DXIV as an example.

• Step 1: Break down the numeral into its components, identifying each letter's value. D = 500, X = 10, I = 1, V = 5.
• Step 2: Apply the addition and subtraction rules. Start with the leftmost character. Add D (500) to X (10), as they are both larger numbers arranged from left to right.
• Step 3: Identify the smaller numeral I before V. Subtract I (1) from V (5), resulting in IV, which equals 4.
• Step 4: Add the calculated values together: 500 (D) + 10 (X) + 4 (IV).
• Step 5: Sum these values to find the Arabic number, which is 514.

This systematic approach ensures accuracy and a clear understanding of how Roman numerals translate into the format we use every day.