Question

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

Short answer

The Arabic number is 19,000.

Step-by-step solution

Understand Roman numeral bars

A bar over a Roman numeral multiplies that numeral by 1,000. Here, the bar is over \overline{XIX}, so this number is \(XIX \times 1,000\). We'll interpret \overline{XIX} as \(XIX \times 1,000\).

Convert XIX to Arabic numerals

Now, convert the Roman numeral \(XIX\) to Arabic numerals. Each letter represents a value: \(X = 10\), \(I = 1\), \(V = 5\). So, \(X + (X - I) = 10 + (10 - 1) = 19.\)

Multiply by 1,000

Multiply the Arabic number obtained from \(XIX\) (which is 19) by 1,000 due to the bar over it. \(19 \times 1,000 = 19,000.\)

Key concepts

Understanding Arabic Numbers

Arabic numbers, or Arabic numerals as they're often called, are the digits 0 through 9 that are commonly used today across the world for most mathematical procedures. These digits form the basis of the decimal system, which is the most widely used numeral system today. Whenever you do simple counting, write your age, or even tally your scores in a game, you are using Arabic numbers. They provide a clean and efficient way to represent numbers and perform arithmetic operations easily. One of the key reasons Arabic numbers gained popularity is due to their simplicity and the ease to perform calculations. Each digit has a place value, allowing for a clear representation of both whole numbers and decimals. This system is a massive step up in terms of efficiency and usability compared to many other numeral systems like the Roman numeral system.

Numeral Conversion: Roman to Arabic

Converting Roman numerals to Arabic numbers might seem tricky at first, but once you understand the basic rules, it becomes straightforward. Roman numerals use combinations of letters from the Latin alphabet: I, V, X, L, C, D, and M.Here's how each letter is interpreted:

• I = 1
• V = 5
• X = 10
• L = 50
• C = 100
• D = 500
• M = 1,000

To convert a Roman numeral to an Arabic number, you read from left to right. If a smaller numeral comes before a larger one, you subtract the smaller from the larger. For instance, XIX is read as 10 (X) plus 9 (IX) because I before X means 1 subtracted from 10.In the exercise, the Roman numeral \(\overline{\mathrm{XIX}}\), indicates XVIII with a bar over it, meaning you first convert XVIII to 19, then multiply by 1,000.

The Step-by-Step Solution Process

Let's dive into the step-by-step approach to convert \(\overline{\mathrm{XIX}}\) to Arabic numerals, which ensures accuracy and understanding. First, understand the role of the bar. A bar placed over Roman numerals indicates multiplication by 1,000, which is a key rule to remember. This transforms our task from converting \(\mathrm{XIX}\) to converting \(\overline{\mathrm{XIX}} \) into something much larger. Next, focus on what's underneath the bar. The Roman numeral XIX consists of three parts: X, I, and another X. You interpret this as 10 (X) plus 9 (IX). Using the rule that a smaller numeral before a larger one means subtraction, XIX equals 19.Finally, apply the bar's effect. With \(\mathrm{XIX}\) equaling 19, the bar over it means we scale up by 1,000. So, you multiply 19 by 1,000, resulting in 19,000.Each step logically follows from the previous one, ensuring that even someone new to numeral conversion can follow along and perform similar conversions successfully.