Convert each of the following from Roman Numerals to Hindu-Arabic numerals
a. CXXVIII
b. XCDCCCXXVII
The correct answer and explanation is:
Correct Answers:
a. CXXVIII = 128
b. XCDCCCXXVII is not a valid Roman numeral due to incorrect symbol arrangement.
Explanation:
Roman numerals are based on combinations of letters from the Latin alphabet: I (1), V (5), X (10), L (50), C (100), D (500), and M (1000). To convert Roman numerals to Hindu-Arabic numerals, each letter is translated into its value, and values are added or subtracted based on their order.
Part a: CXXVIII
- C = 100
- X = 10 (second X is also 10)
- V = 5
- III = 3 (I + I + I)
Now add them:
100 + 10 + 10 + 5 + 3 = 128
This is a valid Roman numeral and converts directly to 128.
Part b: XCDCCCXXVII
This string is not a properly structured Roman numeral. Let us analyze why:
- X = 10
- CD = 400 (C before D means subtract 100 from 500)
- CCC = 300
- XX = 20
- VII = 7
Adding all these:
10 + 400 + 300 + 20 + 7 = 737
But this is incorrect Roman syntax. After CD, which is 400, you should not place CCC, because CD already covers the 400 range. Repeating hundreds after a subtractive hundred violates the Roman numeral rules. Roman numerals must be arranged from largest to smallest unless using a subtraction rule, and a number like CDCCC is not allowed.
Correct Roman for 737 is DCCXXXVII.
Therefore, XCDCCCXXVII is not valid. If someone intended to write 737, they should have written DCCXXXVII.
Summary:
- a. CXXVIII = 128
- b. XCDCCCXXVII = Invalid Roman numeral due to incorrect structure.