The rules for Roman numerals lead to a number of interesting observations:
1. There is only one correct way to represent a particular number as Roman numerals.
2. The converse is also true: if a string of characters is a valid Roman numeral, it represents only one number
(i.e. it can only be read one way).
3. There is a limited range of numbers that can be expressed as Roman numerals, specifically 1 through 3999.
(The Romans did have several ways of expressing larger numbers, for instance by having a bar over a numeral
to represent that its normal value should be multiplied by 1000, but you're not going to deal with that. For the
purposes of this chapter, let's stipulate that Roman numerals go from 1 to 3999.)
4. There is no way to represent 0 in Roman numerals. (Amazingly, the ancient Romans had no concept of 0 as a
number. Numbers were for counting things you had; how can you count what you don't have?)
5. There is no way to represent negative numbers in Roman numerals.
6. There is no way to represent fractions or non−integer numbers in Roman numerals.
There are several simple rules for constructing a Roman
numeral, using the letters M, D, C, L, X, V, and I. Let's review the rules:
1. Characters are additive. I is 1, II is 2, and III is 3. VI is 6 (literally, "5 and 1"), VII is 7, and VIII is 8.
2. The tens characters (I, X, C, and M) can be repeated up to three times. At 4, you need to subtract from the next
highest fives character. You can't represent 4 as IIII; instead, it is represented as IV ("1 less than 5"). 40 is
written as XL ("10 less than 50"), 41 as XLI, 42 as XLII, 43 as XLIII, and then 44 as XLIV ("10 less
than 50, then 1 less than 5").
3. Similarly, at 9, you need to subtract from the next highest tens character: 8 is VIII, but 9 is IX ("1 less than
10"), not VIIII (since the I character can not be repeated four times). 90 is XC, 900 is CM.
4. The fives characters can not be repeated. 10 is always represented as X, never as VV. 100 is always C, never
LL.
5. Roman numerals are always written highest to lowest, and read left to right, so order of characters matters
very much. DC is 600; CD is a completely different number (400, "100 less than 500"). CI is 101; IC is
not even a valid Roman numeral (because you can't subtract 1 directly from 100; you would need to write it
as XCIX, "10 less than 100, then 1 less than 10").
1. There is only one correct way to represent a particular number as Roman numerals.
2. The converse is also true: if a string of characters is a valid Roman numeral, it represents only one number
(i.e. it can only be read one way).
3. There is a limited range of numbers that can be expressed as Roman numerals, specifically 1 through 3999.
(The Romans did have several ways of expressing larger numbers, for instance by having a bar over a numeral
to represent that its normal value should be multiplied by 1000, but you're not going to deal with that. For the
purposes of this chapter, let's stipulate that Roman numerals go from 1 to 3999.)
4. There is no way to represent 0 in Roman numerals. (Amazingly, the ancient Romans had no concept of 0 as a
number. Numbers were for counting things you had; how can you count what you don't have?)
5. There is no way to represent negative numbers in Roman numerals.
6. There is no way to represent fractions or non−integer numbers in Roman numerals.
There are several simple rules for constructing a Roman
numeral, using the letters M, D, C, L, X, V, and I. Let's review the rules:
1. Characters are additive. I is 1, II is 2, and III is 3. VI is 6 (literally, "5 and 1"), VII is 7, and VIII is 8.
2. The tens characters (I, X, C, and M) can be repeated up to three times. At 4, you need to subtract from the next
highest fives character. You can't represent 4 as IIII; instead, it is represented as IV ("1 less than 5"). 40 is
written as XL ("10 less than 50"), 41 as XLI, 42 as XLII, 43 as XLIII, and then 44 as XLIV ("10 less
than 50, then 1 less than 5").
3. Similarly, at 9, you need to subtract from the next highest tens character: 8 is VIII, but 9 is IX ("1 less than
10"), not VIIII (since the I character can not be repeated four times). 90 is XC, 900 is CM.
4. The fives characters can not be repeated. 10 is always represented as X, never as VV. 100 is always C, never
LL.
5. Roman numerals are always written highest to lowest, and read left to right, so order of characters matters
very much. DC is 600; CD is a completely different number (400, "100 less than 500"). CI is 101; IC is
not even a valid Roman numeral (because you can't subtract 1 directly from 100; you would need to write it
as XCIX, "10 less than 100, then 1 less than 10").
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