Roman numerals are the letters that ancient Romans used as their numbers. It is thought that the numerals derived from "tally marks", being scratched lines used to keep count. We use tally marks by counting four lines, and then crossing the first four with another line, going diagonally crosswise, like so:
In a similar manner, the Etruscan civilization (which was later absorbed by the Romans) is believed to have counted by drawing vertical lines. When they got to the fifth, they'd draw a figure with two lines, similar to a "V". When they got to the tenth, they'd draw crossed lines, similar to an "X".
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So, for instance, the number "37" would be tallied as:
IIIIVIIIIXIIIIVIIIIXIIIIVIIIIXIIIIVII
To read a written value, they'd count up the X's (there are three of them in the above) for "thirty"; then they'd see if any V's followed the last X (there is one) for another "five", and then they'd count whatever single lines were left (there are two after the last V) for another "two". Adding, they'd get 30 + 5 + 2 = 37. The Romans copied this method; their major change to the system was to use letters from their alphabet for their numerals.
The Romans expanded this system by adding digits for larger values, and simplified the system for composing numbers in written form. The Roman's numerals are as follows:
Roman 

value 
I (or i) 

1 
V (or v) 

5 
X (or x) 

10 
L (or l) 

50 
C (or c) 

100 
D (or d) 

500 
M (or m) 

1,000 
Customarily, we use the capital letters for Roman numerals, but it isn't "wrong" to use lowercase letters. However, for your class, your instructor may be expecting upperclass letters, so it's probably a good idea to get used to using only them.
What is the difference between "numerals" and "numbers"? "Numbers" are values, such as "three"; "numerals" are the actual written characters which stand for those values, such as the Tamil "௩", the Oriyan "୩", the Ethiopic "፫", the Bengali "৩", the "٣" currently in use in parts of the Arabicspeaking world, the Malayalam "൩", the Punjabi "੩", the Telegu "౩", and the "3" of our socalled "Arabic" system.
Numbers are universal; numerals are culturally specific.
The ancient Romans used letters as their numerals. They also had particular rules for building their written numbernames with these numerals.
(Note: Our modernday rules for Roman numerals are more strict than what the ancient Romans actually used. And some modernday clocks with Romannumeral numbering on their dials don't obey our modernday rules for numbering, especially for the character for "four". But the discussion below will apply in your math class, to movie dates, to outlines in your English class, and other modernday contexts.)
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Roman numerals don't use place value like our numbers do. For instance, when we write the character (that is, the numeral) "5", this digit stands for "five ones" in the number "215", for "five tens" in the number "251", and for "five hundreds" in the number "521". In Roman numerals, on the other hand, the digit "V" only and ever means "five ones".
The method for writing numbers in Roman numerals (other than one, five, ten, fifty, etc) is to add symbols for numbers, starting with the largest value smaller than the target value. Then keep adding units, progressively smaller, until we add up to the target value. But we can't have four of the same character in a row; in such a situation, we have to use subtractive notation instead. It's kind of like making change when you're dealing with money. I'll explain as we go.
Starting at the beginning (namely, at "one", because the Romans didn't have any concept of "zero"), we have:
I = 1
II = 1 + 1 = 2
III = 1 + 1 + 1 = 3
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The next number is 4, which is 1 + 1 + 1 + 1, but using four I's for "4" isn't allowed (because we can't do four of the same numeral in a row). If we were dealing with money, we could pay four dollars with four singles (which is 1 + 1 + 1 + 1 = 4) or we could pay with a five and get a single in change (which is 5 – 1 = 4). In Roman numerals, we "subtract" the 1 from the 5 by putting an I in front of a V: IV is "four" in the sense of "one, from five, leaves four".
IV = (5 – 1) = 4
In Roman numerals, we can do this subtractive thing with Is, Xs, and Cs (that is, with ones, tens, and hundreds) but not with Vs, Ls, or Ds (that is, not with fives, fifties, or five hundreds). So, for instance, we can't use "VL", as "five from fifty", for "45", or "VC", as "five from a hundred", for "95".
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Continuing our counting, we have:
V = 5
VI = 5 + 1 = 6
VII = 5 + 1 + 1 = 7
VIII = 5 + 1 + 1 + 1 = 8
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You may be wondering about that expression for "eight" in Roman numerals. Wouldn't we be using fewer characters if we took "ten" and subtracted "two"? Yes, but another rule in Roman numerals is that we can only subtract one character from another, so we aren't allowed to do "IIX = 10 – 1 – 1 = 8". We instead have to add up from five, as shown above. But to create the expression for "nine", we can subtract one from ten: "IX" is "nine" in the sense of "one, from ten, leaves nine".
IX = (10 – 1) = 9
Continuing our counting, we have:
X = 10
XI = 10 + 1 = 11
XII = 10 + 1 + 1 = 12
XIII = 10 + 1 + 1 + 1 = 13
The next number is fourteen, which we get by adding ten and four, where "four" is written as "one from five": XIV is "fourteen" in the sense of "ten, plus the amount of onefromfive".
XIV = 10 + (5 – 1) = 10 + 4 = 14
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