Introduction to Roman Numerals

=== by Bob Sutherland ===

This web page includes all of the important information you need to know about Roman Numerals.

When do people use Roman Numerals?

For some reason Roman Numerals seem to be slowly disappearing from our school curriculums and Mathematics textbooks in some jurisdictions but we still use Roman Numerals in our society. The date of a movie that goes scrolling by within the screen credits is likely to be in Roman Numerals. The page numbers of the preface section of a book are likely to be in Roman Numerals.

When working with ancient Roman Numerals (i, ii, iii, iv, v, vi, vii, viii, ix, x, xi, xii, xiii, xiv, xv, xvi, xvii, xviii, xix, xx, xxi …) we refer to our modern numbers as Arabic Numerals (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21 …).

Roman Numerals were never intended to be printed quickly for doing arithmetic calculations such as adding, subtracting, multiplying or dividing. Back during the times of the Roman Empire merchants probably had a bag of small stones. They would move the stones from one pile to another to add up the prices of the items you were buying. Tax collectors may also have used stones to calculate the taxes owed by each citizen. The tax collectors would then use Roman Numerals to write down the amount paid by each citizen by carving it into wood, staining it onto an animal skin, or writing it on one of the early forms of paper.

Roman Numerals were designed to be slowly carved and chiselled into wood or stone to make a permanent record of a number. Today we engrave some forms of jewelry, wall plaques, trophies, statues and tombstones with permanent numbers and letters. We number the outside of our homes and buildings with an address. We number the doors of hotel rooms and office buildings. When you visit a large shopping mall you will likely find the sections of the parking lot are numbered on lamp posts to help you later find your car. In sport stadiums each of the entrance ways you have to pass through will have an overhead sign with a letter or number to help you find your way to your seat. To create these types of permanent numbers is what Roman Numerals were designed to do back during ancient times.

Roman Numerals are created using letters

The modern version of Roman Numerals we teach in schools today makes use of the following letters:

I = 1
V = 5
X = 10
L = 50
C = 100
D = 500
M = 1000

You can use small letters instead of capital letters:

i = 1
v = 5
x = 10
l = 50
c = 100
d = 500
m = 1000

I would recommend using capital letters whenever you will be using numbers greater than 49 just because people will find it easier to spot and correctly identify a capital letter "L" rather than its small letter equivalent "l" in a Roman Numeral.

When ancient workmen first chiselled the letter I into a rock they were probably trying to carve a picture of one finger. The number two is represented with II for two fingers. The number three is represented with III for three fingers. The V is a picture of a hand representing five fingers. Stretch your hand out wide and the two lines of the V are your thumb and baby finger. The X is a picture of two hands representing ten fingers.

The stone carvers would have chosen to use straight lines as much as possible for their numbers because it is much easier to chisel a straight line into a rock than a curved line.

In the metric system of measurements centi- means 100, as in 100 centimetres = 1 metre. The letter c is used as an abbreviation for centi-, for example cm means centimetre. This should help you remember that C = 100 in Roman Numerals.

In the metric system of measurements milli- means 1000, as in 1000 millimetres = 1 metre. The letter m is used as an abbreviation for milli-, for example mm means millimetre. This should help you remember that M = 1000 in Roman Numerals.

Converting Roman Numerals to Arabic Numerals

When you convert a number from Roman Numerals to the Arabic Numerals that we use now you just add up the value of each of the letters. Here are some examples:

LXXXII
= L + XXX + II
= 50 + 10 + 10 + 10 + 1 + 1
= 80 + 2
= 82

MMMDCCXVII
= MMM + D + CC + X + V + II
= 1000 + 1000 + 1000 + 500 + 100 + 100 + 10 + 5 + 1 + 1
= 3000 + 700 + 10 + 7
= 3717

ML
= M + L
= 1000 + 50
= 1050

Converting Arabic Numerals to Roman Numerals

When you convert a number from the Arabic Numerals that we use now to Roman Numerals you convert one digit at a time. Here are some examples:

2637
= 2000 + 600 + 30 + 7
= MM + DC + XXX + VII
= MMDCXXXVII

1253
= 1000 + 200 + 50 + 3
= M + CC + L + III
= MCCLIII

805
= 800 + 5
= DCCC + V
= DCCCV

The rules for creating Roman Numerals

There is no such thing as the number zero in Roman Numerals. The number zero was only recently invented just a few centuries ago, which was long after the rise and fall of the Roman Empire. When converting from Arabic to Roman Numerals you just skip over any digits that are zero and do not do anything. When you are converting from Roman to Arabic Numerals you have to carefully watch for missing zeros and insert them into the correct position in the Arabic Numeral.
Roman Numerals are always written with the letters in order from largest value to smallest value. The order is: M, D, C, L, X, V, I. You add up the value of each of the letters in a Roman Numeral to find the total value of the number.
The letters M, C, X and I representing 1000, 100, 10 and 1 can be repeated up to a maximum of three times in sequence in a Roman Numeral.
The letters D, L, V representing 500, 50 and 5 may only appear once in a Roman Numeral.
There are six exceptions to rules 2 and 3. The numbers 4, 9, 40, 90, 400 and 900 are written with a letter having a smaller value appearing before a letter with a larger value. You subtract the smaller number from the larger number in this situation. This may create the situation where the X, C or M could possibly appear four times in a Roman Numeral but one of the repeating letters would not be with the others in sequence.

The early stone carvers would chisel I for one, II for two, III for three, V for five and VI for six. They could have chiselled IIII (four lines) for four but then someone realized that it would be less work to chisel IV (three lines) for four. It took a lot of work to chisel each letter including a simple I. Can you imagine being told to chisel big letters half your height or bigger into the stone walls above each of the many entrances and passage ways to the seats in the Colosseum?

Here is a list of the six exceptions mentioned in rule 5 above:

IV = 5 - 1 = 4
IX = 10 - 1 = 9
XL = 50 - 10 = 40
XC = 100 - 10 = 90
CD = 500 - 100 = 400
CM = 1000 - 100 = 900

Unfortunately you have to memorize the six exceptions but that should not be difficult to do as they are all of the situations where you encounter the Arabic digits 4 and 9.

More examples of converting between Roman Numerals and Arabic Numerals

Here are a few more examples of converting back and forth between Arabic and Roman Numerals:

XLIX
= XL + IX
= (50 - 10) + (10 - 1)
= 40 + 9
= 49

1964
= 1000 + 900 + 60 + 4
= 1000 + (1000 - 100) + 60 + (5 - 1)
= M + CM + LX + IV
= MCMLXIV

390
= 300 + 90
= 300 + (100 - 10)
= CCC + XC
= CCCXC (Notice that there are four C's but only three of them are in sequence.)

MMMDCCCLXXXVIII
= MMM + D + CCC + L + XXX + V + III
= 3000 + 500 + 300 + 50 + 30 + 5 + 3
= 3000 + 800 + 80 + 8
= 3888 (The longest number in Roman Numerals.)

MMMCMXCIX
= MMM + CM + XC + IX
= 3000 + (1000 - 100) + (100 - 10) + (10 - 1)
= 3000 + 900 + 90 + 9
= 3999 (The highest number you can count up to using the above rules for Roman Numerals.)

2012
= 2000 + 10 + 2
= MM + X + II
= MMXII