Roman Numerals and Arithmetic

I’ve always been perplexed by roman numerals.
First of all, they’re just *weird*. Why would anyone come up with something so strange as a way of writing numbers?
And second, given that they’re so damned weird, hard to read, hard to work with, why do we still use them for so many things today?

The Roman Numeral System
I expect most people already know this, but it never hurts to be complete. The roman numeral system is non-positional. It assigns numeric values to letters. The basic system is:
1. “I” stands for 1.
2. “V” stands for 5.
3. “X” stands for 10.
4. “L” stands for 50.
5. “C” stands for 100.
6. “D” stands for 500.
7. “M” stands for 1000.
Standard roman numerals don’t have any symbols for representing numbers larger than 1000. Some modern usages add an overbar, so that “V” with a horizontal line floating over it represents 5000, etc. But that’s a modern innovation.
The symbols are combined in a bizarre way. Take a number symbol, like X. A group of that symbol appearing together are added together, so “III” = 3, and “XXX” = 30. Any symbol *smaller* than it that precedes it is subtracted from it; any symbol smaller than it that *follows* it is added to it. The notation for a number is structured around the *largest* number symbol used in writing that number. In general (though not always), you do not precede a symbol by anything smaller than 1/10th its value. So you wouldn’t write “IC” for 99.
1. IV = 4; V=5, I=1, I precedes V so it’s subtracted, so IV = 5 – 1.
2. VI = 6; V=5, I=1, I follows V so it’s added, so VI = 5 + 1 = 6.
3. XVI = 15. X=10, V = 5, I=1. VI is a number starting with a symbol whose value is smaller than X, so we take its value and add it. Since VI=6, then XVI=10+6 = 16.
4. XCIX = 99. C=100. The “X” preceeding the C is subtracted, so XC=90. Then the IX following it is added. X is ten, preceeded by “I”, so “IX” = 9. Xo XCIX = 99. *(This example was corrected because I screwed up.)*
5. MCMXCIX = 1999. M = 1000. “CM” is 1000-100=900, so MCM = 1900. C = 50, XC = 90. IX=9.
For some reason (there are a number of theories of why), 4 is sometimes written IV, and sometimes IIII.
Where did this mess come from?
The roman numerals date back to shepherds, who counted their flocks by marking notches on their staffs. They didn’t original use roman letters, but just notches on the staff.
So when counting their sheep, they would mark four notches; and then on the fifth one, they would cut a diagonal notch, the way that in tallying we commonly write four lines, and then a diagonal strike-through. But instead of striking through the preceeding notches, they just used the diagonal to turn a “/” notch into “V”. Every tenth notch was marked by a strike-through, so it looked like “X”. Every tenth V got an extra overlapping notch, so it looked sort of like a Ψ; and every tenth “X” got an extra overlapping notch, so it looked like an X with a vertical line through the center.
In this system, if you had 8 sheep, that would be “IIIIVIII”. But the leading IIII are not really needed. So you could just use “VIII” instead, which became important when you wanted to do a big number.
When this system moved to writing, the simple notches became “I” and “V”; the strike-through became “X”, The Ψ-like thing became “L”, Beyond that, they started using mnemonics; so C, D, and M are all based on the latin words for 100, 500, and 1000.
The prefix-subtraction stuff came as it transitioned to writing. The problem with an ordinal system like this is that it involves a lot of repeated characters, which are very difficult for people to read correctly. Keeping the number of repetitions small reduces the number of errors that people make reading the numbers. It’s more compact to write “IX” than “VIIII”; and it’s a lot easier to read, because of fewer repetitions. So scribes started using the prefix-subtraction form.
Arithmetic in Roman Numerals
The most basic arithmetic in roman numerals is actually pretty easy: addition and subtraction are simple, and it’s obvious why they work. On the other hand, multiplication and division are *not* easy in roman numerals.
### Addition
To add two roman numerals, what you do is:
1. Convert any subtractive prefixes to additive suffixes. So, for example, IX would be rewritten to VIIII.
2. Concatenate the two numbers to add.
3. Sort the letters, large to small.
4. Do internal sums (e.g., replace “IIIII” with “V”)
5. Convert back to subtractive prefixes.
So, for example: 123 + 69. In roman numerals, that’s “CXXIII + “LXIX”.
1. “CXXIII” has no subtractive prefixes. “LXIX” becomes “LXVIIII”.
2. Concatenate: “CXXIIILXVIIII”
4. Internal sum: reduce the “IIIIIII” to “VII” giving “CLXXXVVII”; then reduce the “VV” to “X”: “CLXXXXII”
5. Switch to subtractive prefix: “XXXX” = “XL”, giving “CLXLII”. “LXL”=”XC”, giving “CXCII”, or 192.
### Subtraction
Subtraction isn’t any harder than addition. To subtract A-B:
1. Convert subtractive prefixes to additive suffixes.
2. Eliminate any common symbols that appear in both A and B.
3. For the largest remaining symbol in B, take the first symbol in A larger than it, and expand it. Then go back to step two, until there’s nothing left.
4. Convert back to subtractive prefixes.
So 192-69 = “CXCII-LXIX”.
1. Remove prefixes: CLXXXXII – LXVIIII.
2. Remove common symbols. CXXX – VII.
3. Expand an “X” in “CXXX”: CXXVIIIII – VII.
4. Remove common symbols: CXXIII = 123.
### Multiplication
Multiplication using roman numerals is not particularly easy or obvious. You can do the trivial thing, which is repeated addition. But it should be pretty obvious that that’s not practical for large numbers. The trick that they used was actually pretty nifty. It’s basically a strange version of binary multiplication. You need to be able to add and divide by two, but those are both pretty easy things to do. So here goes:
Given A×B, you create two columns, and write A in the left column, and B in the right. Then:
1. Divide the number in the left column by two, discarding the remainder. Write it down in the next row of the left column.
2. Multiply the number in the right column by two. Write it down in the right column next the the result from step 1.
3. Repeat from step 1 until the value in the left column is 1.
4. Go down the table, and cross out every row where the number in the left column is *even*.
5. Add up the remaining values in the right column.
Let’s look at an example: 21 * 17; XXI * XVII in roman numerals
We build the table:
Left Right
XXI(21) XVII (17)
X(10) XXXIV (34)
V(5) LXVIII (68)
II(2) CXXXVI (136)
I(1) CCLXXII (272)
Then strike out the rows where the left hand side are even:
Left Right
XXI(21) XVII (17)
V(5) LXVIII (68)
I(1) CCLXXII (272)
Now add the right hand column:
Why does it work? It’s binary arithmetic. In binary arithmetic, to multiply A by B, you start with 0 for the result, nad then for each digit dn of A, if dn=1, then add *B* with n 0s appended to the result.
The divide-by-two is giving you the binary digit of A for each position: if it’s odd, then the digit there was 1, if it’s even, the digit in that position was 0. The *multiply by 2* on the right is giving you the results of appending the zeros in binary – for the Nth digit, you’ve multiplied by two *n* times.
### Division in Roman Numerals
Division is the biggest problem in roman numerals. There is no good trick that works in general. It really comes down to repeated subtraction. The only thing you can do to simplify is variations on finding a common factor of both numbers that’s easy to factor out. For example, if both numbers are even, you can divide each of them by two before starting the repeated subtraction. It’s also fairly easy to recognize when both numbers are multiples of 5 or 10, and to do the division by 5 or 10 on both numbers. But beyond that, you take a guess, do the multiplication, subtract, repeat.
Some Common Questions
* **Why does a clock use IIII instead of IV?** There are a surprising number of
theories for that. The top contenders are:
* IV are the first letters of the name of Jupiter (I don’t buy this one,
because the romans weren’t particularly concerned about writing down
Jupiter’s name) or the first letters of Jehovah in latin (more convincing,
since early christians did follow the jewish practice of not writing god’s
* IIII is more symmetric with VIII on the clock face.
* IIII allows clockmakers to use fewer molds to make the numbers for the
clock face.
* The king of France liked the way that “IIII” looked better that “IV”.
* Coincidence. Technically, “IIII” is as correct as “IV”. So someone who
started making clocks just happened to be someone who used “IIII” instead of “IV”. In fact, the Romans themselves generally preferred “IIII”.
* **Why do we still use roman numerals?** No *practical* reason. Our society tends to be rather worshipful of the romans, and to consider Latin to be the language of scholars. So anything that wants to *look* impressive has traditionally used roman numerals, because that’s what they used in latin.
* **Is there a roman numeral 0?** Yes, but it’s not authentic. During the middle ages, monks using roman numerals used “N”, for “nullae” to represent 0. But it wasn’t the positional zero of arabic numbers; it was just a roman numeral to fill into the astronomical tables used to compute the date of Easter rather than leaving the column blank.

0 thoughts on “Roman Numerals and Arithmetic

  1. J-Dog

    Ave, and thanks for the lesson. I have always been interested in Roman culture, and even though I was raised to say the Mass in Latin, I have never run across anyone that explained working with Roman numerals in such detail.

  2. Polymath

    That trick for multiplying is very nifty indeed. But note that Roman numerals are SUCH a bad way of representing numbers that the method for multiplying does not rely on the representation in any way. Our normal multiplication, for example, relies on place value, which is encoded in the representation. But the Roman multiplication works for any encoding at all.

  3. Mark Paris

    One small quibble with “Our society tends to be rather worshipful of the romans, and to consider Latin to be the language of scholars.” I don’t think many people today consider Latin to be the language of scholars. In the past Latin was not just considered to be the language of scholars, it actually was the language of scholars. It was at least in part like the use of English in science today. It was a language that every educated person knew whatever his or her native language. Today people use roman numerals because they think it makes whatever they are doing look more important, like numbering the super bowl in roman numerals.

  4. Daniel Martin

    Note that the reason I’ve always heard of for movie copyright dates being written in Roman numerals was obfuscation: when movies were shown essentially only in theaters, the thought was that a copyright date obviously five years old would cause audiences to leave, but if you obscured the copyright date by writing it in Roman numerals, the audience wouldn’t instantly realize that this was five year old stuff you were trying to get them to watch.
    Doesn’t completely make sense now that I write it out, but if you look at old movies from the 40s, 50s, or 60s, (and some movies in the 70s and 80s) the copyright year is always written in Roman numerals.

  5. Dave S.

    Aych! No wonder the Roman Empire declined.
    I wonder if there is any branch of mathematics where the Roman system acually works better than ours? Other than stylistically I mean (e.g. Super Bowl XXXIII is way cooler looking than Super Bowl 33).

  6. Dave S.

    One modern use of Roman numerals that I think justified on purely practical grounds is the numbering of introductory or preliminary material in a book. Those pages numbered, i, ii, iii, iv, etc. – so as not to be confused with the actual text of the book with its Arabic pagination.

  7. Zeno

    We should never forget how a weak background in Roman numerals nearly caused Bart Simpson to be devoured by tigers in the Lemon of Troy episode. Of course, Roman numerals are also handy for pretentious people who can’t think of new names for their children.
    P.S.: It’s “weird”, not “wierd”.

  8. Paul Clapham

    And while we’re on the topic of spelling, you use the word “bizarre” regularly and always misspell it the same way.

  9. Uffe

    This is from my notes from discussions with archeologists in Rome some years ago.
    Subtracting prefix?
    For your first common question, I once checked this out myself at the real colloseum in Rome. The seats there are – of course – numbered in Roman, Roman numerals! Surprisingly you find that some of the numbers use subtracting prefix, most don’t.
    From archeologist there I was explained that the reason is that subtracting prefix was a sort of short-hand custom, but that official, learned scholars always avoided it in official writing. At the colloseum the stone carvers from time to time had slipped into the colloqial writing instead of the official latin.
    Why still use them?
    You’r wrong there. There is in fact a real practical reason for it. Just try to carve indian/arabic numbers in stone and you will understand. Roman numbers are designed for stone carving, whereas in India they painted numbers. And, the international scientific writing was in latin until less that 150 years ago here in Europe.
    Why the mess of multiplication/division?
    They didn’t have that mess. Romans use a 10-base abacus for multiplication and division, the roman numerals where used for official symbolic representation of numbers that resulted. The actual calculations were done base 10 on an abacus.

  10. Xanthir

    Uffe – That’s some pretty interesting information. Thanks.
    Mark – Thanks for the article. Nothing really new to me, but it’s always fun. I thought that the half/double method of multiplication was exclusive to the egyptians, though! I am corrected.

  11. One Brow

    Roman muliplication can work just like ours. You start with your basic tables (I * I = I, V * X = L, etc.) For example, 23 x 56 = 1,288.
    XXIII * LVI:
    Just like for us, division combines multiplicaiton and subtraction.
    However, whether the Romans used these methods is an entirely different question.

  12. Karl

    The best use nowadays for Roman Numerals is for teachers of 7th (or so) grade math. It provides a way to explain the difference between NUMBER and NUMERAL – an EXTREMELY important concept for understanding algebra that is too often ignored.

  13. Mark C. Chu-Carroll

    The example using 99 is fixed; so are the spelling errors. As usual, thanks for the corrections.

  14. clamboat

    How about fixing example #3 at the beginning of the post?
    You wrote “XVI = 15”. Actually XVI = 16.

  15. Joe in Australia

    Has anyone come across any especially-nifty algorithms for translating decimal integers to Roman numerals?

  16. Xanthir

    Joe – the only one I know of is the obvious.
    Divide the number by 1000 (drop remainder) to get the number of Ms.
    Modulo by 1000, store as new number.
    Divide by 500.
    Modulo by 500, store.
    Divide by 100.

  17. ekzept

    Aych! No wonder the Roman Empire declined.
    while these manipulations could be done, i believe there’s evidence people like Roman accountants used an abacus for frequent calculation. it’s possible other systems were used, too, since Romans were adept at making loans and collecting interest.
    it seems to me Roman numerals are also good because they can be readily chiseled into stone, for reasons similar to why cuneiform was popular: even though cuneiform was written on wet clay tablets with blunt reeds, the scribes probably had penmanship like mine.

  18. Trisha

    Great post! Thanks
    A number of years ago I was in a restaurant and overheard a conversation from a nearby table about Roman numbers that was really interesting. I don’t remember the details now and have no idea if those people had any idea of what they were talking about but they claimed that Roman science and engineering were only able to progress so far because of how difficult their number system was to work with.

  19. Elohite

    The BBC still uses roman numerals in the final copyright tagline at the end of all its programs showing the year it was produced. I very much enjoyed the closing of the New Year’s program at the Millenium when the tagline changed from MCMXCIX to MM and knowing the numeral would never be as short as that again with regular roman numerals. Just a nerdy thing and an example of where it’s still used. Ya never know though, maybe the Beeb will still be around at the year 5000AD when its tag will be a V with a bar across it

  20. dsfa

    If you take the first six Roman numerals and arrange them in order, biggest to smallest, thus: DCLXVI. What is this number? I read this in an essay by Martin Gardner.

  21. DouglasG

    The Roman Republic followed by the Roman Empire is the longest continuous civilization in history. The Republic went from approximately 560BCE to 27BCE, and depending upon how you define it, it went until 476 (Western) or 1461 (Eastern). We have a LONG way to go before we should talk about the decline of the Rome.

    Roman numerals are nowhere near as cool as Greek numerals.

    He he he… Using Greek Numbers π = 80.

  22. Jonathan Vos Post

    I love this blog!
    Some of my published play with Roman numerals:
    Product of digits of Roman Numerals.
    “Long” prime Roman numerals. Smallest prime whose Roman numeral representation has n characters.
    Numbers n such that n divides number of letters in Roman numeral for n.
    Roman numeral complexity of n.
    Number of symbols in the Roman Fibonacci number representation of n.
    Positive integers whose representation as Roman Fibonacci numbers has exactly two symbols.
    Positive integers whose representation as Roman Fibonacci numbers has exactly three symbols.
    Positive integers whose representation as Roman Fibonacci numbers has exactly four symbols.
    Jonathan Vos Post
    former Adjunct Professor of Mathematics, Woodbury University
    former Adjunct Professor of Astronomy, Cypress College
    over 15,000,000 hits/year

  23. Jonathan

    We were studying Peano’s axioms a few years back and the professor asked us to make up a successor algorithm for the Roman numerals. I did it, but it was not particularly easy.
    Robert Kaplan touches a lot of neat stuff about computation in his “The Nothing that Is: A Natural History of Zero.” There’s probably better out there, but Kaplan is quite accessible.

  24. Stephen Frug

    Why do we still use roman numerals? No practical reason.
    I don’t think that’s true. Dave S. pointed out above that there is a practical reason in the openings of books. I think this can be generalized: it’s sometimes useful to have two different ways of writing the numbers, for differentiation purposes. It’s sort of like using capital letters versus lower case. Now, I grant you, many uses don’t fall into this category; and there are other ways to achieve the same effect in many cases. Still, there is a type of use that makes a lot of sense. (Anyone want to invent “capital” Arabic numerals?)

  25. Markus

    Excellent article, but two small mistakes remain:
    XVI = 15 – should be 16
    C = 50 should be 100
    3. XVI = 15. X=10, V = 5, I=1. VI is a number starting with a symbol whose value is smaller than X, so we take its value and add it. Since VI=6, then XVI=10+6 = 16.
    5. MCMXCIX = 1999. M = 1000. “CM” is 1000-100=900, so MCM = 1900. C = 50, XC = 90. IX=9.

  26. Dub

    5. MCMXCIX = 1999. M = 1000. "CM" is 1000-100=900, so MCM = 1900. C = 50, XC = 90. IX=9.
    sp. C = 100

  27. Spongebrain Madcowpants

    The origin of I and V as tally marks makes no sense, because it doesn’t explain why you’d group the tally marks in fives rather than some other number.
    The more convincing explanation is this: I represents a finger. V represents a hand (five fingers). X represents two hands. The remaining symbols are initials for the Roman words: C for centum, M for… um, whatever the Latin word for 1000 is, which begins with “mil.”

  28. Grady

    I’m glad you’re working on this, because 123456789 are just as much an enemy of freedom as fried pieces of potato. It’ll just be a matter of time until we stop using those terrorist-loving arabic numerals.

  29. Alon Levy

    Why would IV be considered an improper use of God’s name? In Latin, the Christian god’s name is written IAHVEH.
    Grady, Arabic numerals aren’t Arab. They were invented in India, which is sort of an ally in the War on Terra (R). They just came to the West via Arabia; they don’t even have the same shape in Western languages as in Arabic.

  30. Andrew Cooper

    This post reminds me of one of the best tag lines I’ve seen on the subject of different ways to respresent numbers – although it doesn’t involve Roman Numerals, I’m afraid.
    “There are 10 types of people in this world, those who understand binary, and those who don’t.”
    Now back to the topic at hand…

  31. Alon Levy

    You can do the same in handwriting in English-speaking countries, “There are II types of people in this world, those who understand Roman numerals and those who don’t.” In the rest of the world it tends to break down because even people who use Western numerals usually write 1’s like typewritten 1’s, as opposed to as simple vertical bars.

  32. Xanthir

    Just to chime in with another number-representation thing:
    “You and dead people can understand hex. How many people can understand hex?”
    The answer, of course, is 5700610. ^_^

  33. Heath McKay

    The Roman numeric system used IIII for 4. IV came about from the Swiss because it was cheaper to make IV out of gold than IIII on their timepieces.

  34. Munawar Butt

    how roman numeral symbols originated?
    only one site in the World.
    the science and technology of numerals symbols.
    very hard subject.
    12 pages of my book. The science and languages of God.
    (if u do not understand, contact me)
    numerical science researcher
    munawar butt

  35. Jonathan Vos Post

    Added to the Online Encyclopedia of Integer Sequences since my last posting in this blog.
    A123054 Numbers whose Roman numeral representation, reversed, is a Roman numeral.
    1, 2, 3, 4, 5, 6, 9, 10, 11, 19, 20, 30, 40, 49, 50, 51, 60, 90, 100, 110, 190, 200, 300, 400, 500, 600, 900, 1000, 1100, 1900, 2000, 3000
    COMMENT A subset of this is A078715 Palindromic Roman numerals. Not “Old style” Roman numerals (where 4 = IIII).
    {n such that Reversal(Roman(n)) = m and
    Roman^-1(Reversal(m)) = j for some integer j}.
    {n such that Roman^-1(Reversal(Roman(n))) is an element of {Roman(k)}}.
    EXAMPLE a(1) = 1 because Roman(1) = I, and Reversal(I) = I, which is Roman.
    a(4) = 4 because Roman(4) = IV, and Reversal(IV) = VI, which is Roman.
    a(10) = 19 because Roman(19) = XIX which is a palindromic Roman numeral.
    a(27) = 900 because Roman(900) = CM, and Reversal(CM) = MC, which is Roman.
    1999 is not in the sequence because “MIM” is not a well-formed Roman numeral for 1999, although it looks like one; see Schildberger.
    CROSSREFS Cf. A078715.
    KEYWORD base,easy,fini,full,nonn
    Jonathan Vos Post, Sep 26 2006

  36. Tim Giffey

    There is a teacher that says that by moving one line this expression is true. Can you help me solve this?
    IX + VI = XI (you can’t move or alter the + or = signs)

  37. Ray Greaves

    There is an even more rational approach to Roman arithmetic. Instead of performing calculations with the unweildy Roman numerals, the Romans used the abacuse. The Roman abacus allowed values specified using Roman numerals very easily into a value displayed on a bi-quinary base ten place value system. The calculations can then be performed on the abacus and the result translated back into the written Roman notation. To se the type of abacus used see .

  38. Kelley

    I think thats kind of weird.
    “I” stands for 1.
    “V” stands for 5.
    “X” stands for 10.
    “L” stands for 50.
    “C” stands for 100.
    “D” stands for 500.
    “M” stands for 1000.
    Removing the “M” from th list would just give you 666.
    If the letters give you 666 I woulder if it would give you a date just using :I,V,X,L,C, and D.

  39. Jonathan Vos Post

    A036786 Length of Roman notation for n is less than length of decimal representation.
    10, 50, 100, 101, 105, 110, 150, 200, 400, 500, 501, 505, 510, 550, 600, 900, 1000, 1001, 1002, 1004, 1005, 1006, 1009, 1010, 1011, 1015, 1020, 1040, 1050, 1051, 1055, 1060, 1090, 1100, 1101, 1105, 1110, 1150, 1200, 1400, 1500, 1501, 1505, 1510, 1550
    1000 = M is shorter in Roman numerals.
    Cf. A036787, A036788.
    J. H. Conway (conway(AT)
    Corrected by Larry Reeves (larryr(AT), Sep 25 2000

  40. imd

    I believe the answer to solving the problem
    IX+VI=XI in one move with out altering
    the positioning of the stick,+ or = and still equaling 11
    is X+NI=XI. The N represents a zero in roman numerals. If that’s not it I would like to know as well!!!

  41. Jonathan Vos Post

    Not lines, line segments. Use matchsticks.
    Pick up the left half of the “V” and stricke it, let it burn to ash.
    What remains:
    IX + /I = XI
    The slanted matchstick can be read as if nearly vertical. As if:
    IX + II = XI.

  42. Anonymous

    wft… can any one tell me the real reason for roman numerals like i know its saposed to be for mathimatical purposes but i wanna know than meets than eye…..

  43. Robert

    To be honest, considering the history, I think Roman Numerals are extremely logical. Numbers are originally used to count, not to do more complicated forms such as calculating interest. If you’re counting the simplest number system is: I, II, III, IIII, IIIII, etc… Once you get larger numbers it becomes convenient to group things, or perhaps invent new symbols for larger numbers. If you ignore the prefix notation, (so 4 = IIII) then addition in roman numerals is actually easier than in our decimal system.
    1) Combine the numbers
    2) Sort (optional, since we abandoned prefixes)
    3) Group (optional, since everyone understands IIIII = V = 5)
    If you’re a shepherd, and you want to combine two herds of sheep, this is all you need.
    Outside of possible math-related work you guys have, how often do you need to multiply in real life?

  44. rx1

    There is an even more rational approach to Roman arithmetic. Instead of performing calculations with the unweildy Roman numerals, the Romans used the abacuse.


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