8 ms·

I always thought that roman numerals would be a simpler way to do basic arithmetic and might lend itself more to simple commerce. For example: III represents 3

• kbenson 9y ago
  > For example: III represents 3 things, so III + II = IIIII For simple commerce application that is simpler, I just have to then remember that IIIII = V, and VV = X and XXXXX = L, LL = C. Armed with just those simple rules I could probably get by in the market place in Rome. I'm not sure having to look at all the possible conversions of a context free grammar to reduce the state to its minimum is actually easier than some simple arithmetic in our current system.
  • radarsat1 9y ago
    In the general case maybe not, but to get by in a marketplace you'd need just those one he listed... like.. 3 or 4 things. 5, 10, 100, 1000. Most people can remember 3 or 4 things. Children can do it, let alone market vendors who are used to considering that 5 chickens = 1 goat. (Inflation these days!)
    • kbenson 9y ago
      That's just addition. Do you only need addition in a marketplace? Subtraction requires you convert to another form prior to actually subtracting, so you have to know what form you want. Multiplication and division? Harder.
      • mywittyname 9y ago
        You'd use the take-away method. XX - III = XVIIIII - III = XVII It's literally just addition in reverse. Instead of having reduction passes where you convert IIIII to V, etc., you'd have an expansion passes to convert V to IIIII.
        • kbenson 9y ago
          Yes, that's what I said. But you chose a simple case. Try subtracting more complex numbers, specifically something more complex for the subtrahend. E,g, 42 - 13, which is XLII - XIII. Not only to you have to convert, but you have to convert both sides, and to either a similar format that can be subtracted, or to a lowest common format as suggested above (which is not roman numerals, BTW, so you need to know two systems, Roman numerals, and the alternate format with no prefixes to use to arithmetic). And that still isn't event touching multiplication or division, which are very common in simple commerce. 100 coins for 8 chickens, or 60 coins for 5 chicken. What's the better deal? I want 7 of these 3 coin breads. You say that's 30 coins? Guess I should check that...
          • drostie 9y ago
            For many cases one can ignore division, but yes, as the article said that generally required an abacus. Multiplication doesn't in general require an abacus if you're trained in roman numerals. Let's take your "more complicated numbers", 42 * 13. Expand XLII * XIII = X * XLII + XLII + XLII + XLII, the first requires you to know a times table to see that it's CDXX, the rest you'd add mentally, LLL - XXX = CXX, so it's CDXX + CXXVI = DXLVI. Five hundred forty six. Like, I'm not saying it's as clean as 42 * 13 = 420 + 126 = 546, it's not. But it's not educationally prohibitive either.
  • hinkley 9y ago
    Just thinking about how they work here, it looks like Roman numerals are effectively a base 5 notation, with a special behavior for 5n - 1. In base 10 math there are 18 outcomes for adding two digits, and you have to do carry operations for 9 of them. With carries there are 20 and you have special cases for 10. In Roman numerals there are 10 outcomes, you have to do carries for 5 of them, special casing for two (4 and 9), the numbers are at least one digit longer and non uniform, so it's harder to line up the columns to do an addition in the first place. No wonder algebra was invented in Arabic.
    • bdamm 9y ago
      And I'm betting that the 5n-1 behavior was probably only used by those with education, and may have even been a class distinguishing numeric feature, with the benefit of making numerals slightly impenetrable to those who didn't know the trick.
    • JadeNB 9y ago
      > Just thinking about how they work here, it looks like Roman numerals are effectively a base 5 notation, with a special behavior for 5n - 1. This doesn't seem quite accurate. Roman numerals are organised around multiples of 5, whereas a base-5 system should be organised around powers of 5.
• malnourish 9y ago
  I'm not 100% sure, but through anecdotal experience and from what I've read (although haven't vetted), we lose track of counters at ~8 of them. That is, III is easily discernable as 3 'I's, but take IIIIIIIII, and try to recognize how many 'I's are there immediately.
  • tathougies 9y ago
    That's why you group five I's into a V?
    • lou1306 9y ago
      This, and another theory states that V is kind of similar to an open hand, i.e. five fingers, making it a convenient abbreviation. Consequently, X is just two V's joined at the vertex.
  • saghm 9y ago
    > we lose track of counters at ~8 of them I must be below average in that regard; I never have trouble being able to distinguish the number of things when there are three or fewer, but it's somewhat difficult for me to tell four and five apart, and I essentially have to resort to manually counting for five and six.
• deleted 9y ago
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• riffraff 9y ago
  you skipped over III + I = IV, and VI+III = IX. Subtractive notation is confusing.
  • bllguo 9y ago
    I believe some Roman inscriptions were found to use additive, i.e. III + I = IIII, VI + III = VIIII. Certainly I feel that nobody would look at IIII and be confused as to what it represents.
  • bluehawk 9y ago
    They mention this in the article, and say there is very little evidence of this being common in ancient Rome.
    • drostie 9y ago
      Also it's not "really" hard. Think of subtractive notation as a way to write down what's really an int1 - int2 expression in normal roman numerals, with the out-of-order characters representing int2. To add these together, (A1, A2) + (B1, B2) = normalize(A1+B1, A2+B2) where those additions are a little easier. The normalize step is not very complicated; it basically says "if a letter is repeated on both sides eliminate one instance of it; if it is not but there are too many on one side, adjust the other side to compensate." The only nuance is that "too many" for the left hand side is "four" whereas "too many" for the right hand side is "two."
  • mywittyname 9y ago
    According to the article, that notation didn't come into use until the middle ages. So using IIII instead of IV was almost certainly acceptable in the context of transactions. Even if it wasn't, doing the conversion before addition/subtraction would be trivial (and the first step).
  • zw123456 9y ago
    Lets say I am a farmer and I take eleven pigs to market. I start with XI pigs and sell three III. I know X = VV and I know V = IIIII so I expand XI = VVI = V IIIII I, then take way III sticks and am left with VIII. It is easier for simple primitive thing because the stick I represents a thing and the V represents 5 I's and X represents 2 V's. It is perhaps easier for someone like a farmer in Roman times to use than remembering arithmetic rules. It is more directly related to the physical world. At least that was my hypothesis.
• sanityUnbounded 9y ago
  Curious, why do you have to remember that 2 + 2 = 4 & 3 + 2 = 5? Once you know the values the symbols represent, at that point isn't it similar in simplicity to roman numerals? II + II = IIII 2 + 2 = 4 Don't see how the latter problem lends itself to any more memorization beyond symbols
  • jdminhbg 9y ago
    The equivalent to II + II = IIII isn't 2 + 2 = 4, it's 2 + 2 = 22. Simply jamming the symbols together gives you the right answer.
    • dllthomas 9y ago
      Right, much like I + V is IV. (There is more help; it's not as simple as people have been portraying, though...)
      • Spivak 9y ago
        Sure, but that notation was likely born out of convenience by people who wanted a shorthand for situations where it's easier to represent a number as a subtraction but it does make the number system more complicated.
        • jacobolus 9y ago
          Personally I wish we had a good notation and separate single-syllable pronunciation for negative digits –1 through –9, and were willing as a society to accept numbers written with a mix of positive and negative digits. Decimal arithmetic becomes quite a bit easier if you normalize your numbers to the digits from –5 to 5, and it would help prepare students for algebra.
          • moefh 9y ago
            An ugly thing about using negative digits with base ten (or any even base) is that you either have ambiguity on how to write numbers, or you have a different number of positive and negative digits. For example, if you do base ten with digits EDCBA012345 (E=-5, D=-4, etc.) -- that's 11 digits -- then five can be written as 5 or 1E; fifteen is 15 or 2E, etc. So you either: - Accept that some numbers can be written in more than one way, which seems a huge disadvantage over what we have today. - Rule out certain sequences of digits (E can only follow a negative digit, 5 can only follow a positive digit, if 5 follows a 0, then it depends on the digit before the 0, etc.). Not only is this inelegant, but it adds seemingly arbitrary rules for people to memorize when they're first learning how to write numbers. - Remove one of the digits (e.g., remove E). Then things aren't symmetric around zero -- e.g. five is 5, but minus five is A5.
• nawgszy 9y ago
  >1 is one thing, 2 represents 2 things, 3 represents 3 things and so on I follow you thus far. >Then I have to remember that 2 + 2 = 4, and 3 + 2 = 5 No you don't. You have to remember that 1 + 1 = 2; 2 + 1 = 3; ...; 9 + 1 = 10; and then the rules repeat themselves, respecting columns for addition. All mathematics between 1 and 10 like 4 + 5 are already known at this point. Roman numerals, on the other hand, give you no easily repeatable pattern to follow as the order of magnitude increases. I + I = II, III + II = V, V + V = X, X + X + X + X + X = L, LL = C, and I only got this far because of what you said. What comes next? C * 5 = x_1, x_1 * 2 = x_2, x_2 * 5 = x_3, ..., x_n * 5 = x_n+1, x_n+1 * 2 = x_n+3 The complexity is unbounded. Sure, if you constrain yourself to "getting by in the market place in Rome", that's one thing, but even then I would imagine arguments around arithmetic could go either way.
  • jacobolus 9y ago
    You’re speaking from a position of bias and ignorance (we all do this sometimes, but it’s important to be aware of), with a lifetime of familiarity with Hindu–Arabic number notation and almost no experience reading/writing Roman notation or translating back and forth between written numbers and pebbles or other tokens on a counting board, which is how most calculations were done in Roman times (persisting to this day in our word calculate, from the Latin word for pebble). If you spent several decades working with a counting board and had only occasionally seen Hindu–Arabic arithmetic, you would likely feel the opposite (as, indeed, people did for the first few centuries after written arithmetic was introduced to Europe – for example our word cipher, meaning secret code, comes from the word for the Arabic 0, reflecting people’s early confusion about pen and paper arithmetic). The Romans (and others in the Roman empire) were the premier engineers, merchants, bureaucrats, astronomers, etc. of their era and region. They didn’t have any problem doing extremely complex computations. As for your specific concerns: the easy pattern is that the letter for a group of five literally looks like half of the letter for a group of ten. V = X/2, L = C/2, D = ↀ/2, ↁ = ↂ/2, etc. So you need to remember the meanings and relations for the symbols for I, X, C, ↀ, ↂ and then just count them. The patterns that: IIIII = V; XXXXX = L; CCCCC = D, ↀↀↀↀↀ = ↁ are really not that hard to remember. The groups of five are mostly there as a shorthand, because writing nine of the same symbol in a row is harder to count and takes up more space. These patterns are certainly no harder than remembering the English words ten, hundred, thousand (or Latin words unus, decem, centum, mille), which are also arbitrary symbols. Since the numbers are always written in order, you can learn to separate them by digits. People would have “chunked” a long string of these symbols into the word for each each digit, and pronounced them using words pretty much like modern languages. Just like in our natural languages, the system is not strictly positional – you just skip writing/pronouncing any lines on the counting board with no pebbles. So if you see DCCCXXXXVII you think of it / read it as “eight hundred forty seven”, first split into the groups DCCC XXXX VII, with the digit represented by the pattern and the order of magnitude represented by the symbols used. Or alternately, you would have seen them as the visual patterns “three pebbles on the hundred line and one in the space above; four pebbles on the tens line; two pebbles on the one line and one in the space above”. When you’re thinking of the meaning of these symbols, you’re going to be fluently translating them in your head back and forth between three representations: verbal, counting board, and written. Once these have all been worked with extensively, there’s not much friction. It’s just like learning to read, or learning music notation. Someone experienced can read a musical phrase on a score and hear the sound of the whole expression in their head, rather than trying to count which line each note is on, count out the tempo, etc. Objectively, the Roman system is easier to teach up to a basic level, especially to someone illiterate. Basic calculations on a counting board are straight forward and easy to explain and motivate. Multi-digit multiplication gets a bit annoying in both cases because it involves the summation of many partial partial products. Long division and square roots get nasty in both systems. Where the Hindu–Arabic system really shines is when people need to frequently work with very large numbers, very precise numbers (though remember there were no decimal fractions per se in Europe until >1600), or numbers of different orders of magnitude, have access to cheap and abundant paper, and can spend years training to do basic arithmetic. The biggest advantages of pen and paper methods for basic arithmetic are that it’s easy to see the whole work process, and therefore more easily check for mistakes, and that writing the final answer doesn’t take as much space. It’s also much easier to write down and explain pen and paper arithmetic methods in a printed book. The counting board methods are often faster to perform. But more importantly still, pen and paper arithmetic is easy to generalize to more sophisticated mathematical notation for fractions, algebraic equations, etc.
• allencoin 9y ago
  I remember learning Roman numerals in 1st or 2nd grade and finding them to be a lot easier for basic arithmetic. I mean, all of the examples we used in school to learn basic arithmetic had things like "(picture of 4 apples) + (picture of 2 apples) = (picture of 6 apples)". Roman numerals are kind of a pictograph in a way.
  • rtpg 9y ago
    but if you're in first grade then you do IV + II = IVII...?? To each their own, but I'm unsure of the relative ease of roman numerals
    • tree_of_item 9y ago
      One of the points the article made was that IV wasn't really a thing when the Romans were actually using these numerals.
• gozur88 9y ago
  You'd think you could just collect all the numbers and then translate, i.e. if I add some things and get XXVVVVVIIIIIIIIIII, I know that's XXVVVVVVVI = XXXXXVI = LVI. The problem is IV isn't 6, it's 4.