>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.

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.

>You're speaking from a position of bias and ignorance

I made a fairly concise point, but you're right about one thing

>we all do this sometimes

Because you most certainly just did

>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.

That's not the issue with the "easily repeatable pattern". The issue is this:

>V = X/2, L = C/2, D = ↀ/2, ↁ = ↂ/2, etc.

Do you see that "etc" you post? That's because, to represent 10^n, you need to know 2*n + 1 characters.

This is an obvious downfall that Hindu-Arabic numerals obviously do not have. Even if you ignore the shorthand (which would be foolish, parsing 5-9 characters in a row can take a serious moment of focus), you still need to know an unbounded amount of characters to represent the integers alone, and let's not even discuss the obvious fallacy of fractions etc..

Also,

>Objectively, the Roman system is easier to teach up to a basic level

Even this is an unfounded statement. Make an argument. Prove it. Objectivity of teachability can't possibly be something you just throw out casually in an argument, let alone an argument where you told someone they were biased and ignorant for, as far as I can tell, no reason at all.

That was not a personal judgment, just a straight-forward factual assertion. (Unless you have spent a decade doing calculations on a counting board...?) Pretty well everyone, including myself, is speaking from a position of bias and ignorance when discussing Roman numerals and counting boards, because these are not pervasively used for basic calculation in our society. We should bear that in mind and try to get outside of our preconceptions before making off-hand judgments.

It’s very seldom that anyone in the Roman empire – or medieval Europe – would need bigger numbers than, say, one million, especially for writing final answers down. But at least one counting board we know about from 300 BCE probably had lines for 10 orders of magnitude, and it’s not like more couldn’t easily be added if necessary. http://www.akg-images.fr/archive/-2UMDHU23V18G.html

As for fractions: Roman fractions were mostly twelfths (think inches and ounces).

Roman numerals and counting boards are obviously not an ideal system for writing down very large or very precise numbers, as I said. I’m not here claiming that they are a better basis for society’s numeration than more strictly positional Hindu–Arabic numerals.

My point is that we shouldn’t be so hasty to dismiss them out of hand or exaggerate their flaws. They were a highly effective system for doing complex calculations: precisely estimating the positions of stars and planets centuries into the past/future, running a bureaucracy overlooking an empire of 60+ million people, building large-scale engineering projects, and so on.

When it comes to teaching, I can only go by my own anecdotal experience trying to teach young children about numbers, the writings of various elementary math teachers, and the fragmentary remnants of debates in medieval Europe. I don’t know of any modern peer-reviewed research about Roman or medieval European counting boards, sorry.

(There is lots of evidence that learning place value using Hindu–Arabic numerals is very difficult for children, compared to other concepts and skills, requiring several years of study before primary school students really figure it out.)

Seems like counting boards might make a partial comeback sometime though: https://vimeo.com/204368634

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.

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

One of the points the article made was that IV wasn't really a thing when the Romans were actually using these numerals.

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.

That's why you group five I's into a V?

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.

> 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.

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

Because if you don't memorize your 'primitive' algebraic rules you'll end up just doing a pullback with roman numerals in the middle. Integers don't have any values, they're mathematical objects with certain properties. Asking about 'the value of 5' doesn't make sense unless you're trying to convert to another, already known, number system.

What is 2 + 5? Well 2 is II and 5 is V which is IIIII. So then we have IIIIIII which is (IIIII)II or VII which is 7.

I just want to point out that it's humorously appropriate for a user named "Spivak" to mention pullbacks.

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.

Right, much like I + V is IV.

(There is more help; it's not as simple as people have been portraying, though...)

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.

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.

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.

That numbers have interesting relationships to each-other is not an “ugly thing”, but rather a beautiful thing. The relationships exist whether we pay attention to them or not, so giving people a way to write and pronounce those relationships is a nice way to improve numeracy.

Writing numbers in multiple possible ways is really not that big a problem for most uses (for record-keeping in business transactions maybe). Personally I recommend becoming familiar with two different normalizations: all same-signed digits (the form we use now) or [–5, 5] with digits before a terminal 5 rounded away from zero. But in general for personal scratch work numbers don’t need to be normalized unless you feel like it. As long as you understand that 20–3 is the same as 10+7, it doesn’t really matter which one you think of as primary.

We use multiple number representations all the time, with e.g. the vulgar fraction 11/8 alternately representable as the “mixed number” 1 + 3/8, or as the decimal fraction 1.375, the simple continued fraction [1; 2, 1, 2], the percentage 137.5%, or the common logarithm ~10^(0.1383). Figuring out that 1.375, 1.385̅, 1.42̅5̅, 1.43̅5, and 2.6̅2̅5̅ are the same number isn’t that hard, with some practice.

You just need to add a sort to to get VI from that -- something explained in the article regarding multiplication.

V + I = VI though.

You just have to put the larger number first for the addition.

That might be enough to prevent you from getting any well-formed wrong answers.

It's not enough of the story, though - consider:

    VI + V = VIV
    V + IV = VIV

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.

> 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.

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.

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.

> 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.

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!)

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.

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.

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...

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.

I think we can all agree addition is fairly easy. Multiplication, since it can be explained in terms of addition, it also not really that hard. Subtraction and division are harder, as I originally noted, and by extension division, are harder, as there is not one simple rule for conversion before subtraction that won't require additional conversions unless you go to the simplest form, which is unwieldy (along string of I's).

From my example, which was meant for subtraction, not addition or multiplications, we get the following:

XLII - XIII

so we convert to an optimized form for arithmetic (which is, as I've noted a second format you need to know and keep track of, since it's not valid Roman numeral format

XXXXII - XIII

then we note that we have to borrow from the X "place", so let's do that

XXXVIIIIIII - XIII

now we can actually subtract

(XXXX - X) + V + (IIIIIII - III) = XXX + V + IIII = XXXVIIII

Now we have to convert back to valid Roman numerals

XXXVIV = XXXIX

This is using common current arithmetic techniques. I'm not sure whether this is the method they would have employed, or whether it would have been a different method that could have been easier or harder depending on how it combines with the attributes of this number system. Important things to note about the method shown are a pre and post conversion step (or steps, if you don't get every sub-conversion right initially or don't go to a lowest common format), and then the arithmetic itself requires more borrows.

Well, we can eliminate notations like XL, IV, IX, etc because, as noted in the article, those were not in common usage until much later. Even if we don't, then the first step is to expand these values, much like you would with fractions (i.e., convert to lowest common denominator).

Subtraction is as easy as addition (unless you go into negative numbers). The stategy is to walk the subtrahend left-to-right, eliminating from both as you go. If you run into a value that you don't have, then you expand the minuend from the right until you do.

For your "difficult" example of 42 - 13:

XXXXII - XIII

1 Eliminate an X from both.

Now you have XXXII - III.

2 Eliminate I from both

Now you have XXXI - II. Repeating (2) gives you XXX - I.

3 Since there is no I to eliminate, expand the minuend from the right to create one.

Now you have XXVV - I, repeat since you still don't have a I, repeat (3) which gives you XXVIIIII - I

Eliminate the from I both, giving the final result: XXVIIII.

Optional, you can "reduce" this to XXVIV, like you would if you had a fraction, like 3/9, but as stated in the article, this notation was not common place in ancient Rome.

XLII – XIII = XXLIX, of course ;)

you skipped over III + I = IV, and VI+III = IX.

Subtractive notation is confusing.

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).

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.

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.

They mention this in the article, and say there is very little evidence of this being common in ancient Rome.

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."