Comment by butisaidsudo | Hacker News Reader

Comment by butisaidsudo | Hacker News Reader

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For Roman numerals to make sense you need to remove access to convenient writing tools. In a world without paper and pens, Arabic numerals are the ones that seem unwieldy.

I was on a multi-day hiking trip with a friend where we just had a pack of cards and some spiced rum to keep ourselves entertained in the evening. The first night we decided to play cribbage, but we didn't have paper or the like to keep score on.

At first I tried using a stick to scratch out numbers in the dirt. It was doable, but very awkward. The low light of the camp fire made it hard to see, and I quickly ran out of undisturbed soil within arms reach.

I then gathered a few twigs to shape into numbers, using the patterns you'd see on an LED clock. That worked reasonably well, but took more effort than I liked.

I switched to Roman numerals thinking that the simpler shapes would be easier to work with. This turned out to be true, but I discovered it had the added benefit of being really easy to increment numbers.

In most cases, incrementing is incredibly simple. To go from 0 -> 1 -> 2 -> 3, you simply add a stick each time. To go from 3 -> 4, you pinch together the bottoms of the 2nd and 3rd sticks. Then you take the first stick away, then later drop it on the other side of the V. Add another stick, cross some sticks, etc.

There are a lot of things that seem poorly done at first glance, but make total sense when you understand the environment they were developed in. I've heard younger folks wonder why old TV shows are so poorly written. When you've always been able to watch on demand (or rent episodes on DVD), it's hard to understand the restrictions on writing when there was no way for your audience to watch previous episodes if they hadn't seen them when they had aired. And of course this is true for most software I've ever worked on.

I like your thought experiment to illustrate why Roman numerals make sense for the time. I think you're right! It is more practical given the tools than Arabic numerals.

To respond to the hiking story; I would opt to use binary if I were in the same situation. I think it's a superior system to use especially out in the woods. It's hard to read for many folk unexposed to binary but it allows you to encode lots of base 10 numbers using a few items. For example, a twig rotated 0 degrees could signify 0, and a twig rotated 90 degrees could be 1. 5 twigs gets you 0 - 31!

To bring it back to the Romans... I realize Binary wasn't commonly used in the Ancient World and that is something I find a little baffling. Base 2 seems like it would be more useful in a pre-pen and paper world, especially when you have to etch something. The closest thing I can find is the Inca Quipu ("talking knots") but even that encodes base 10 numbers into base 4 knots.

If base 2 was widely used by human beings our fingers could have encoded up to 1024 digits.

How did base n > base 2 evolve?

Natural binary takes a lot of work to increment, but there's Gray code: https://en.wikipedia.org/wiki/Gray_code

Gray code takes a lot of work to learn. It's not very likely that people would have discovered it as a way of counting on their fingers (or with other objects) prior to the development of other number systems. If they had, it would have been extremely hard to teach, learn, or verify that someone was using it properly, without other number concepts to use as a reference!

I learned to count in binary on my fingers in high school, and I can confirm that it's more physically challenging than counting the conventional way, and also harder to remember and communicate numbers without translating them into language.

Although you can only represent 10 values instead of 1024 values, remembering one of those values is also correspondingly about 100 times easier.

Plus, it's easy to teach the direct correspondence between fingers and objects even to children or to illiterate or innumerate adults. We can imagine people using fingers this way for a straightforward matching task: "I saw this many dogs!" "I'll give you this many melons!"

Indeed, the unary system is the easiest for very small numbers.

And then it leads to separate "uninterpreted" names for each number of fingers that a person can hold up. After that there's a natural route toward place value with units of either one or two hands.

Even the shapes of the Roman numerals may have originated from chunking "hands" as units:

https://en.wikipedia.org/wiki/Roman_numerals#Hand_signals

(although the Romans did use further multiplicative combinations of 5s and 10s, they didn't come up with implicit ways to continue the process indefinitely)

> To go from 0 -> 1 -> 2 -> 3, you simply add a stick each time. To go from 3 -> 4, you pinch together the bottoms of the 2nd and 3rd sticks.

The subtractive part of Roman numerals is an innovation the Romans generally didn't use. (Which is why clocks say IIII and not IV.) It adds significant complexity for no benefit.

> Which is why clocks say IIII and not IV

There seems to be a bit of uncertainty on that [1].

[1] https://www.electrictime.com/news/roman-iiii-vs-iv-on-clock-...

Interesting. There's definitely a benefit, though:

MCMLXXIX -vs- MDCCCCLXXVIIII

> incrementing is incredibly simple

So Roman numerals are like a decimal analogue of Gray codes?

Hadn't seen those before! Yeah I suppose they are, although could you call Roman numerals decimal? I don't know enough about the technical definition to say. I'm assuming "decimal" and "base 10" are synonymous. And the base is the number of distinct digits you have to work with. Roman numerals don't really work that way though. Man, I remember when I first learned about different bases, it took me a bit to wrap my head around the idea that base 10 is completely arbitrary. It doesn't make more sense than the other bases, it's just what we standardized on. Now I'm realizing that maybe the idea of bases themselves aren't some rule of the universe.

It's not a decimal system, no. The decimal system is a positional system (just like binary, hexadecimal, sexagesimal ...). In a positional system, part of the information of a symbol is in its position in the whole string, and it always represents an 1-to-(base-1) part of its base. That's why every base is base-10 when represented in itself.

However, most human languages encode numbers as an algebraic expression: "one {times} thousand {plus} three {times} {one} hundred {plus} thirty {plus} eight" = 1338. Some languages are more strict/consistent in this than others. English has its quirky unique words for 11 and 12 (instead of one-ten and two-ten), Danish and French enjoy adding multiplications and divisions to the mix (due to retaining more of their 20-based origin): "quatre-vingt-dix-sept" = 4 * 20 + 10 + 7 = 97, "halvtreds" = halvtredsindstyve = (3 - 0.5) * 20 = 50.

The Roman numeral system works likewise, it's just a shortened form of saying the number. Which means you need a symbol for every word for a magnitude, and your expression range is limited to the words your language has for ever greater quantities. But for most humans, in our daily lives we rarely need to precisely number more than, say, a few ten-thousand things. Beyond that "myriad"/"a lot"/"uncountable" suffices.

The simple form of the Roman system is just additive: MDCLXVI. You could add subtraction: XC instead of LXXXX to make it faster to write. But then you're already on the path to a positional system.

Romans used a device called a "calculator" which was a flat plate with vertical channels cut or pressed into it. It resembles what we would now call an "abacus".

get rid of the subtractive rule, and roman numerals are just a way of writing the state of a calculator, with each letter representing a calculi, and the value of the letter representing which column it is in.

most abacuses are base 10, with a special segment in each column for a stone that represents "5" in that position.

Thus, each roman numeral is either a power of 10, or half a power of 10.

with that context in place, the method of writing it isn't a positional system, exactly, but it's a way of serialising a positional system by capturing positions by naming them explicitly.

edit:

here's an image of a replica calculator, note how the columns are labeled.

https://i.pinimg.com/originals/9c/79/d6/9c79d6681331953e09be...

edit2: it’s also easy to see in this context how this relates to our system of money, and coin values and names such as “cent”, if roman numerals aren’t decimal then neither are coins.

> English has its quirky unique words for 11 and 12 (instead of one-ten and two-ten)

Of course, consistency with other number names above 20 would lead to "ten-one" and "ten-two", not "one-ten" and "two-ten"; or consistency with other number names below 20 would lead to "oneteen" and "twoteen", or some variant (as with "thirteen" in place of "threeteen").

Well, it's certainly based on thinking in terms of powers of ten. That is, CXLV (145) is 100 and 40 and 5, as opposed to, say, 1 gross and one.

OK, valid point of about whether it's fair to call Roman numerals decimal. They're similar because all the symbols (except I) are multiples of 5 and 10, but it's not really the same thing as true base 10.

the abacus that they are based off of was in fact, base 10, positional. See my other comment.

Swiss Jass games (which typically require adding up three digit scores until you reach a four digit goal) use a specialized writing system that is easy to increment as well:

https://www.earthli.com/jass/manual.php?page=scoring#writing...

How many numbers do you have to keep track of simultaneously during a cribbage game? What range is required for these numbers?

Normally you have a board with pegs for each player, and rounds have you move your pegs up a number of holes for your score. First to 121 wins. Increments are anywhere from nothing to 29 (excluding 19, 25-27).

Personally I'd just make a few differently sized sticks/rocks to represent 5s, 1s, and 30s. Maybe 2s since those are common. Maybe -1s as well, and consolidate if I want during shuffling/dealing. (30s being convenient because if you lose by multiples of 30 it counts as a skunk / an extra loss.) (Edit: Actually I might just make a board in the dirt with sticks to designate every 5 points, like a real board, and each player just puts down 1-5 rocks in the segment they are in. But that gets dangerously close to just finding 120 small rocks... On the plus side, that's almost enough rocks for one side in a game of Go.)

People invented positional number systems that were also easy to write and increment well before Roman numerals. Those look good on monuments but that's about it.

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