How did anyone do math in Roman numerals? (2017)

https://www.washingtoncitypaper.com/columns/straight-dope/article/20854121/how-did-anyone-do-math-in-roman-numerals

by pmontra • 6 years ago

128 124 6 years ago

dhosek2• 6 years ago

When I got my math teaching credential, there were bunches of interesting historical things we learned along the way including Egyptian Fractions https://en.wikipedia.org/wiki/Egyptian_fraction

Never actually used any of it so most of it has evaporated from my memory along with calculating square roots by hand, but it's nice to know at least enough to be able to look up the information if I want it.

shagie• 6 years ago

Egyptian fractions have an interesting property that may have been useful back then.

Consider the problem "How do you divide five things for eight people?"

Simple - cut everything into 1/8 and give each person five.

But Egyptian fractions give an even easier way. 5/8ths is 1/2 + 1/8. Divide four wholes into halves - give each person a one half. Take the unit and divide it into eights and give each person one. The answer to this problem is the way you write the number itself. This makes division of the items easier and simpler.

jrootabega• 6 years ago

That's how I cut up bell peppers into the right sizes. They're either 4 loved or 3, so cutting them into 3 or 4 equal sized piles is a challenge

msla• 6 years ago

Is it blogspam when it's called syndication?

Here's the "original" source of the column, which is on a less... determinedly fashionable website, so it might be friendlier to mobile users and people who dislike fixed headers:

https://www.straightdope.com/columns/read/3330/how-did-anyon...

Also, previously:

https://news.ycombinator.com/item?id=14818633

gowld2• 6 years ago

Fixed headers are better than fixed autoplay ad videos.

forinti3• 6 years ago

Roman numerals are very economical.

The base with best radix economy (except for e) is 3. But roman numerals are better still.

For representing 0-999, you would need 19 base-3 digits, but only 15 roman numerals (plus a symbol for zero).

monadic23• 6 years ago

...in terms of digits, which is a very odd quality to optimize for.

gowld2• 6 years ago

...in terms of only the worst case numeral.

Common numerals like II III VII VIII have worse length, while I IV VI are same, and V X IX are better.

And factoring in per-digit cost, Roman numerals up to 999 have 5 distinct digits, 46% more cost per digit than base 3, making it worse than base 3 in almost every case, information theoretically. (You could win some back with a huffman encoding, though)

jbay808• 6 years ago

A good one to optimize for if you need to carve them into stone columns :)

Animats• 6 years ago

Add and subtract on a good abacus (like a Japanese soroban) is quite fast. Multiply and divide are miserable, but in ordinary trade, it's mostly add and subtract with the occasional multiply.

With a soroban, a slide rule, and a book of tables, you can do most classical engineering math. Slowly.

dwheeler• 6 years ago

In general, with an abacus. Roman numerals for generally used for recording information, not for calculating with them. The article emphasizes how easy it is to add and subtract with Roman numeral notation, but everything else I've read emphasizes the Abacus even for that. After all, most people today calculate with calculators as well, and we have a snazzy Hindu-Arabic system for numbers.

hencq• 6 years ago

It's actually somewhat surprising to me that the Romans didn't invent the concept of zero, when they used the abacus for daily calculations. With the benefit of 20:20 hindsight you'd think that the concept of zero would follow quite naturally from the concept of an empty column on the abacus.

Sharlin2• 6 years ago

But that's the thing about Roman numerals: you don't need a placeholder number to represent empty columns. And for "what is XVI subtracted from XVI" they could just use a word meaning "nothing", such as nihil or nihilum. The need for the concept of zero as we understand it really only arises together with a place-value system.

andrewl• 6 years ago

In An Introduction to Mathematics (1911) Alfred North Whitehead wrote:

By relieving the brain of all unnecessary work, a good notation sets it free to concentrate on more advanced problems, and, in effect, increases the mental power of the race. Before the introduction of the Arabic notation, multiplication was difficult, and the division even of integers called into play the highest mathematical faculties. Probably nothing in the modern world would have more astonished a Greek mathematician than to learn that ... a large proportion of the population of Western Europe could perform the operation of division for the largest numbers. This fact would have seemed to him a sheer impossibility ... Our modern power of easy reckoning with decimal fractions is the almost miraculous result of the gradual discovery of a perfect notation. [...] By the aid of symbolism, we can make transitions in reasoning almost mechanically, by the eye, which otherwise would call into play the higher faculties of the brain. [...] It is a profoundly erroneous truism, repeated by all copy-books and by eminent people when they are making speeches, that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilisation advances by extending the number of important operations which we can perform without thinking about them. Operations of thought are like cavalry charges in a battle—they are strictly limited in number, they require fresh horses, and must only be made at decisive moments.

John Allen Paulos discusses the power of notation in his book Beyond Numeracy:

A German merchant of the fifteenth century asked an eminent professor where he should send his son for a good business education. The professor responded that German universities would be sufficient to teach the boy addition and subtraction but he would have to go to Italy to learn multiplication and division. Before you smile indulgently, try multiplying or even just adding the Roman numerals CCLXIV, MDCCCIX, DCL, ANDMLXXXI without first translating them.

Numbers may be eternal and invariant, but numerals, the symbols used to represent numbers, are not, and the above anecdote illustrates how easy it is to take for granted the Hindu-Arabic numerals we use today. The history of numeration systems is a long one extending from prehistoric times to the adoption in the Renaissance of our present system. The heroes of the story are the nameless scribes, accountants, priests, and astronomers who discovered the principles of representing numbers systematically.

bhntr32• 6 years ago

Thank you for the quotes. I think this essentially answers (or at least provides a hypothesis for) the question I posted about how the numerical system influenced their math.

sixstringtheory• 6 years ago

Nice quotes. The first one is how I think about programming languages as well. We've moved from binary/punchcards, to assembly, to imperative and declarative layers, to OO/functional code that can be statically analyzed, and systems designs that can be automatically verified. There's all sorts of memory management tools from manual allocation and freeing, to garbage collection and reference counting, and optionals to avoid nulls that can be statically checked. Each thing has solved entire classes of problems and we're getting more and more software solving different kinds of problems, faster.

morelisp2• 6 years ago

Whiggish bullshit. Functional code and GC were invented in 1959. Some level of static typing was de rigeur in most application-development languages after assembly and before the scripting boom starting in the late 80s. In Coders At Work Frances Allen bemoaned the effect C's popularity had on automated program analysis since 1970:

C has destroyed our ability to advance the state of the art in automatic optimization, automatic parallelization, automatic mapping of a high-level language to the machine.

"Progress" in CS remains fad-driven pop culture churn.

aswanson• 6 years ago

Amazing, the nth level effects a superior notation can have in problem solving. The same consequences of a higher-order programming language.

rozgo• 6 years ago

This is absolutely the case for programming languages. From Haskell to Rust. Programming languages shape the way we think and reason about the problem. Sometimes it does this so well, it leaves us starring into the abyss. Facing dead on the problem domain—after all sides quests have been removed. Once you can delegate away all the busy work, you can focus on bigger and better abstractions.

blub4• 6 years ago

Both Haskell and Rust are poor examples because one has to focus more on the notation (i.e. types, declarations, etc).

Python is better.

boogies• 6 years ago

Our modern power of easy reckoning with decimal fractions is the almost miraculous result of the gradual discovery of a perfect notation.

Great quote. I wonder if this process will continue with adoption of duodecimal numbers.

chrismarlow9• 6 years ago

I'm reading a book called Mathematics from the Birth of Numbers by Jan Gullberg and it covers these systems from various cultures in depth. Great book and easy to read even for the non mathematician. It reads like a cross between a history book and a technical reference. This is literally the first topic in the book.

philwelch6• 6 years ago

Adding Roman numerals, at least, isn’t hard. If it was I doubt Roman numerals would have ever lasted. Arabic is still, in my opinion, easier to add—from the perspective of a lifetime spent exclusively doing arithmetic in Arabic numerals—but it’s not much easier until you’ve memorized all 50 unique sums of one-digit numbers. Multiplication, though, that’s the real difference maker.

Jedd2• 6 years ago

Adding Roman numerals, at least, isn’t hard.

I find it insanely difficult - but as you note, a lifetime of arabic numerals, and a lack of skill in appropriate tools (eg. an abacus) will skew that comparison.

... but it’s not much easier until you’ve memorized all 50 unique sums of one-digit numbers.

This doesn't feel right.

I don't think I have memorised the sum of all pairs of 1-digit numbers - but contemplating this now, it's impossible for me to be sure. I'm not sure where multiplication kicks in for breaking down larger numbers into quotient and divider for me, let alone 'most people'.

Given the maximum value (sum) of two x 1-digit numbers is 18, a naive assumption is the permutations don't really number 50 (I get 45) - and given that roman numerals didn't have a zero, a fairer comparison would be [1-9][1-9] (36 unique combinations)

Either way, in any counting system there's presumably a similar 'memorisation' gate you have to pass for the fundamental set. With roman numerals there was historically 7 I think - I, V, X, L, C, D, M - and summing those wouldn't be anywhere near as straightforward as summing sets of single-digits, so I don't think the comparison of this requirement is as skewed against arabic numerals as you suggest.

User232• 6 years ago

Please explain the easy method for adding, for example MCMLXVII and LXV. I mean that seriously, I'm curious what the trick is.

d2v2• 6 years ago

This is only tangentially related, but until now I never really thought about how I sum single digit numbers, but it's not by having memorized all combinations. I'd say I have all combinations that sum to 10 or less memorized, which adds up to 25 unique combos (plus the rule that number + 0 = number).

If I can tell it sums to more than 10, I break it up mentally into [larger number] + [smaller number] = [larger number] + ([smaller number] - remainder) + remainder, where [larger number] + ([smaller number] - remainder) = 10

You could break it down further by just memorizing what each digit less than 10 is when you add 1. Then you can do addition like 5 + 4 = 1 + 1 + 1 + ... = 2 + 1 + ... = 8 + 1 = 9. Then you'd only have to memorize 9 things (10 if you include 0). I guess this assumes that you know the order numbers go in though, whereas memorizing all of the combinations doesn't require that.

rland• 6 years ago

I wonder if arabic numerals really are the best system.

Too much inertia is invested in them now but I wonder if a yet easier representation exists. For example, I know that there was a brief push to use quaternions in physics.

ThrustVectoring• 6 years ago

Base 12 is likely better than base 10. Twelve is the number of phalanges on your hand that you can touch with your thumb, so if that counting system had caught on and stuck we'd likely have a better base for divisibility.

jacobolus2• 6 years ago

Paulos in the quoted snippet is arguing against a straw-man version of the past, set up as a foil to prove some unrelated point, rather than taking it on its own terms and trying to understand it. This is a common conceit of modern writers, but it does a disservice to readers.

Roman numerals by and large were not used for calculating, but for recording calculations. Those calculations were done (as the very word “calculate” suggests) using pebbles or other tokens, on some kind of counting board.

The point of Roman numerals is to be as direct as possible a representation of the state of the counting board. They are a serialization format.

Think of it as the JSON (or s-expressions if you like) of the ancient world. You don’t run your algorithm by writing and rewriting JSON literals over and over with pen and paper. They are just a record; your computation is done using a different automatic tool.

When someone writes down the state of a graph structure as a big nested JSON literal, you don’t laugh at them for having a uselessly unwieldy written expression.

bhntr32• 6 years ago

I feel like the article misses the most interesting question about Roman numerals and Roman (Greek really) math. How did the numerical system influence the math that they developed and used?

The Greeks were really into geometry using the compass and straight edge so they actually did a lot of math without really needing numbers at all. They viewed calculation as less worthy of mathematicians and my understanding is that we don't have a lot of evidence for how merchants and engineers did basic calculations since most of the great Greek math texts ignored it.

Algorithms and algebra probably existed in some informal way but they weren't really formalized until the Arabs did it with the help of Arabic numerals.

So, while you can do some calculations in roman numerals or using an abacus, the interesting question to me is: Did the Greeks (and the Romans) not develop algebra or use Arabic numerals because they weren't that into numbers as compared to geometry? Or was it the other way around? Did the clumsiness of doing calculations in Roman numerals keep them from developing more complex systems of numerical calculation?

I'm not an expert on the subject at all but it's always interested me. It makes me think of Bret Victor's Media for Thinking the Unthinkable (http://worrydream.com/MediaForThinkingTheUnthinkable/)

k__• 6 years ago

"Did the clumsiness of doing calculations in Roman numerals keep them from developing more complex systems of numerical calculation?"

Probably? I mean, look what the world achieved after it left roman numerals behind.

goatinaboat2• 6 years ago

Probably? I mean, look what the world achieved after it left roman numerals behind.

Yet the Romans were able to construct aqueducts that are still standing, and a road network spanning thousands of miles, and many other great feats of civil engineering.

DNied2• 6 years ago

look what the world achieved after it left roman numerals behind.

Look what the world achieved after we started wearing button shirts.

setr• 6 years ago

Did the clumsiness of doing calculations in Roman numerals keep them from developing more complex systems of numerical calculation?

The book Mathematics for the Million[0] suggests a number of limitations were created from the roman numeral system, largely due to the difficulty of division, making infinite series and even large (and extremely small) numbers difficult to work with, intuit and even see outright. As a specific example, it suggests the Achilles and the tortoise paradox[1] is trivially intuited and resolved in the decimal system, whereas no relationship between each division is made clear in the roman numerals

[0] https://archive.org/details/HogbenMathematicsForTheMillion/p...

[1] https://en.wikipedia.org/wiki/Zeno%27s_paradoxes#Achilles_an...

pvg• 6 years ago

but it's always interested me.

There's Morris Kline's Mathematical Thought from Ancient to Modern Times which is 3 glorious fat volumes of just this stuff.

tjoc• 6 years ago

The Greeks had several competing numeral systems (https://en.wikipedia.org/wiki/Greek_numerals) including a decimal based system. What I do not get is how come the Romans adopted (if they adopted from the Greeks) the most unwieldy one.

jacobolus2• 6 years ago

The Greeks were really into geometry using the compass and straight edge so they actually did a lot of math without really needing numbers at all. They viewed calculation as less worthy of mathematicians and my understanding is that we don't have a lot of evidence for how merchants and engineers did basic calculations since most of the great Greek math texts ignored it.

Greek society was awash with arithmetical calculations. It wasn’t written about probably mostly because it was considered so obvious and commonplace. (Though it’s also a bit hard to say quite what was written about, since we’ve lost the vast majority of the books from the time.)

http://worrydream.com/refs/Netz%20-%20Counter%20Culture%20-%...

* * *

Did the Greeks (and the Romans) not develop algebra or use Arabic numerals because they weren't that into numbers as compared to geometry?

Algebra and arabic numerals really took off with the introduction of cheap paper and printed books.

It’s really hard to transmit the oral culture of skilled use of a counting board via printed book (we might call it tacit knowledge), but you can pretty straight-forwardly print out a pen-and-paper arithmetic algorithm.

Grustaf2• 6 years ago

Why would it be significantly harder than using some other system of numerals?

shultays2• 6 years ago

They probably had algorithms for it, but even then it sounds challenging.

Addition sounds easy and works mostly like how we do base 10 addition. I imagine they would first go for sub 5 part which is a bit exceptional and had to be manually. And then start grouping letters together like we do and create carries if they reach the next letter.

Subtraction sounds harder. It sounds close enough to our base 10 system but borrowing from next digit is much more complicated. Like subtracting D-I (500-1). The answer is CDXCIX but I am not sure how I can go there. And now imagine this with more complex numbers

But multiplication and division? in that weird base format? I won't even try

Keep in the mind that they don't convert or even think numbers in base 10 system like we do.

Grustaf2• 6 years ago

How numbers are written down does not necessarily correspond to how you do calculations. Given that 499 was called 499 I’m pretty sure they thought in base 10, so subtracting 1 from 500 would be trivial. Writing it down took a few more symbols but so what?

kccqzy• 6 years ago

When I was learning programming, one of the beginner's exercises was to implement addition, subtraction, and multiplication of natural numbers "by hand" (division wasn't covered). There you learn to formalize the grade school algorithms you know by heart into concrete code, a necessary mental process for beginning programmers.

At the end of the session, the teacher asked us to try the same with Roman numerals. That was when I knew it's significantly harder.

Koshkin2• 6 years ago

Well, you can try and then you'll appreciate the difference, I am sure. Even if implemented in computer hardware, operating with Roman numerals would either be slow or take many more transistors (or both). As to why in a philosophical sense, it is because the positional system was invented specifically as a computational device, which only happened many years after people learned how write numbers down. Optimization effort takes time, and a random solution is not guaranteed to be optimally suited for a particular application (such as performing calculations).

082349872349872• 6 years ago

We all do the same thing. Our calculators work in binary, and we only do i/o in decimal. Their calculators (slaves who pushed pebbles, calculi, around) worked in a decimal place-value system (or used egyptian/russian peasant multiplication), and only the i/o was done in roman numerals.

dan-robertson• 6 years ago

I am very wary of just about anything published on the history of science or mathematics. Much of it tends to be very low quality full of just-so stories, or at best whiggishness. Unfortunately much of the work by academics on the subject is not much better as the field is full of former mathematicians or scientists with an interest in history rather than lots of historians with an interest in mathematics or science.

phonon• 6 years ago

Romans essentially used the same binary arithmetic computers use today for multiplication; doubling and halving.

http://www.phy6.org/outreach/edu/roman.htm

MayeulC• 6 years ago

Yeah, this was discussed previously: https://news.ycombinator.com/item?id=13636277

And division basically works the same.

Discussed article is https://thonyc.wordpress.com/2017/02/10/the-widespread-and-p...

Nowhere as complex as it's made out to be.

frandroid• 6 years ago

The IIII-for-4 notation survives today on the faces of clocks.

What? On which clocks?

paragraft• 6 years ago

When I search for "clock with Roman numerals" on Google images I see about 50% use IIII.

gerikson• 6 years ago

On this new Omega Constellation, just as a random example

https://www.omegawatches.com/watch-omega-constellation-omega...

oblio• 6 years ago

I was as baffled as you when I first saw IIII instead of IV...

sn412• 6 years ago

It is well-known that the place-value system introduced into Europe from India via the Arabs played an invaluable role in modern arithmetic.

But, if you read the "Sand Reckoner" by Archimedes, what he lays out are the rudiments of a place-value system. He essentially describes the modern notation, but not rules for addition, subtraction, multiplication and division using this notation.

Another tidbit: if you see the recent movie about Shannon "The Bit Keeper", he shows the journalist a mechanical device (he designed?) which can do calculations using Roman numerals.

jbit• 6 years ago

I'd never heard of this documentary, so I took a look. I assume you meant "The Bit Player"? ( https://www.imdb.com/title/tt5015534/ ). It looks interesting, and it's on Amazon prime video so I'll give it a watch. Thanks!

sn412• 6 years ago

Sorry, you are right.

reidacdc• 6 years ago

I have read that the uptake of Arabic numerals was actually fairly slow and fraught in Europe, but of course can't put my hands on any references.

There's a reddit thread [0] that might be some of what I saw, and my wife does paleography work where she runs across books of accounts that are rendered in Roman numerals, because that's how formal accounts were prepared, even if the actual accounting was done by other means.

That same reddit thread has link to an "algorists vs abacists " article [1] which purports to back this up, but I can't confirm because the article is paywalled for me.

Edit: Moved/fixed links.

[0] https://www.reddit.com/r/AskHistorians/comments/12m0vp/how_a...

[1] https://www.jstor.org/stable/pdf/2686479.pdf?seq=1

siraben• 6 years ago

Contrast Roman numerals with the rod calculus[0] invented in Ancient China. Wikipedia has a list of algorithms for calculating with rods, from the usual arithmetic operations to fractions, division, square and cube roots, Gaussian elimination, and solving polynomials.

It would seem that such tasks would be extremely difficult for someone working with the Roman numerals.

[0] https://en.wikipedia.org/wiki/Rod_calculus

zapzupnz• 6 years ago

Not really related to the article per se but I always find it interesting how one may become tempted to say "this alternative to a thing I already know makes so much sense, why don't we always use it?"

I felt the same way when encountering Chinese numbers via Japanese. If 二 is two, 十 is ten, 四 is four, and twenty-four is 二十四, that's so clear! Two tens and four!

I quickly decided that this number system, though something I'd obviously need to learn and become acquainted with if my Japanese learning were ever to progress, wasn't necessarily as easy as I initially imagined. Yes, there are no places, but numbers in this system are grouped at different boundaries — not every thousand but every ten-thousand.

So 六十七億八千三百一万五千四百二十一 breaks up as sixty-seven hundred-thousands eight-thousand-three-hundred-and-one ten-thousands five-thousand-four-hundred and two-tens-and-one — and obviously, that's not quite how we would represent six-(billion/thousand-million) seven-hundred-and-eighty-three-million fifteen-thousand-four-hundred-and-twenty-one, or rather 6,783,015,421.

I post this not to discuss the positives or negatives on the Chinese number system compared to the Arabic one or vice-versa. Rather, just how one's imagination can be so easily captured by the apparent simplicity of an alternative to that with which one is familiar, almost to the point of wanting to adopt it altogether. The realisation of where things get tricky for oneself, often not coming until quite a bit later, sometimes doesn't come until later.

For myself, I tried using roman numerals for my own math for a long time but stopped when I found division too brain-breaking!

earthboundkid• 6 years ago

Living in Japan, I became accustomed to using numbers for up to around 10,000 yen ($100USD) due to interactions at the stores and around town, but when I would hear the price of a car (1,000,000+ yen or 100 myriad yen) or a house, it would just confuse me and not register at all. It’s all just based on your personal experience, I think.

pjmlp• 6 years ago

I have the same experience with German reversed way of speaking numerals, yet I speak the language fluently, including a good understanding of Swiss German as well.

But reversing back the numbers into my Portuguese brain, just doesn't work after a certain size.

madcaptenor• 6 years ago

Similarly, Indians express large numbers in terms of lakh (10^5) or crore (10^7) which is confusing to people using the thousands-based system.

mixmastamyk• 6 years ago

I'd previously read that Fibonacci had helped popularize the hindu-arabic notation in Europe. Wikipedia says "In 1202, he completed the Liber Abaci (Book of Abacus or The Book of Calculation)," which included lessons and examples.

Interestingly, he grew up with a merchant father based in Northern Africa and had internalized it.

jankotek• 6 years ago

Most merchants at Roman times used Greek nimeral, it has much easier algebra comparable to arabic.

nprateem• 6 years ago

Or feet and inches. I've been watching videos online of people making things with the added challenge of using imperial measurements. 7 29/32 less a margin of 5 3/8 plus a 1/16 offset... Is metric too easy for Americans?

scott31• 6 years ago

A millenia later, humans will look back and wonder the same about how we did math on decimal system, while binary is far superior. I expect they will even genetically remove one finger from each hand so they can do binary math better with their fingers

How did anyone do math in Roman numerals? (2017)

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