
How did anyone do math in Roman numerals? (2017) - pmontra
https://www.washingtoncitypaper.com/columns/straight-dope/article/20854121/how-did-anyone-do-math-in-roman-numerals
======
andrewl
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.

~~~
philwelch
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.

~~~
User23
Please explain the easy method for adding, for example MCMLXVII and LXV. I
mean that seriously, I'm curious what the trick is.

~~~
jeffcoat

      MCMLXVII + LXV
       = MCCCCCCCCCLXVII + LXV (canonicalize)
       = MCCCCCCCCCLXVIILXV    (concatenate)
       = MCCCCCCCCCLLXXVVII    (sort)
       = MCCCCCCCCCLLXXXII     (combine, VV => X)
       = MCCCCCCCCCCXXXII      (... keep combining, LL => C)
       = MMXXXII               (... C{10} => M, nothing left to combine)
       = MMXXXII               (optionally, look for ways to re-write with the subtraction rule)

~~~
philwelch
I originally thought canonicalize was an important step, but it actually isn’t
for humans. For humans, CCCCCCCCC requires a lot more tedious counting than CM
or even DCCCC, leading to more errors than simply allowing the human to notice
that CM+C=M.

------
bhntr3
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/](http://worrydream.com/MediaForThinkingTheUnthinkable/))

~~~
k__
_" 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.

~~~
goatinaboat
_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.

~~~
AnimalMuppet
Sure. But they did those things by experience and rules of thumb. They didn't
do a real stress analysis on those aqueducts, for instance.

~~~
mdiesel
For those interested in reading further, the ideal (unloaded) shape of an arch
isn't a semicircle, but a catenary.

It sounds so simple: so hangs the chain, stands the arch. Took until Hooke in
the 17th century before that was written down, though there are earlier (15th
century) examples in architecture.

The Romans were still working on the Greek ideology that the circle was the
perfect shape. Not to belittle what they did, but the key advances were really
in concrete and having an authoritarian empire giving unprecedented resources
to public works.

~~~
ahazred8ta
"Ut pendet continuum flexile, sic stabit contiguum rigidum inversum -- As
hangs a flexible cable, so inverted stand the touching pieces of an arch."
(although they figured out at some point that this curve was close to being a
parabola)

------
dwheeler
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
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.

~~~
Sharlin
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.

~~~
crazygringo
Thanks for explaining that clearly. I've always been so baffled by people who
claim that some society didn't have a concept for zero, as if "inventing" zero
marks some major advance in intelligence.

Every culture has a concept of "nothing" which works for zero. The ancient
Greeks debated over whether nothing was a _number_ or not, but that's just a
semantic splitting of hairs.

At some point a _symbol_ for nothing becomes useful so you invent a number-
like notation for it. But that's just a matter of convenience. It's not some
great conceptual leap.

~~~
dwohnitmok
I think that for the majority of people throughout history, numbers were
inseparable from numerals, i.e. notation for numbers. This would explain why
people are far more comfortable with the notion of real numbers (which despite
their name are very very strange in a lot of ways) than imaginary and complex
numbers. Even their names betray the difference. However, one has a common
notation that everyone has learned whereas the other has a more confusing and
less well-known notation.

Therefore I think ancient arguments over whether 0 is a number (and acceptance
thereof) are representative of a greater paradigmatic shift, similar in
essence to the arguments over whether the square root of -1 is truly a
"number."

Viewed that way 0 is the first step in a journey of an understanding of
numbers from purely counting discrete entities, to abstract parts of
computation.

So basically I would posit that it is in fact a great conceptual leap (just as
the negative numbers are) that only seems like an obvious fluke of notation
when every schoolchild has learned it.

~~~
crazygringo
I would clarify that to say numbers were inseparable from _words_ for them.
Writing has only existed a short time of our history, so numerals are pretty
recent.

And the "meaning" of zero as a number like others along a number line, rather
than as mere notation for "nothing", I assume only ever became necessary with
the invention of negative numbers.

With addition, multiplication and division, zero simply does nothing or
annihilates a number, and so doesn't need to be treated like other numbers.
AFAIK, zero as a number arises first in figuring out how to "get to" negative
numbers, e.g. what is two minus four (one, zero, negative one, negative two),
where zero is required as a _numeric_ concept.

Negative numbers were a big step forwards. Zero, I still don't see it --
either it was just convenient notation for "nothing", or part and parcel of
the shift to negative numbers. Unless I'm missing something in the historical
record?

~~~
jcranmer
From my understanding (and I well could be wrong here), Europeans inherited
from Greek a mathematics system that favored geometry and tended to abhor
algebra. Concepts like integers, rationals, and irrational numbers are all
pretty easy to explore and explain with geometry. By contrast, zero, negative
numbers, and imaginary numbers create absurdities in geometry (how can a line
have length 0? -2? 3 + 4i?). Moreover, even as algebra is introduced to
Europeans via the Arabs, I can see people resisting algebra in part because it
introduces these absurdities and paradoxes that need explanation.

As far as I can tell from the historical record (and it doesn't help that
modern histories tends to describe historical mathematical discoveries in
modern terms, meaning it's difficult to work out as a lay person in what terms
the historical discoverer understood their own work), it looks like the
acceptance of zero, negative numbers, and complex numbers are more or less
concurrent, and this also seems to coincide with the shift in mathematics from
being predominantly geometric to algebraic.

------
zapzupnz
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
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
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.

~~~
loxs
There is (for me at least) something very deep (in the brain) regarding
numerals and basic math. I am quite proficient in English, but always do
arithmetic in my native Bulgarian and then have to translate the result
(unlike other speech, which flows freely). And not for the lack of trying. And
that is even though Bulgarian numerals do translate 1:1 to English.

~~~
spiralx
I read a fascinating article once I've never been able to find again that said
the brain actually has internal biological representations of zero and one
along with a rudimentary representation of two, but nothing above that.

------
msla
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...](https://www.straightdope.com/columns/read/3330/how-did-anyone-do-
math-in-roman-numerals/)

Also, previously:

[https://news.ycombinator.com/item?id=14818633](https://news.ycombinator.com/item?id=14818633)

~~~
gowld
Fixed headers are better than fixed autoplay ad videos.

------
dhosek
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](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
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
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

------
sn41
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
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/](https://www.imdb.com/title/tt5015534/)
). It looks interesting, and it's on Amazon prime video so I'll give it a
watch. Thanks!

~~~
sn41
Sorry, you are right.

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

[http://www.phy6.org/outreach/edu/roman.htm](http://www.phy6.org/outreach/edu/roman.htm)

~~~
MayeulC
Yeah, this was discussed previously:
[https://news.ycombinator.com/item?id=13636277](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...](https://thonyc.wordpress.com/2017/02/10/the-widespread-and-
persistent-myth-that-it-is-easier-to-multiply-and-divide-with-hindu-arabic-
numerals-than-with-roman-ones/)

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

------
dan-robertson
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.

------
nprateem
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?

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

~~~
shultays
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.

~~~
Grustaf
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?

~~~
shultays
Hmm, I was assuming they didn't have a concept of base 10. If they do have it
I wonder how they couldn't make the connection and write stuff in base 10 as
well instead of a mixed base with weird rules

Then I guess the real challenge is converting numbers around?

------
forinti
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).

~~~
monadic2
...in terms of digits, which is a very odd quality to optimize for.

~~~
gowld
...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)

~~~
forinti
Suppose there was a basketball game between Athens and Rome (go Athens!) and
you have a 3 digit scoreboard.

You would need 19 base 3 digits and only 15 roman numerals (plus N for zero)
in order to represent every number.

In base three, you sometimes have all three digits equal. With roman numerals,
you can reuse the same digit in different positions.

~~~
dmurray
I don't understand this. 999 is less than 3^7, so you can represent any number
up to 999 with just seven base-3 digits. Where does the 19 come from?

~~~
mdiesel
I play village cricket, and the scoreboard has cards with numbers on, then
hooks to hang them up depending on the score.

The problem is: Given the full range of possible (or at least plausible)
scores, how many of the cards do we need for a full set?

So let's simplify it to just a run tally. You could be 111, so you'd need at
least 3 of the 1 cards etc. Allow for scoring up to 999 (unlikely) and that's
29 cards to keep somewhere (only 2 zeros needed)

In base 3, you need 7 digits, but only 3 cards per, so we are doing better
with 19 cards needed (21=3*7, but don't need all zeros, and that gets you to
1093 so for 999 you could save another)

In roman numerals, You'd need an M, a D, 3 Cs, 3 Ls, 3 Xs, 1 V and 3 Is. Total
is 15 cards.

Can we do better? Good question.

~~~
mdiesel
The answer is: It reduces to a question of permutations. 7! > 999, so we could
do it with different arrangements of just 7 cards.

Can we do better? Good question...

~~~
mdiesel
Now if we arranged them on a 2d grid, could use 4 cards, then the different
shapes (even discounting the similar looking shapes) would get you to >1000\.
You'd have to allow more than just the standard tetrominoes.

------
siraben
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](https://en.wikipedia.org/wiki/Rod_calculus)

------
mixmastamyk
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.

------
Animats
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.

------
scott31
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

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

What? On which clocks?

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

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

------
reidacdc
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...](https://www.reddit.com/r/AskHistorians/comments/12m0vp/how_and_when_did_the_arabic_numbers_took_over_the/)

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

