
How Did Anyone Do Math in Roman Numerals? - iamjeff
http://www.washingtoncitypaper.com/columns/straight-dope/article/20854121/how-did-anyone-do-math-in-roman-numerals
======
WalterBright
It's interesting how the notation used can encourage or retard progress. For
example, Leibniz's calculus notation was vastly superior to Newton's, and
calculus theory advanced much more quickly where Leibniz's notation was used.

[https://en.wikipedia.org/wiki/Leibniz%27s_notation](https://en.wikipedia.org/wiki/Leibniz%27s_notation)

~~~
nnq
About this particular case: why hasn't Euler's notation
([https://en.wikipedia.org/wiki/Notation_for_differentiation#E...](https://en.wikipedia.org/wiki/Notation_for_differentiation#Euler.27s_notation))
become _the clear winner?_

I mean, its the only one that is both _not confusing for beginners_ (because
it doesn't trick them into the "cool, let's `simplify` the dx at the
denominator with the next one" mindset) and it also translates easily to code
(or other 1D encoding), like you can write "second derivative of f with
respect to x" as D[f, x, 2] and "integral of a with respect to t" as "D[a, t,
-1]".

My point is mainly that _there is nothing rational at all in how humans choose
notations..._ without some obscure historic events, we'd probably still be
using some derivative of roman numerals or maybe even sexagesimals!
([https://en.wikipedia.org/wiki/Sexagesimal](https://en.wikipedia.org/wiki/Sexagesimal))

(I imagine the reason is because "highly performing" individuals have their
own "internal" language to think in, so general language is just a for
communication, so a political decision... unfortunately for _education :(_ )

~~~
bonoboTP
I think a lot of my confusions when I first learned calculus would have been
eliminated with a notation that clearly expressed that derivatives operate on
_whole functions_ , not on values. So the derivative of f(x) at x=0 is not
some function of f(0), but it is derivative(f)(x). Also, even without
derivatives, sometimes the expression f(x) refers to the whole function,
sometimes just a particular value of it at a specific x.

I don't know if higher-order functions are really that tough to
teach/understand but I think it would actually simplify and demistify many
things.

Currently, of the widespread notations I like this one best:

f(x0) = d/dx (sin(x)*cos(x)+x^2) | x=x0 (with the last part in subscript)

I also miss variable scoping from math writing and it disturbs me that
variable names often carry semantics, like p(x) and p(y) can be the
probability density functions of random variables X and Y (so p is a different
function depending on the name of the input variable that you substitute so it
doesn't actually operate on real numbers, but (string, number) pairs). I'd
prefer to explicitly mark the functions as p_X(x) and p_Y(y).

Similar things come up a lot with differential equations where you don't
really know whether something (like y or u) is supposed to be a function (of x
or t) or "just" a variable.

Despite the general opinion among laypeople that math notation is very precise
and unambiguous, I find that it's often very sloppy and unless you already
understand the context very well, it can easily be misleading. Math notation
is somewhere between normal natural language and programming languages, and
depending on the writer it may be closer to one or the other.

One can argue that this is necessary for compactness.

~~~
noir_lord
I've noticed that I understand what is going on much more when a function is
written in code than in its mathematical form.

A lot of that is familiarity but I don't think all of it is.

~~~
ashark
Code's easy for me (unless "mathy" in appearance like Haskell) but
mathematical notation's always made me feel dyslexic.

I'd love to see this beauty or clarity or whatever that people find in
mathematics, but I've never caught even a hint of it. Seems like it needs a
good IDE to make up for deficiencies in its language.

~~~
noir_lord
Have you see SageMath[0]?

That allows you to use Python inside a mathematical environment and it's
amazing.

Saved my bacon a few times when I need to translate maths to code and I need
to poke it a bit to get an understanding of how it works.

[0][http://www.sagemath.org/](http://www.sagemath.org/)

------
zw123456
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 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.

With Arabic numbers, I have to learn that 1 is one thing, 2 represents 2
things, 3 represents 3 things and so on. Then I have to remember that 2 + 2 =
4, and 3 + 2 = 5, there is more memorization required.

Where Roman Numerals for someone with little or no education could get by in
the market square with some simple rules and even use twigs as a primitive
calculator. It is not until you get to much more complex ideas that the Arabic
notation wins out.

So perhaps, different notations lend themselves better or worse depending on
the application ?

Just some random thoughts that this very interesting post brought to mind.

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

Subtractive notation is confusing.

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

~~~
drostie
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."

------
ewanm89
Okay so the article is wrong about some points. We have evidence for both IIII
and IV notation in classic roman archeological finds. The obvious example is
the entrance fee doors around the colliseum in Rome, they're are engraved with
numbers, in both forms of notation.

Seriously I figured they used abacus for everything they just figured out the
notation to write it down in at the end some would convert to the if notation
while others would not.

~~~
wklauss
> We have evidence for both IIII and IV notation in classic roman
> archeological finds.

The article doesn't say subtractive notation was not used in classic roman
times, it says only that it was rare.

------
flipp3r
> the Romans greatly preferred the simpler IIII to IV, XXXX to XL, and so on.
> (The IIII-for-4 notation survives today on the faces of clocks.)

 _looks at watch_

Well, damn, it's IIII. However my watch does use IX over VIIII, what's up with
that?

~~~
ldjb
There's a theory[0][1] that it's mostly down to aesthetics. It looks visually
more pleasing that way when split into three groups of four numbers:

    
    
       I, II, III, IIII (consisting of I only)
       V, VI, VII, VIII (consisting of I and V)
      IX,  X,  XI,  XII (consisting of I and X)
    

[0] [http://mathtourist.blogspot.co.uk/2010/08/iiii-versus-iv-
on-...](http://mathtourist.blogspot.co.uk/2010/08/iiii-versus-iv-on-
clocks.html)

[1] [https://www.hautehorlogerie.org/en/encyclopaedia/glossary-
of...](https://www.hautehorlogerie.org/en/encyclopaedia/glossary-of-
watchmaking/s/roman-numeral-iiii-on-dials-1/)

~~~
sharpercoder
Personally, I find distinguishing III and IIII quite hard in several fonts.
However, the difference between [II and III] and [III and IV] is easier to
read.

------
interlocutor
Fun fact: What is known as "arabic numerals" in the west is known as "indian
numerals" in middle-eastern countries. That's because it came from India.
[https://en.wikipedia.org/wiki/History_of_the_Hindu%E2%80%93A...](https://en.wikipedia.org/wiki/History_of_the_Hindu%E2%80%93Arabic_numeral_system)

~~~
MattLeBlanc001
I'm arab and never heard that arabic numerals are known as indian numerals in
middle eastern countries.

~~~
interlocutor
Here's an example: [http://www.chrono24.ae/ulyssenardin/marine-chronometer-
manuf...](http://www.chrono24.ae/ulyssenardin/marine-chronometer-manufacture-
abu-dhabi-edition--id4350609.htm)

The digits are on the watch are standard arabic digits but the description
calls it "indian numerals"!

------
oaktowner
You know what I've always wondered? How did they _pronounce_ each number? I
mean, I read it "eye, eye-eye, eye-eye-eye, eye-vee, vee, vee-eye."

But of course, there's no way they actually _said_ it that way.

Right? Right??

~~~
madcaptenor
There are words for numbers in Latin. From Wiktionary
([https://en.wiktionary.org/wiki/Appendix:Latin_cardinal_numer...](https://en.wiktionary.org/wiki/Appendix:Latin_cardinal_numerals)):
unus, duo, tres, quattuor, quinque, sex...

Just like an English speaker sees "1, 2, 3, 4, 5, 6" and pronounces "one, two,
three, four, five, six". Sure, the Roman numerals look like letters, but that
doesn't mean you pronounce them that way.

~~~
fela
But that is very interesting, that means that when saying the numbers they
were basically using something similar to Arabic numbers!

~~~
jacobolus
In my opinion the important innovation of Hindu–Arabic numbers is not having
different symbols for each digit, but rather using the same ten symbols (and
no others!) for representing larger groups, which gives the system a lovely
uniformity across numbers of different scales.

There aren’t many (if any?) human languages that duplicate this feature of
being a purely positional system.

------
thunderbong
I had, for some reason, never thought of that directly. Although I do remember
some joke on arithmetics in one of the Asterix comic books.

But reading this article and trying out a few things, I found it was great
fun!

~~~
sideshowb
...which is funny (the Asterix thing), because spoken French isn't exactly
brilliant with numbers either: 99 for example is expressed as "eighty
nineteen", 70 as "sixty ten".

My friends lived in France a while and said their landlady could _never_ count
their rent (paid in cash) correctly first time. Always stumbled somewhere
between 100x+60 and 100x+100 for integer values of x.

The Swiss have corrected this in Swiss French.

~~~
sk0g
Same thing with Belgian French, much to my dismay.

I speak a language that does this as well (Georgian), where say 54 is
ormotsdatotxmeti (two times twenty and fourteen.)

I didn't find maths particularly different difficulty wise, when thinking
about it in Georgian or not. What did trip me up, was the times! Up to x:29
it's 29 minutes past x, but at x:30 it is 30 minutes to ++x. Weird.

~~~
vidarh
It survives in part in quite a few other languages as well. E.g. Danish
("fjerds" is a contraction of "four into 20" = 80).

Most other European languages at least (I'm sure it's present elsewhere too,
but I don't know enough non-European languages to say) have some remnants of
more widespread counting in 12's or 20's or both (e.g. in English a "dozen" is
12, a "gross" is a dozen dozen (144), and a "score" is 20, hence "four score
and seven years ago" in Lincolns Gettysburg Address - 87 years).

------
hrehhf
Off topic rant:

It bugs me when people mix our standard numerals with Roman numerals, such as
12MM to mean twelve million. They are different numerals and the meaning is
not defined when they are used together.

And Roman numerals are not like SI suffixes, meaning they are not
multiplicative; Roman numerals are additive, so MM is two thousand, not one
million. Also, M is an SI suffix, so 12M means twelve million and 12MM just
looks like a typo.

Obviously people do not use SI suffixes may not feel the same way, this is
just my pet peeve because I use SI suffixes in science.

------
squid_ca
Here's the article on its own site:
[http://www.straightdope.com/columns/read/3330/how-did-
anyone...](http://www.straightdope.com/columns/read/3330/how-did-anyone-do-
math-in-roman-numerals)

------
MichaelBurge
In 2017 we just do:

    
    
        $ perl6 -e 'say Ⅻ / Ⅲ'
        4

~~~
fahadkhan
care to explain why this works?

~~~
yorwba
Unicode has codepoints for roman numerals 1 through 12 and then the base
symbols up to one hundred thousand (ↈ) which I just saw for the first time.

That Perl allows them in integer literals is strange, but also logical in a
weird way. (Now I wonder what other number systems it supports.)

EDIT: Ok, so I installed perl6 (rakudo) just to test this out, and it
apparently doesn't work out of the box. Pity.

~~~
MichaelBurge
I'm not a compiler dev, but it looks fairly new and appears to have many other
special Unicode ranges too, including camels and beer mugs???

[https://github.com/rakudo/rakudo/blob/beec02a6fa69e3ac290b4d...](https://github.com/rakudo/rakudo/blob/beec02a6fa69e3ac290b4dd24c07d87a9f248b13/tools/build/makeMAGIC_INC_DEC.pl6)

~~~
zoffix222
That's just a helper script to generate tables for for .succ/.pred string
increment/decrement methods. Nd and No chars as numeric literals were
available since first stable release in December 2015

------
maxxxxx
Since we have a lot of math experts here I thought I'd ask a question I was
always wondering about: Is there an inherent advantage or disadvantage to
using the decimal system as we do? Somehow I think octal or hexadecimal would
be easier but I am not sure.

~~~
jeffwass
I was about to post that in my opinion base-12 is superior to base-10. But
someone beat me to it. In a six-fi novel I'm writing, an advanced alien
civilisation uses base-12.

As to your question specifically regarding base-16 instead of base-12, it
depends.

Decimal itself is just a bizarre choice, most likely due to humans having
literally ten digits. In decimal we can represent exact fractions of 1/2, 1/5,
and 1/10 (without repeated decimals like 0.33333 for 1/3). Counting by fives
(and twos) is very easy.

But choosing prime factors of 2 and 5 is a strange choice in itself. Why skip
3? Why is it more useful to easily represent fraction 1/5th as 0.2 instead of
1/3rd? How often do we use fifths?

Hexadecimal in one sense is easier, all prime factors are two. So we can
represent 1/2, 1/4, 1/8, and 1/16 exactly.

Duodecimal (Base 12) is very convenient for having a high proportion of exact
fractions. Eg - 1/12, 1/6, 1/4, 1/3, and 1/2 can all be represented exactly.
I'd argue in everyday use we're more likely to consider 1/3rd of something
than 1/5th. Counting by twos, threes, fours, and sixes is easy. Watch, let's
count to 20 (24 in decimal) by 3's : 3, 6, 9, 10, 13, 16, 19, 20. In 4's : 4,
8, 10, 14, 18, 20. By 6 : 6, 10, 16, 20.

Base-12 offers _four_ handy subdivisions (excluding 1) instead of two for
decimal or three for hexadecimal. That beats hexadecimal using fewer unique
digits. It beats decimal by two using only two extra unique digits.

And I think it's these reasons it was chosen for various historical
subdivisional units (inches per foot, pence per shilling).

The other item to consider is the relative number of unique values per digit.
I'm not sure of the utility of having 10, 12, or 16 here.

At one extreme, while binary is useful for discretising signals in digital
logic, using only zeroes and ones becomes cumbersome for daily use at higher
numbers.

Once we're at base 10 and higher, I'm not sure how much here extra digits help
or hurt.

~~~
tzs
> In a six-fi novel I'm writing, an advanced alien civilisation uses base-12

Six-fi? Typo or genre I'm not aware of?

~~~
jeffwass
Haha, yeah, sci-fi.

Luckily I'm not yet at the copy-editing stage ;-)

~~~
maxxxxx
Make them use base-23 and somehow this will allow them find the secrets of the
universe because the question to life, the universe and everything can be
solved easily with base-23 math.

------
barking
The greeks used the method on this page
([http://www.trottermath.net/algebra/multsqs.html](http://www.trottermath.net/algebra/multsqs.html))
to do multiplication so I suppose the romans did too.

------
aphextron
Perhaps this is why there were really no Roman mathematicians. For how much
they admired and emulated the Greeks, they themselves were never really able
to contribute to math and science in the same way. Practically everything we
think about today in western civilization in terms of Law, Architecture,
Engineering, and Urban Planning comes directly from the Romans. Yet they never
produced an Archimedes or a Pythagoras. Euclid remained the height of
mathematical sophistication in the West through their entire reign until the
rise of Arab/Islamic mathematics in the 800s.

~~~
jacobolus
Seems unlikely. Greeks and Romans probably did calculations in pretty much the
same way, using a counting board. The early Greek way of writing numbers (look
up Attic numerals) was the direct ancestor of the Roman method.

The alternative later way the Greeks (e.g. Euclid, Archimedes) wrote numbers
was using their alphabet for 1–9, using separate letters for 10–90 and
100–900, then writing 1000–9000 with a “thousand” symbol plus the letter for
1–9 on top. Overall also a big pain in the butt.

------
MayeulC
The article is wrong on a couple of counts (for multiplication and division).

Here is a more complete one: [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/)

(Surfaced on HN with
[https://news.ycombinator.com/item?id=13636277](https://news.ycombinator.com/item?id=13636277)
)

~~~
IshKebab
Huh so I've wondered how the Babylonian "base 60" system worked. Looking at
the image in that article it's not base 60 at all! It's more like a mix of
base 6 and base 10. The factor increase for the columns are not:

60x 60x 60x 60

The are

6x 10x 6x 10

Much simpler than it sounds from "base 60".

------
JadeNB
For what it's worth, you can find this material on the Straight Dope site
itself; no need to go through the WCP site (which doesn't let you read the
article without Javascript).
[http://www.straightdope.com/columns/read/3330/how-did-
anyone...](http://www.straightdope.com/columns/read/3330/how-did-anyone-do-
math-in-roman-numerals)

------
EGreg
I heard they divided by two on one side and multiplied by the other, adding 1
where they needed.

In fact they did this in ancient Egypt too!

[http://www.mathnstuff.com/math/spoken/here/2class/60/egyptm....](http://www.mathnstuff.com/math/spoken/here/2class/60/egyptm.htm)

------
vok
Claude Shannon created the THROBAC I to do this.
[https://www.flickr.com/photos/synesthete/2591624549](https://www.flickr.com/photos/synesthete/2591624549)

------
Garet_Jax
Some aspects of 'math' came along with arabic, but Roman Numerals work fine
for their situation, at that time.

In conversation, it's just the I-M/0-1000 range for instance, learning how to
notate that number is a different issue.

Notation is somewhat straight-forward: When you get to 4, the number undergoes
a state change to 'subtract from the next number, dropping everything behind
it, and we're in a new state so "the" number is changed to 5' so III becomes
IV. Otherwise, just keep adding 1. That should work, adding to the previous
rules as the state changes. Going to X leaves the V rules in place.

So it would be a XXXII-bit computer, for example.

