## Re: OT: Math with roman numerals (Was: Do people ever make variant numerical systems for non-primitive cultures?)

Expand Messages
• If you require permitting four symbols in a row and forbid subtractive symbols then Roman numbers could be rewritten as positional notation like this:
Message 1 of 17 , Mar 5, 2013
• 0 Attachment
If you require permitting four symbols in a row and forbid subtractive
symbols then Roman numbers could be rewritten as positional notation like
this:

MMDCXXXXIII -> 02 11 04 03 where each two-digit group represents a 5's and
units digit for the next power of ten.

In other words, the rightmost digit of each group is base 5 (M, C, X, I)
and the leftmost digit is base two (V, D, L, V), and the groups are base
ten. (I..VIIII, X..LXXXX, C..DCCCC, M..VMMMM).

From there it's an easy matter to construct a multiplication table for
pairs of two-digit groups multiplied together since each two-digit group
represents one decimal digit.

01 = 1 or 10 or 100 or 1000 (I, X, C, M)
02 = 2 or 20 or 200 or 2000 (II, XX, CC, MM)
03 = 3 or 30 or 300 or 3000 (III, XXX, CCC, MMM)
04 = 4 or 40 or 400 or 4000 (IIII, XXXX, CCCC, MMMM)
10 = 5 or 50 or 500 or 5000 (V, L, D, V)
11 = 6 or 60 or 600 or 6000 (VI, LX, DC, VM)
12 = 7 or 70 or 700 or 7000 (VII, LXX, DCC, VMM)
13 = 8 or 80 or 800 or 8000 (VIII, LXXX, DCCC, VMMM)
14 = 9 or 90 or 900 or 9000 (VIIII, LXXXX, DCCCC, VMMMM)

--gary

On Tue, Mar 5, 2013 at 1:14 PM, James Kane <kanejam@...> wrote:

> I think I read somewhere that the subtraction thing, eg XL for forty, is
> in fact a later invention and the Romans themselves were quite happy to
> have four I's or four V's in a row.
>
> Would this make the maths easier?
>
>
> On 6/03/2013, at 3:43 AM, Daniel Burgener <burgener.daniel@...>
> wrote:
>
>
• ... Basically, correct. ... Four Xs that should be. You will never have four Vs in a row. ... Yes, IMO. When I played with doing math(s) using Roman numerals,
Message 2 of 17 , Mar 6, 2013
• 0 Attachment
On 05/03/2013 21:14, James Kane wrote:
> I think I read somewhere that the subtraction thing, eg
> XL for forty, is in fact a later invention

Basically, correct.

> and the Romans themselves were quite happy to have four
> I's or four V's in a row.

Four Xs that should be. You will never have four Vs in a row.

> Would this make the maths easier?

Yes, IMO.

When I played with doing math(s) using Roman numerals, the
first step I did was to change the post-Roman subtractions.

The system seems to have developed from what we find on
funeral inscriptions where, e.g. IIXXX = duodetriginta =
two-from-thirty, i.e. 28.

From 18, 19 until we reach the hundreds the +8 or +9
numerals were designated in the spoken language a "duode-"
and "unde-" respectively, i.e.
duodeviginti = 18
undeviginti = 19
duodetriginta = 28
etc.

I suspect, however, that when doing math(s) this was not
done. At any rate, AFAIK 'four' was always written as IIII
in the Classical period, and 'forty' as XXXX etc.
========================================================

On 05/03/2013 21:45, Gary Shannon wrote:
> If you require permitting four symbols in a row and
> forbid subtractive symbols then Roman numbers could be
> rewritten as positional notation like this:
>
> MMDCXXXXIII -> 02 11 04 03 where each two-digit group
> represents a 5's and units digit for the next power of
> ten.
>
> In other words, the rightmost digit of each group is
> base 5 (M, C, X, I) and the leftmost digit is base two
> (V, D, L, V), and the groups are base ten. (I..VIIII,
> X..LXXXX, C..DCCCC, M..VMMMM).

Exactly! That's the way the old Roman abacus worked. The
system is bi-quinary.

PS - I think I referred to the abacus as 'calculus' in an
earlier email. 'Twas an error. Of course the thing was an
'abacus', and _calculi_ ("pebbles") were use as counters.
Hence the verb _calculāre_ = "to calculate", and the nouns:
_calculātor_ = "one who calculates" or "a teacher of arithmetic"
calculō (gen: calculōnis) = "an accountant"

--
Ray
==================================
http://www.carolandray.plus.com
==================================
"language … began with half-musical unanalysed expressions
for individual beings and events."
[Otto Jespersen, Progress in Language, 1895]
• ... numerals were designated in the spoken language a duode- and unde- respectively, i.e. duodeviginti = 18 undeviginti = 19 duodetriginta = 28
Message 3 of 17 , Mar 6, 2013
• 0 Attachment
> From 18, 19 until we reach the hundreds the +8 or +9
numerals were designated in the spoken language a "duode-"
and "unde-" respectively, i.e.
duodeviginti = 18
undeviginti = 19
duodetriginta = 28
etc.

This may be a remnant of a change from octal to decimal system.
Similarly, a duodecimal system could be expressed using only decimal digits:

10[A] = A = twofrom dozen = IIЖ
11[A] = B = onefrom dozen = IЖ
12[A] =10[C] = dozen = Ж
13[A] = 11[C] = dozen one = ЖI
...
21[A] = 19[C] = dozen nine = ЖΨIII
22[A] = 1A[C] = twofrom twodozen = IIЖЖ
23[A] = 1B[C] = onefrom twodozen = IЖЖ
24[A] = 20[C] = twodozen = ЖЖ
25[A] = 21[C] = twodozen one =ЖЖI

Or a duodecimal system expressed using only octal digits:

8 = foufrom dozen = IIIIЖ
9 = threefrom dozen = IIIЖ
10[A] = A = twofrom dozen = IIЖ
11[A] = B = onefrom dozen = IЖ
12[A] =10[C] = dozen = Ж
13[A] = 11[C] = dozen one = ЖI
...
19[A] = 17[C] = dozen seven = ЖΨI
20[A] = 18[C] = fourfrom twodozen = IIIIЖ
21[A] = 19[C] = threefrom twodozen = IIIЖ
22[A] = 1A[C] = twofrom twodozen = IIЖ
23[A] = 1B[C] = onefrom twodozen = IЖ
24[A] = 20[C] = twodozen = IЖ
25[A] = 21[C] = twodozen one = ЖI

> I suspect, however, that when doing math(s) this was not
done.

It is essentially a system with digits -2 to 7 instead of usuall 0 to 9.
The algorithms for countig with negative digits are the same as the usual;
in both cases the last step is a normalisation of the number.

(1)(9) + (2)(8) = (3)(17) = (4)(7)
XVIIII + XXVIII = XXXVVIIIIIII = XXXXVII

(2)(-1) + (3)(-2) = (5)(-3) = (4)(7)
IXX + IIXXX = IIIXXXXX = XXXXVII
• ... Yes, Roman numbers representations are representations of sums themselves (except for the preffix), that s why sum is so simple. In the Indo-Arabic
Message 4 of 17 , Mar 7, 2013
• 0 Attachment
2013/3/5 Daniel Burgener <burgener.daniel@...>:
> On Tue, Mar 5, 2013 at 6:52 AM, Leonardo Castro <leolucas1980@...>wrote:
>
>> 2013/3/4 Matthew George <matt.msg@...>:
>> > I was thinking about Roman numerals, and how terrible for performing
>> > mathematics that whole system is.
>>
>> I once affirmed that to a friend and he told me "Let's check!" and
>> started trying to make basic operations with Roman numerals. To my
>> surprise, he succeeded! He succeeded and said: "It's not that
>> difficult...". I watched he performing the operations while analyzing
>> developing the methods himself and it really didn't looked very
>> difficult.
>>
>> I don't have time to repeat what he did now, but I challenge you do so.
>>
>
> Interesting! I just tried some basic addition and multiplication with
> roman numerals and it was in fact quite easy. Multiplication got tedious
> quickly and I didn't go far enough to figure out really complicated
> problems. (I could do, for example MXIII * V with little trouble, but I
> haven't tackled MXIII * XVII yet). Addition was easy though.
>
> Here's a simple example of the method I came up with for addition:
>
> CCLXXIV + CXCVII
>
> I did the work in two phases: First, "clump everything together", then
> "rewrite".
>
> Since each roman numeral represents a quantity, I can add them by simply

Yes, Roman numbers' representations are representations of sums
themselves (except for the preffix), that's why sum is so simple. In
the Indo-Arabic system, a 9 is not simply nine.

> grouping those quantities together, with special rules for cases like "IX"
> etc. I, X and C can be prefixed to larger numbers to indicate
> subtraction. In this situation, there are two cases. If there is a
> positive numeral of the same value in the other number, they cancel. If
> not, you can carry the prefix down.
>
> So here's the "clumping" for the above example:
>
> CCCCLXVVI
>
> The prefixed X and I each cancel one in the other number. Now this result
> isn't good because it has four Cs and two Vs, which should become a "CD"
> and an "X" respectively. That's what the "rewrite" phrase is for.
>
> So the end result is:
>
> CDLXXI.
>
> (Note that the second phase may require multiple iterations, if for example
> our "VV" -> "X" transformation resulted in four Xs. There are also
>
> So to check in base 10, we did: 274 + 197, and got 471, which is the
>
> I would wager that once one became familiar with the method, this could be
> faster than addition in modern numerals (although not worth the time to
> convert back and forth to Roman numerals). Multiplication seems easier
> with the modern way though based on my brief experimentation with it.
>
> -Daniel

I don't remember if my friend found a simple way of performing
divisions. I think division requires "guessing" quotients and checking
if you have to raise or lower it in the next trial. That's how we do
with Indo-Arabic numerals too, isn't it?
• ... I don t believe that s how I learned it. I learned a method like this: http://www.mathsisfun.com/long_division.html The guess and check method sounds
Message 5 of 17 , Mar 7, 2013
• 0 Attachment
On Thu, Mar 7, 2013 at 7:34 AM, Leonardo Castro <leolucas1980@...>wrote:

>
> [snip]
>
> I don't remember if my friend found a simple way of performing
> divisions. I think division requires "guessing" quotients and checking
> if you have to raise or lower it in the next trial. That's how we do
> with Indo-Arabic numerals too, isn't it?
>

I don't believe that's how I learned it. I learned a method like this:
http://www.mathsisfun.com/long_division.html

The guess and check method sounds especially tedious in roman numerals
where multiplication is tedious.

-Daniel
• ... But I think this method also involve guess and check: the operation 42 ÷ 25 = 1 remainder 17 is presented as a single step, but it involves guessing 1
Message 6 of 17 , Mar 7, 2013
• 0 Attachment
2013/3/7 Daniel Burgener <burgener.daniel@...>:
> On Thu, Mar 7, 2013 at 7:34 AM, Leonardo Castro <leolucas1980@...>wrote:
>
>>
>> [snip]
>>
>> I don't remember if my friend found a simple way of performing
>> divisions. I think division requires "guessing" quotients and checking
>> if you have to raise or lower it in the next trial. That's how we do
>> with Indo-Arabic numerals too, isn't it?
>>
>
> I don't believe that's how I learned it. I learned a method like this:
> http://www.mathsisfun.com/long_division.html

But I think this method also involve guess and check: the operation
"42 ÷ 25 = 1 remainder 17" is presented as a single step, but it
involves guessing 1 and checking that 25x1 is less than 42, then
guessing 2 and checking that 25x2 is greater than 42. If you can do it
very fast, it's another matter. Dividing (e.g.) 343 by 42 would not be
as easy to do in one's head.

Besides, arithmetic usually involves know a lot of simpler results by
heart and using them to perform more complex ones. Maybe you could
skip a lot of intermediate calculations with Roman numerals too.

>
> The guess and check method sounds especially tedious in roman numerals
> where multiplication is tedious.
>
> -Daniel
• ... Fair enough. -Daniel
Message 7 of 17 , Mar 7, 2013
• 0 Attachment
On Thu, Mar 7, 2013 at 9:17 AM, Leonardo Castro <leolucas1980@...>wrote:

> 2013/3/7 Daniel Burgener <burgener.daniel@...>:
> > On Thu, Mar 7, 2013 at 7:34 AM, Leonardo Castro <leolucas1980@...
> >wrote:
> >
> >>
> >> [snip]
> >>
> >> I don't remember if my friend found a simple way of performing
> >> divisions. I think division requires "guessing" quotients and checking
> >> if you have to raise or lower it in the next trial. That's how we do
> >> with Indo-Arabic numerals too, isn't it?
> >>
> >
> > I don't believe that's how I learned it. I learned a method like this:
> > http://www.mathsisfun.com/long_division.html
>
> But I think this method also involve guess and check: the operation
> "42 ÷ 25 = 1 remainder 17" is presented as a single step, but it
> involves guessing 1 and checking that 25x1 is less than 42, then
> guessing 2 and checking that 25x2 is greater than 42. If you can do it
> very fast, it's another matter. Dividing (e.g.) 343 by 42 would not be
> as easy to do in one's head.
>
> Besides, arithmetic usually involves know a lot of simpler results by
> heart and using them to perform more complex ones. Maybe you could
> skip a lot of intermediate calculations with Roman numerals too.
>
> >
> > The guess and check method sounds especially tedious in roman numerals
> > where multiplication is tedious.
> >
> > -Daniel
>

Fair enough.

-Daniel
• ... This is where the multiplication table comes in. A table of values already calculated for easy reference. It s also why kids are taught the times tables.
Message 8 of 17 , Mar 7, 2013
• 0 Attachment
On Thu, Mar 7, 2013 at 9:17 AM, Leonardo Castro <leolucas1980@...>wrote:

> 2013/3/7 Daniel Burgener <burgener.daniel@...>:
> > On Thu, Mar 7, 2013 at 7:34 AM, Leonardo Castro <leolucas1980@...
> >wrote:
> >
> >>
> >> [snip]
> >>
> >> I don't remember if my friend found a simple way of performing
> >> divisions. I think division requires "guessing" quotients and checking
> >> if you have to raise or lower it in the next trial. That's how we do
> >> with Indo-Arabic numerals too, isn't it?
> >>
> >
> > I don't believe that's how I learned it. I learned a method like this:
> > http://www.mathsisfun.com/long_division.html
>
> But I think this method also involve guess and check: the operation
> "42 ÷ 25 = 1 remainder 17" is presented as a single step, but it
> involves guessing 1 and checking that 25x1 is less than 42, then
> guessing 2 and checking that 25x2 is greater than 42. If you can do it
> very fast, it's another matter. Dividing (e.g.) 343 by 42 would not be
> as easy to do in one's head.
>
> Besides, arithmetic usually involves know a lot of simpler results by
> heart and using them to perform more complex ones. Maybe you could
> skip a lot of intermediate calculations with Roman numerals too.
>
> >
> > The guess and check method sounds especially tedious in roman numerals
> > where multiplication is tedious.
> >
>
This is where the multiplication table comes in. A table of values already
calculated for easy reference. It's also why kids are taught the times
tables. (They still are, right?)

stevo

> -Daniel
>
• ... I don t believe that s how I learned it.  I learned a method like this: http://www.mathsisfun.com/long_division.html
Message 9 of 17 , Mar 7, 2013
• 0 Attachment
--- On Thu, 3/7/13, Daniel Burgener <burgener.daniel@...> wrote:
I don't believe that's how I learned it.  I learned a method like this:
http://www.mathsisfun.com/long_division.html
=========================================================

That's pretty much what I learned back in the Dark Ages (1940s) and what I still use when I don't have a calc. or comp. handy ;-)  except for the first step (25 into 4 = 0). Our rule of thumb was: if the divisor doesn't go into the first digit, start with the first two digits (25 into 42 in their ex.).
• ... But I think this method also involve guess and check: the operation 42 ÷ 25 = 1 remainder 17 is presented as a single step, but it involves guessing 1
Message 10 of 17 , Mar 7, 2013
• 0 Attachment
--- On Thu, 3/7/13, Leonardo Castro <leolucas1980@...> wrote:

2013/3/7 Daniel Burgener <burgener.daniel@...>:
>
> I don't believe that's how I learned it.  I learned a method like this:
> http://www.mathsisfun.com/long_division.html

But I think this method also involve guess and check: the operation
"42 ÷ 25 = 1 remainder 17" is presented as a single step, but it
involves guessing 1 and checking that 25x1 is less than 42, then
guessing 2 and checking that 25x2 is greater than 42. If you can do it
very fast, it's another matter. Dividing (e.g.) 343 by 42 would not be
as easy to do in one's head.

Besides, arithmetic usually involves know a lot of simpler results by
heart and using them to perform more complex ones. Maybe you could
skip a lot of intermediate calculations with Roman numerals too.
=====================================

Yes, but.... that's where knowing the multiplication table helps. Doesn't anyone learn that anymore???

42 is obviously bigger than 25; but you _know_ that 2x25 is 50, so 1 is the obvious answer..... Same with 343 by 42-- you _know_ that 6x40 is 240 which would leave 83 (and deduct 7x2=14 from that gives 69, which also contains 42) so you'd try 7 or 8 (and 8 turns out to be right).

Yes, it can involve a certain amount of trial and error, but I suspect even the (cleverer?) ancient Romans knew things like this too........ But is there any alternative????
>
> The guess and check method sounds especially tedious in roman numerals
> where multiplication is tedious.
>
> -Daniel
• ... I don t think so; however, you can eliminate all of the trial and error by simply trying all possible options in order. The radix/base of your numeral
Message 11 of 17 , Mar 7, 2013
• 0 Attachment
On 7 March 2013 10:02, Roger Mills <romiltz@...> wrote:
> --- On Thu, 3/7/13, Leonardo Castro <leolucas1980@...> wrote:
>
> 2013/3/7 Daniel Burgener <burgener.daniel@...>:
>>
>> I don't believe that's how I learned it. I learned a method like this:
>> http://www.mathsisfun.com/long_division.html
>
> But I think this method also involve guess and check: the operation
> "42 ÷ 25 = 1 remainder 17" is presented as a single step, but it
> involves guessing 1 and checking that 25x1 is less than 42, then
> guessing 2 and checking that 25x2 is greater than 42. If you can do it
> very fast, it's another matter. Dividing (e.g.) 343 by 42 would not be
> as easy to do in one's head.
>
> Besides, arithmetic usually involves know a lot of simpler results by
> heart and using them to perform more complex ones. Maybe you could
> skip a lot of intermediate calculations with Roman numerals too.
> =====================================
>
> Yes, but.... that's where knowing the multiplication table helps. Doesn't anyone learn that anymore???
>
> 42 is obviously bigger than 25; but you _know_ that 2x25 is 50, so 1 is the obvious answer..... Same with 343 by 42-- you _know_ that 6x40 is 240 which would leave 83 (and deduct 7x2=14 from that gives 69, which also contains 42) so you'd try 7 or 8 (and 8 turns out to be right).
>
> Yes, it can involve a certain amount of trial and error, but I suspect even the (cleverer?) ancient Romans knew things like this too........ But is there any alternative????

I don't think so; however, you can eliminate all of the trial and
error by simply trying all possible options in order. The radix/base
of your numeral system determines how many options there are- given
two numbers with the same number of digits, in a base-10 system the
smaller can go into the larger a maximum of 9 times, so you have only
have 10 options, and on average it will take 5 attempts
(test-multiplies) to find the right match going through the
possibilities linearly, or 3.3 attempts if you use bisection. Knowing
multiplication tables lets you cut down the average number of attempts
by constraining the range you need to check to something smaller than
0-9 by comparing only the leading digits.

This only works if you can get things into a positional notation with
a consistent base, of course, but it's already been noted that Roman
bi-quinary numerals can be grouped into decimal positions.

Using a smaller base means that generating each digit of the result
will be faster, because there are fewer options to try, but that's
balanced out by the fact that you just have to do it more times. A
binary system is nice for division because there are only two options,
and the next result digit depends solely on whether one number is
bigger than the other or not, so division can be done entirely by
shifting digits and subtracting; memorized multiplication tables for
a binary notation help essentially by simulating a quaternary, octal,
or hexadecimal system and generating multiple digits at a time.

Note that the division algorithm is strongly dependent on the fact
that we use a place-value notation. Different representations can make
different operations easier or harder in surprising ways, such that
sometimes the easiest way to perform some calculation might actually
be to change your notation, do the calculation in a different,
specialized, notation, and then translate back to your standard
notation- that's what we'd be doing by grouping Roman numerals into
simulated decimals. But for an example where Indo-Arabic numerals are
particularly bad, it turns out that calculating roots is really easy
when you represent things in an exponential notation (scientific
notation, or IEEE 754 floating-point format).

-l.
• ... We did (1970s). Does anyone know if the Romans had multiplication or division tables? It would be a trivial matter for an educated slave to crunch numbers
Message 12 of 17 , Mar 7, 2013
• 0 Attachment
--- On Thu, 3/7/13, Roger Mills <romiltz@...> wrote:

> > > http://www.mathsisfun.com/long_division.html
>
> > Besides, arithmetic usually involves know a lot of simpler results by
> > heart and using them to perform more complex ones. Maybe you could
> > skip a lot of intermediate calculations with Roman numerals too.
>
> Yes, but.... that's where knowing the multiplication table
> helps. Doesn't anyone learn that anymore???

We did (1970s). Does anyone know if the Romans had multiplication or
division tables?

It would be a trivial matter for an educated slave to crunch numbers and
then for other slaves to copy the results into usable tables. Thereafter,
not much need for heavy crunching -- just look up the answer in the
Liber Magnum Ruber Calculonum.

Much like the various sine and log books of yore -- why reinvent the wheel
every time you want to multiply 12 by 143, when all you have to do is look
up the answer in a book?

I just did the experiment of how to multiply in RomNum -- trivially easy,
by breaking down large numerals into smaller constituents, using a table
or memorised answers and then toting up all the small constituents. (The
bias to *think* in arabic numerals is not easy to overcome!)

Division proved to be a rather harder mountain to climb. A table would be
extremely handy, but I don't know if the Romans would have thought simialrly.

• Why bother creating a system to make less work for slaves? They re *slaves* . Or so I presume the Romans would have thought. Matt G.
Message 13 of 17 , Mar 7, 2013
• 0 Attachment
Why bother creating a system to make less work for slaves? They're *slaves*
.

Or so I presume the Romans would have thought.

Matt G.
• ... because slaves are *expensive*, expecially the well-educated ones that are able to do arithmetics. Also, they were the kind of slave who tended to have
Message 14 of 17 , Mar 8, 2013
• 0 Attachment
On 2013-03-07 at 15:12:15 -0500, Matthew George wrote:
> Why bother creating a system to make less work for slaves? They're *slaves*
> .
>
> Or so I presume the Romans would have thought.

because slaves are *expensive*, expecially the well-educated ones
that are able to do arithmetics.

Also, they were the kind of slave who tended to have some
freedom of action, and they could have created the system
for themselves.

Roman slaves were somewhat more similar to today's wage workers
(ranging from abused menial laborers to respected professionals
with good chances to retire / buy their freedom and get rich)
than modern-era slaves (who I believe were mainly in the
menial-work area of jobs).

--
Elena ``of Valhalla''
• On Fri, Mar 8, 2013 at 3:14 AM, Elena ``of Valhalla
Message 15 of 17 , Mar 8, 2013
• 0 Attachment
On Fri, Mar 8, 2013 at 3:14 AM, Elena ``of Valhalla'' <
elena.valhalla@...> wrote:

> On 2013-03-07 at 15:12:15 -0500, Matthew George wrote:
> > Why bother creating a system to make less work for slaves? They're
> *slaves*
> > .
> >
> > Or so I presume the Romans would have thought.
>
> because slaves are *expensive*, expecially the well-educated ones
> that are able to do arithmetics.
>

You also probably don't want to skimp on comforts for educated slaves,
since they'll probably be tutoring your children.

> Also, they were the kind of slave who tended to have some
> freedom of action, and they could have created the system
> for themselves.
>

Makes sense

> Roman slaves were somewhat more similar to today's wage workers
> (ranging from abused menial laborers to respected professionals
> with good chances to retire / buy their freedom and get rich)
> than modern-era slaves (who I believe were mainly in the
> menial-work area of jobs).
>

I think most people think of the slavery that occurred in the Americas up
until the Civil War, which was definitely focused on menial labor, as well
as some use as household servants. What I hear about most as regards
slavery right now is in the sex trade, but I don't know how big a part of
the whole black market of human trafficking that is.
Your message has been successfully submitted and would be delivered to recipients shortly.