# RE: [OT] Roman numeral arithmetic

From: Ayers, Mike (Mike_Ayers@bmc.com)
Date: Tue Oct 02 2001 - 11:24:44 EDT

> From: Edward Cherlin [mailto:Edward.Cherlin.SY.67@aya.yale.edu]
> Sent: Saturday, September 29, 2001 05:55 PM
>
> If we omit the later use of subtractive notation (iv=4, xc=90
> etc.), the original Roman numerals are exactly equivalent to
> the Chinese abacus where each wire holds four beads below the
> bar (value I, X, C, M) and one above (value V, L, D, U+2181).
> It is well known that practiced abacists could beat users of
> subtraction. The same technique is taught (under the Korean
> name Chisanpeop) for two-column finger arithmetic, using the
> thumbs for the five beads and the other fingers for the one beads.

Maybe, but I think you may be missing a point: subtractive notation
was an improvement (or so I believe). I will use the same examples and my
finest ASCII graphics. Note: fixed width font required.

> > From: unicode-bounce@unicode.org
> > [mailto:unicode-bounce@unicode.org]On
> > Behalf Of James Kass
> > Sent: Sat, September 22, 2001 4:52 PM
> >
> > Doug Ewell wrote:
> >
> > > >> I would be fascinated to see some sort of evidence
> > > >> subtraction is easier in Roman numerals than in
> > Hindu-Arabic ("European")
> > > >> numerals.
> > > >
> > > > I + I = II
> > > > X + X = XX
> > > > X + X + X = XXX
> > > > C + X = CX
> > > > CX - X = C
> > >
> > > For these carefully chosen examples, sure, but what about:
> > >
> > > III + IX = XII
>
> III+IX = III + VIIII = VIIIIIII = XII

First, subtractive notation gives us the opportunity to perform
preoperations on each final higher order place, so that some calculations
occur solely during the simplification:

+----Cancel from the right
V
+---+
| |+---+
| || |<---Keep
III + IX = XII
|| ||
|+----------+|
+------------+
^
+-----Keep

> > > XXIV + XXVII = LI
>
> XXIV + XXVII = XXIIII + XXVII = XXXXVIIIIII = XXXXXI = LI

Next, we see that like types are handled individually for the first
part of the operation, followed by combination:

++-----++------++++
||+----||-+<---||||---Cancel
||| || | ||||
XXIV + XXVII = XXXXVVI = XXXXXI = LI
| | | |||
+-----+-|-------++|
+---------+

> > > C - I = XCIX
>
> C - I = LXXXXX - I = LXXXXVIIIII - I = LXXXXVIIII = XCIX

Now, a bit that seems quite tricky at first. Each symbol can
(during computation only) be expanded into a subtractive-additive form,
using the next lower level symbol (ignoring groupings of five):

C - I = XCX - I = XCIXI - I = XCIX

>
> Let's get serious. Try 1984 + 1066.
>
> MCMLXXXIV + MLXVI = MDCCCCLXXXIIII + MLXVI = MM DCCCCLL XXXXVIIIII
> = MMML = 3050

Ugh. That was an anticase. Try:

MCMLXXXIV + MLXVI = MMCMLLXXXXVV = MMML
| ||
| ++
Cancel--->| |
| C
| |
+--+

>
> > > etc. This is no better than European digits, and it
> > feels a little like
> > > doing math with pounds, shillings, and pence.

Actually, get a little used to it and you'll find it easier than
decimal addition and subtraction. This is the force of habit at play.
Decimal mathematics are done purely by rote memorization of the tables, then
using combining techniques. It is those combining techniques that give it
power and flexibility, especially where higher order operations are
concerned, and make up for the poor results of the memorized tables.

> Lsd is a simple mixed base.

Says you and Timothy Leary.

;-)

/|/|ike

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