> I am not convinced. The fact is that John has had to type three
> separate lines: one labeled "in the units place", one labeled "in the
> tens place", etc. For the sake of completeness, this would need to
> continue indefinitely (and one would quickly run out of alphabet letters).
But that's what actually DOES happen with Roman numerals, so is
no objection. The system was continued by doubling up the outer strokes
of M and the right stroke of D to multiply by 10, giving symbols like
I)) ((I)) I))) (((I))) ...
for 5000 10000 50000 100000 ...,
and at a later date by placing bars over letters to multipy their
values by 1000, but neither of these conventions increases the
difficulty of multiplication.
> More importantly, what this means is that one would need to
> memorize not just one "times table" (as in Arabic notation, up to 9 by
> 9) but rather two, three, indefinitely many "times tables".
This ignores my remark that every arithmetician would know
the transformation I -> X -> C -> M -> ((I)) -> (((I))) -> ... and
V -> L -> D -> I)) -> I))) that multiplies by 10. So knowing that
III times VIII is XXIV entails knowing its images:
XXX times VIII is CCXL, as is
III times LXXX.
> And that, it seems to me, is the whole point. The invention of
> place-value ciphering was a leap beyond tallying (and Roman numerals are
> an upgraded version of tallies, it seems to me) because it meant that
> counting and computation could be reduced to a small set of basic facts.
Well, I agree of course that place-value was a great invention,
and that multiplication in Roman numerals is a bit harder. What
the abacus gave them was really a way to use "III times VIII" to
mean any of its images at will.
> Perhaps those who think that Roman numeral arithmetic is not so
> difficult could (say) try to multiply CMLXXXIX by itself, and then
> report back to us whether their opinion remains the same?
I don't think that this is a fair test, because none of us has
been taught the multiplication table in Roman numerals. An arithmetician
who had would find it just as hard to multiply 989 by 989 after
having been taught the conversion rule.
John Conway