## Arithmetic with Roman Numerals

It is fairly obvious that Roman numerals are more difficult to use than our Hindu-Arabic numerals. They are fine for recording a date or indicating a chapter number. But, they make arithmetic somewhat difficult.

Some arithmetic is easy:

CXXI + CXII=CCXXXIII

XVI + VII=XXIII

XVII - VI=XI

CXII x II=CCXXIV

We can even cancel some symbols:

CIV + VI=CX

Here, the I's cancelled, since one of them was negative.

More complicated arithmetic mainly involves more "carries," "borrows," and conversions from one form of a four (IIII) or a nine into a more accepted form (IV):

MCMXCVI + XIV=MCMXCXX=MCMCX=MMX

I got a preliminary answer (MCMXCXX). I didn't like its form, so I combined symbols to get the next answer (MCMCX). Then I did a similar combining to get my final, more acceptable answer (MMX).

And we also notice that years of the 21st Century will have less interesting Roman numeral representations, at least for a while.

Complicated multiplications are actually just as easy, but much more tedious:

XXVII x XIV=LLXXVVV - XXVII + CCLXX=CCCLXXVIII

I didn't simplify any of my partial sums, because I suspected that some cancelling of some symbols was going to be possible.

Division is more difficult. See my article, Division With Roman Numerals.

An observation can be made: The Japanese Abacus (the soroban) is ideal for calculating with Roman numerals. This abacus, unlike the more ambiguous Chinese abacus, uses four beads to represent ones, one bead to represent a five, four beads for tens, one for 50, etc. It is simple to convert from numbers represented on the abacus to Roman numerals and back again.

Another observation is that Egyptian Numbers are essentially Roman numerals, with no fives, or 50's, or 500's:

=1996

So, in Egyptian, numbers are longer than they are in Roman numerals. But, it turns out that arithmetic is simpler. There are fewer carries and no borrows (when adding). So I deduce that calculation was done more often in Egypt than in the Roman Empire. And I thought that maybe the Romans were more interested in displaying their numbers, than they were in calculating. This deduction, of mine, is a little shaky, as Egyptian arithmetic was done in hieratic (script hieroglyphics), which was not as easy to manipulate. In hieratic, the number seven is one symbol, which represents seven vertical lines.