The beauty and logic of Roman numerals

by Karel Thönissen

Every now and then I hear people complain about the Roman numerals and how complicated it is to perform arithmetic with these. People will then grab a piece of paper do some fancy calculation and wonder how the Romans would have done that in a reasonable amount of time. How did the Romans do without the digit zero? How did they trade without our efficient numeric system and our hand-held computing devices.

Well: they had their own hand-held computing devices! For most practical purposes these devices were quicker than any ordinary modern person with pen and paper. Okay, electronic computers beat the Romans. How come that the Romans performed so well with arithmetic?

We often forget that our numeric notation and our calculation technology go hand in hand. New technology affords new notations. Hexadecimal notation became useful with the invention of the binary digital computer, paper afforded the storage of intermediate results of calculations (something that is impossible on an abacus), etc.

From archeology we know that the Romans used the abacus for their arithmetic. The abacus has several rows with beads that can be shifted to represent the unit, the unit of 5, the unit of 10, the unit of 50, 100, 500, 1000, etc. It is no coincidence that these correspond exactly with the Roman numerals: I, V, X, L, C, D and M. Given a number on an abacus it is very simple to write it down with this notation: there is a more or less one to one correspondence.

I wrote down 'more or less' because there are two interesting questions that need to be answered. Firstly, why did they use beats to represent the units of 5, 50, 500 at all? Why not do without these and offer 10 beats in the row for the units, for the units of 10 and for the units of 100, etc. We all know that the human mind can instantly recognise the count of small collections of items. When collections of items get larger, we really need to count. So when adding 8 to a number it is simpler to add a 'fiver' and three 'oners' than eight 'oners' because in the former case we can recognise the beats immediately.

Secondly, why did the Romans represent some numbers as a subtraction rather than a simple addition, e.g. why is 4 represented as IV rather than IIII or 9 as IX rather than VIIII? The anser is that this notation was far more efficient on the technology of those days, i.e. the abacus.

When we want to add 4 on our abacus, we can either add 4 units to the current accumulent, or subtract 1 and add 5 (since 4 = !1+5 ). Notice that the latter requires only the shifting of 2 beats, whereas the former requires the shifting of 4 beats. So adding IV is less work than adding IIII. But there is another problem. There are five beats on the abacus for the units of 1. When the accumulent already has 2, 3, 4, or 5 beats shifted in for the units, then adding IIII is not directly possible since there are only 5 beats on the row. Now remember your school days: in a case like this you would carry over to the higher unit. In this case, one would mentally add 4, ignore the fivers on the row for the units, and shift in one fiver on the row for the fivers. Now this corresponds exactly with IV: adding 5 subtracting 1. So in 5 out of 6 cases (when the row for the units has 1, 2, 3, 4, 5 beats) adding IV is not only shorter but also avoids a carry-over.

This notation only makes sense for the 4 as a numeral. Representing 3 as IIV instead of III gains nothing as far as the number of beat shifts is concerned: both require 3 shifts when performing arithmetic. Moreover III can be added without carry-over to 0, 1 and 2 beats, but not to 3, 4 and 5 beats. So either notation requires a carry-over in half the cases, so both notations are equally efficient.

Romans had abacuses as hand-held devices and with a little exercise it is possible to perform simple math faster than with pen and paper. The nice property of the abacus is that it can remember the intermediate results for you. During multiplication one only has to shift in the partial multiplications and when these are done the result is already there. We use pen and paper technology to store the intermediate results and then add these at the end, giving the final result. But even during the multiplications we must store partial results. We do this mentally by remember the carry-over. The Romans did not have to do this and they would easily outperform us on everyday arithmetic.

So technology and language/ notation go hand in hand in ways often misunderstood.