The Roman Arithmetic

We all know that the Roman numeral system is cumbersome and inappropriate for developing rigorous and complex mathematics. It is even unsuited for simple operations such as addition, subtraction and multiplication. Yet the Romans used it efficiently to deal with all the complexities of their Empire. So, how did they do it?

Some probable explanation for the origins of Roman numerals

The Roman numbering system originated from ancient Etruscan numerals.  There are several theories as to how it was conceived. One of them is that it derives from the tally sticks used by shepherds. They used to cut notches in the sticks. They would have, thus, symbolized the unit by a notch which later became I, and each fifth unit by a double cut (Λ or V), then each tenth (X) by a transverse cut…

Another possibility is that it represented the system used to count with the hands: I, II, III, IIII corresponding to each of the fingers put forward; V representing all the fingers of the hand held tightly together and the thumb spread out. The numbers 6 to 9 would then have been represented using a V with one hand and I, II, III or IIII with the other one, while 10 (X) would have been represented crossing the two thumbs while holding the other fingers of both hands straight as we would represent a bird today.

These are of course hypotheses that are probably far from the truth, but not necessarily unfounded. It is also likely that we will never know the true origin of Roman numbers, assuming that it is unique.

This numerical system was, however, flawed in many ways, not the least of which was that there was no symbol for the zero … and no real method of counting beyond several thousand, other than adding lines around the digits to indicate multiples.  This is all the more surprising coming from a people who had so much regard for technological advances. Indeed, unlike the Greeks, the Romans were never interested in pure mathematics. There was no innovation in this field during the Republic and the Empire. They had no mathematicians of note, and the Romans acted as if they had no need for it, but only for their practical applications.

Nevertheless, this did not prevent their scholars and architects from building the largest empire that mankind has ever known relative to the world population. Mathematical skills were therefore necessary to govern a complex society and economy, as well as to build vast monuments like the Colosseum or the network of aqueducts crisscrossing the Empire.

So how did the Romans go about performing simple operations?

Addition and subtraction

The difficulty comes from the fact that the Roman numerical system is not directly positional and did not include the zero. It was a clumsy system for obvious arithmetical and logical reasons. It was based on letters of the alphabet (I, V, X, L, C, D and M) that are combined to represent the sum of their values (e.g. XII = X+II = 10 + 2 = 12).

What did the Romans ever do for maths? Very little
Roman abacus

At the beginning the system was additive in the sense that 9 was written VIIII, later and to simplify the writing it became also subtractive, meaning that 9 could then be written IX. This certainly simplified the writing a bit, but made the calculation even more difficult, and required the conversion from subtractive to additive notation at the beginning of any sum. Because of the difficulty of this arithmetic, calculations were usually made on an abacus, of which Chinese abacuses can give us today a fairly accurate representation.

But how did they do it by hand? It was done by reproducing the process used on the abacus. Let’s imagine the sum of 359+1029. This is written CCCLIX+MXXIX in Roman numerals. The numbers had to be rewritten in an additive way , thus becoming CCLVIIII+MXXVIIII.

Then the operation had to be performed and added column by column.

CCCLVIIII
+MXXVIIII
MCCCLXXVVIIIIIIIIII

The result had then to be simplified starting from the right. IIIIIIII (8) is rewritten as VIII and the V is moved to the next column on the left and the operation is repeated. VVV (15) is written XV where X goes into the left column… in total we find the number MCCCLXXXVIII (1388). The process is rigorous, but tedious.

Subtraction was done in the same way, but with even more difficulty without the zero.

Multiplication: an exercise in ingenuity

Multiplication would have seemed impossible if human ingenuity had not found a solution. The Romans used a method to reduce it to a series of additions using only simple operations like doubling or approximately halving. This method is known today as the Russian method.

How was it done?

Write the two numbers to be multiplied next to each other; then double one and halve the other in two independent columns. Just write the whole numbers without the decimals. Repeat the process until the number in the reduce column equals to 1. Then, in the duplicate column, cross out all the numbers that are opposite an even number in the reduce column, and add the rest of the numbers in the duplicate column together. The sum is exactly the required product.

For example: 19 x 321=6099

19        321

9          642

4          1284

2          2568

1          5136

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6099

In the meantime, one will have used the previous method to do the tedious additions.

What about the division? This is a good question, but one for which there is no satisfying answer yet.

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