Dear Cecil:

How did the Romans actually do any mathematical calculations with Roman numerals? Without the concept of places (units, tens, etc.) how did they add, subtract, multiply, divide, sell slaves, and build aqueducts?

Illustration by Slug Signorino

Cecil replies:

Let me toss that question right back at you, Leonard: How do *you* solve complex math problems? You’re probably not working through them in your head, or even on paper. If you need to figure something unwieldy or tricky — say, the square root of 41,786 — you reach for a calculator. And so did the ancient Romans. Their counting devices weren’t electronic, of course, but the tech was high enough for them to establish and administer an empire of nearly 2 million square miles without even coming up with a notation for zero.

The Romans’ contributions to the arena of mathematics weren’t exactly mind-blowing, especially compared to their cultural forebears across the sea in Greece — the Pythagorean theorem is a hard act to follow. When it came to manipulating numbers, the Romans were pragmatists, not theoreticians. As you suggest, conquest, commerce, and engineering were their domains, all fields that do require a certain computational acumen. But the average Roman citizen learned only basic arithmetic in school, under the tutelage of a *calculator*, as math instructors were often called, unless he or she (but almost always he) needed greater knowledge for professional purposes.

And basic Roman arithmetic is largely rather simple, even for those of us spoiled by Arabic notation. Addition is no sweat, because complex Roman numbers already use what math pros call additive notation, with numerals set beside one another to create a larger number. VI is just V plus I, after all. To add large numbers, simply pile all the letters together, arrange them in descending order, and there’s your sum. CLXVI plus CLXVI? CCLLXXVVII, or CCCXXXII. And one of the advantages of the Roman system is that you don’t need to memorize multiplication tables. What’s VI times VI? Six Vs and six Is, which converts to three Xs, a V, and an I: XXXVI.

You can do all this because of the limitation Leonard pointed out above. Roman numerals don’t have what’s called place value, or positional value, the way digits in our system do. The value represented by the Arabic numeral 5 changes depending on its placement within a figure: it can mean five units, or five tens, or five hundreds. But to a Roman, V always meant just plain five, regardless of position. And before you chime in with “What about in IV?” keep in mind that the Roman numerals we use aren’t necessarily the ones the Romans used. Subtractive notation — expressing a value as the difference between a larger number and a smaller one set to its left — was rare in classical Rome and didn’t take off until the middle ages; the Romans greatly preferred the simpler IIII to IV, XXXX to XL, and so on. (The IIII-for-4 notation survives today on the faces of clocks.)

You’ll notice I haven’t mentioned long division — that’s where positional value really pays off. What’s CCXVII divided by CLI? The pile-and-sort method isn’t going to work here. For this one, as well as for the multiplication of larger numbers, you need an abacus. Not too much physical evidence survives, but judging from references in poems by Catullus, Juvenal, and others, and from contemporary devices found in Greece, the standard Roman abacus used glass, ivory, or bronze counters placed on a board marked off into rows and columns. (The counters were at first made of stone and called *calculi* or “pebbles,” the obvious root of several math-related words in English.) A later, more portable version (and this one we’ve found examples of) consisted of a metal plate with beads that slid back and forth in slots.

In either case, the columns or slots were labeled I, X, C, etc., corresponding to the ones column, the tens column, the hundreds, and so on up to millions; the counters or beads kept track of how many you had of each. Essentially, abacuses allowed you to convert Roman figures into a place-based system, do your calculations, then convert back. Some, at least, could even handle fractions, using other specialized slots: though the Romans kept to base ten for whole numbers (as we ten-fingered creatures are wont to do), for smaller values they had a separate base-12 system, making it easier to work with thirds and quarters.

These devices remained in use for centuries after Rome fell. I’ve been talking about the Romans, but remember, their numbering system was still the only one that numerate medieval Europeans had at their disposal. After arriving on the Iberian peninsula in the eighth century, the Arabs introduced their own snazzy notation system (more accurately referred to as Hindu-Arabic), which made its written debut in Christian Europe courtesy of some Spanish monks in 976. Resistance to these foreign ciphers was fierce until the 15th century, when the invention of the printing press spread them widely enough that their utility could no longer be denied, sparking a mathematical revolution. And that’s why I’m able to tell you today that the square root of 41,786 is 204.41624201613725978.

Send questions to Cecil via cecil@straightdope.com.