How did the Romans Calculate?

If the Roman Empire is really so far removed from us in time, why is it that Roman numerals were still in commercial use until the 14th century? Before we throw our own guess into the debate, let us look at the nature of these much maligned numerals. How could anyone calculate with them? Well, how can anyone compute "three hundred and seventy-six times two hundred and thirty-seven". You type these data into you pocket calculator and press the "x" button, that's how. You certainly would not fill page after page with number words. Neither did the Romans: they would load CCCLXXVI and CCXXXVII onto their counting board or abacus and manipulate the pebbles and beads until they had the result. We shall presently do such a multiplication, but first we'll look at addition and subtraction.

The counting board shown here is divided into two vertical strips, the right hand one for addition, the other for subtraction. The top number shown on the right is MDCCCCLXV, the lower one is MCCCCXXV. To add them, we just pile everything together into a mess which is shown in third field from the top. To make it readable we have to reduce it: any five beads on a line are converted to one button in the space to the left of that line, any two buttons in a space turn into one bead on the next line over. The answer is MMMCCCLXXXX as shown in the bottom field.

Note: were are using the word "beads" to remind you of an abacus; our "buttons" would be found in the separate top compartment (called "heaven" by the Chinese) of the abacus. We are also ignoring the medieval convention of writing IV, XL, CD instead of the longer but clearer IIII, XXXX, CCCC, used by the ancients.

In the subtraction on the left strip, the first number MCCCCXXV must first be expanded in order to have enough tokens on every line and space to allow the second number DCLIII to be subtracted. The expansion, which is reduction in reverse, is shown in the second field from the top. It need not be done all at once, but can be performed as needed for subtracting. Answer: DCCLXXII.

The power and flexibility of the Roman system is best shown in how it handles multiplication: because of the numbers V, L,D, etc. you need not memorize any multiplication table beyond five. But five itself is just ten halves , and halving is an easy operation. Doubling is another easy operation, quadrupling is doubling twice -- so the hardest multiplier is three. If you do happen to know the ten-by-ten table, you can read every line together with its preceding space as a single decimal digit, and thus increase your speed.

The multiplication shown here is CLXXXXVIIII times DCLIII. There are four partial products (in the blue and yellow fields) corresponding to the four digits of the multiplier: by three, by five (shifted), by one (shifted twice), and by five (shifted twice). As you pile all that into the first of the fields marked green, something special happens on the M-line: three sets of four. Since there is no space for that many, you turn them into a twelve (cf. blue beads) and carry on. After reducing this you get C*X*X*V*MMMMDCCCCXXXXVII, as shown in the bottom field. If you find this too long, compare it to "one hundred twenty-nine thousand nine hundred and forty-seven". By the way, we have changed the Roman bars to asterisks: they mean "thousand".

A Roman wine merchant would have done this in his head: CLXXXXVIIII is one less than CC, so double DCLIII to MCCCVI, shift to C*X*X*X*DC, and subtract DCLIII, and you'll be LIII short of C*X*X*X* -- factus est.

After all this, you must be dying to see a division, and here it is: MMMMDCXXVIIII divided by XIII (the divisor is not written in). It goes just as you expect. Since XIII takes up two lines, you look at the first two lines (plus spaces) of the number to be divided, and you see XXXXVI, which can accommodate three times XIII. So you write a III on the line where your XXXXVI had its I. Then you subtract III times XIII and are left with VII, which is really DCC in disguise. Then you repeat the game, this time taking aim at what looks like LXXII -- and so on, always wandering toward the smaller values on the right.
To appreciate the ease and freedom of this simple gadget, you owe it to yourself to try one. For starters, why not take a chessboard and a supply of pennies? You can start your calculations on the right or on the left, change direction when you spot an opportunity for an easy move -- as long as you keep track of where you are in the calculation, it cannot go wrong. You can add or subtract tokens to undo a lousy move -- you never need an eraser.

The Indo-Arabic numeral system was supposedly introduced to Europe in the early 13th century by a book called Liber Abaci (book of the abacus) written by the widely travelled Leonardo di Pisa (alias Fibonacci) himself no mean mathematician. Present-day scholars say that it was known in the West much earlier -- though still regarded as a Levantine curiosity -- but that the 13th century introduction of paper from China, as a cheap medium for writing, made it the system of choice for all auditors and tax-collectors who wanted to see the details of every calculation.

The pen-on-paper computation with Indo-Arabic numerals -- including the famous zero (originally a punctuation mark) -- made it possible to check calculations for errors, but also penalised false starts and other trivial mistakes by ugly and confusing erasures. To avoid these, you had to follow certain very tight algorithms, which to this day make elementary arithmetic an incomprehensible and unpleasant discipline to many people. As Scott Carlson points out in the article preceding Kasparov's, the paper method makes little sense when a calculator is at hand -- although mental arithmetic is something he evidently likes. To build the bridge between the two, how about re-introducing the counting board?

This ancient and user-friendly tool was still being used in Europe long after people had begun writing numbers in the more compact Indo-Arabic style. As late as 1550, a German textbook was published by Adam Rise, in which the multiplication shown above would be written as 199 times 653 equals 129947, but the intermediate steps would be left as unnamed patterns on the board. Even the Chinese and Japanese write input and output of their abacus work in this style, and this would probably be the right way to bridge the gap between mental arithmetic and the calculator.

In conclusion: the counting board survived (at least) until the 16th century, and for a while (we guess) just carried the Roman numerals along with it. The fact that they are harder to falsify may also have helped.