Origins: Adding Up

As I write this, the National Football League season is underway, and U.S. football fans are looking forward to Superbowl XL. Some of us are old enough to have watched all the Superbowls, starting with Superbowl I back in MCMLXVII, when Vince Lombardi’s legendary Green Bay Packers defeated the Kansas City Chiefs XXXV to X in the old Los Angeles Coliseum. These days I root for the Carolina Panthers, who lost Superbowl XXXVIII to the New England Patriots, XXXII to XXIX, in a contest remembered more for Janet Jackson’s wardrobe malfunction than for the football game.

In terms of mathematical technology, the Roman numerals in the last paragraph have been obsolescent for more than a thousand years, but we still use them to mark the hours on clock faces, the pages of a book’s preface or the elements in a sequence such as King Henry VIII or Superbowl XL. Students in elementary schools have to put down their calculators to learn a number system designed to be chiseled in stone. The rules they learn are very old, but oddly enough many of these rules don’t come from the Romans but from medieval scribes who codified a process the Romans themselves used much less formally.

Roman numerals are certainly not the oldest system for representing numbers. The oldest numbering system probably involved using one’s fingers, as children do even without any prompting. It’s not much of a step from counting on your fingers to making finger-like marks in the dust: |, ||, |||, ||||. When we get to five, we usually draw a slash through the symbol ||||, creating a new symbol. This system is probably as old as any form of writing.

Mathematicians say these early number systems are additive: There are symbols for certain numbers, symbols that can be combined, and when they are combined the numerical values of the symbols are added. In Superbowl XXXVIII, the numeral is XXX + V + III = 30 + 5 + 3 = 38.

The Egyptians had a simple additive system of numerals with symbols for one, ten, hundred, thousand and ten thousand. If the Romans had done this, their number for 89 would be XXXXXXXXIIIIIIIII. It worked better for the Egyptians, though, because the hieroglyphic symbols of their written language did not all have to be written in line. When they wrote a nine, they could stack the nine vertical marks in a 3 x 3 array, like the spots on a domino.

The Greeks had two systems for representing numbers. The older of these systems was an additive system somewhat similar to the Egyptian system. One was represented by I, as always. For ten, hundred, and thousand, the symbols were the first letters of the Greek words for those numbers: Δ (or delta, the initial letter in deka) for ten, H (hekto) for hundred, and X (chilo) for thousand.^a

To make numerals more compact, the Greeks introduced a symbol for five, using either π (pi), the first letter of the Greek word penta (meaning five), or Γ, which served as a kind of shorthand for π. To represent 50, 500 or 5,000, the symbol for five was combined in some fashion with the symbol for ten, hundred, or thousand. Some of these compound symbols look almost like modern superscript notation: Γ^Δ for 50, Γ^H for 500 and Γ^X for 5,000.

It’s clear that the Roman system has much in common with its Egyptian and Greek forebears. We all know its symbols: I for one, V for five, X for ten, L for fifty, C for hundred, D for five hundred and M for thousand.

Where did these familiar symbols come from? Although it might appear that C and M derive from the Latin words for hundred (centum) and thousand (mille), the fact is that all the Roman numerals, including C and M, evolved over a period of centuries from ad hoc markings used in counting, first by the Etruscans and then by their Roman successors.

In Etruscan times, one was represented by a single stroke | and five by a double stroke, originally written Λ in most cases but eventually inverted by the Romans to V. Ten was represented by two double strokes ΛΛ, and eventually one of the double strokes migrated on top of the other to be represented as X.

The Etruscan symbol for 1,000 was a vertical stroke with two curved strokes on either side attached at the top of the vertical stroke. At some point, this became the Latin M. For 500, half a thousand, the Etruscans would omit one of the curved strokes, and that symbol evolved into the Latin D.

The Etruscan symbol for 100 was likely a horizontal bar with a short vertical stroke on top of it, like an upside-down T. For 50, half of the horizontal bar was eliminated, producing something like an L or a backwards L. That explains the L symbol, but the upside-down T had to turn on its side and lose its crossbar before becoming C, so it remains possible that the initial phoneme (sound) of the Latin word for 100 (centum) had some influence in this case.

Most Romans wrote numerals in a strictly additive way, so four was written IIII, 40 was written XXXX, and 949 was written DCCCCXXXXVIIII. It’s simple enough, and they didn’t seem to mind the length of the numerals. Occasionally they used the subtractive convention, in which, if a smaller number precedes a larger one, then the smaller is subtracted from the larger. This allows nine to be written as IX instead of VIIII, and 94 as XCIV rather than LXXXXIIII.

However, this was not common, and it conflicted with other conventions also in use. Sometimes a multiplicative notation was used, in which VM represented 5,000 and not 995, and CM represented 100,000 and not 900. There was no one in the Roman Empire who could enforce syntax standards consistently, so conflicting notations arose in different places and at different times. Disputes and misunderstandings about the numerals in documents could and did occur.

It was not until medieval times that the system now in use was completed. The rules now are that only certain subtractive combinations are allowed: IV for 4, IX for 9, XL for 40, XC for 90, CD for 400, and CM for 900. Thus 949 can be shortened from the 14-letter form DCCCCXXXXVIIII to the 6-letter form CMXLIX, but it can’t be shortened further to the 4-letter form CMIL. This was particularly noticeable during the year 1999, which was written MCMXCIX rather than the simpler MIM.

The persistence of these ancient numerals in modern life is somewhat surprising. It seems that they serve several purposes. On the face of a clock or the cornerstone of a building, they are an explicit reference to tradition. Sometimes they provide a number sequence distinct from the main sequence; that’s why they are used to number the pages of the preface of a book. Sometimes they are used as ordinal numbers, as in the name of the late Pope John Paul II, where the number symbol means “second” rather than “two.”

And sometimes they are used deliberately to obscure a number. For example, laws require that the copyright dates of movies and television shows be shown as part of the work, but distributors would prefer that the viewers not reflect on whether the work is new or not. So they state the dates in Roman numerals, “Copyright MCMXCVII, all rights reserved …” because they know the audience can’t compute that date before it fades from the screen.

Mathematically, Roman numerals are ox carts in a rocket-powered world, but it isn’t necessary to be modern and efficient all the time. Sometimes it’s nice to turn off the TV and bake your own bread. The survival of the Roman numeral system represents the appeal of the traditional and reminds us of our cultural origins, and for those reasons the system probably will be in use for C’s of years to come.

As I write this, the National Football League season is underway, and U.S. football fans are looking forward to Superbowl XL. Some of us are old enough to have watched all the Superbowls, starting with Superbowl I back in MCMLXVII, when Vince Lombardi’s legendary Green Bay Packers defeated the Kansas City Chiefs XXXV to X in the old Los Angeles Coliseum. These days I root for the Carolina Panthers, who lost Superbowl XXXVIII to the New England Patriots, XXXII to XXIX, in a contest remembered more for Janet Jackson’s wardrobe malfunction than for the football game. In terms of mathematical […]