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Roman numerals

Modern use of Roman numerals

Roman numerals combine features of a tally system and a numeral system. In modern usage:

• Numerals are written with the largest values to the left: MCI is one thousand plus one hundred plus one, or 1101.
• The values of two identical adjacent numerals are added. MMCI is one thousand plus one thousand plus one hundred plus one. XXI is twenty-one.
• A numeral that is “out of order,” that is, that appears to the left of a numeral with a larger value, has its value subtracted from the value of the larger numeral. So for example:

• The switch to using subtraction generally is made at the point where four adjacent identical numerals would be needed if only addition were used. So:

• Roman numerals are very decimally oriented. Reading from left to right, a Roman numeral consists of distinct subgroups, each of which represents what would be a place in a Hindu-Arabic number. Only a particular set of numerals are proper in each subgroup: those that describe values in that range.
• Roman numerals encountered today usually begin with, at most, M's and only M's. Counting them gives the value of the thousands place in a decimal number.

Place the mouse cursor over the Roman numeral below to split it into its decimal groups. Place the mouse cursor over the yellow areas to make the numbers count up, and see how they change.

The effect is that only I's, X's and C's are subtracted, and only from, at most, the next two larger numerals.  This convention greatly eases the reader's burden, by substituting recognition for calculation. For example, anyone who has been reading copyright dates on many books immediately recognizes a date beginning “MCM...” as something from the 1900's. If “MIM” were permitted, the reader would have to actually do the subtraction.

• In the past Roman numerals included symbols other than those in the above table, some until quite recently. As late as the 1960's the United States Government Printing Office used the convention that a bar over a letter multiplied its value by 1000, so V with a bar is 5000, and M with a bar is 1,000,000. This usage is now very rare.
• Today only addition and subtraction are employed in reading a number written in Roman numerals. But see below.

Today, Roman numerals are used mainly as an alternative to the Hindu-Arabic numerals in outlines and other instances in which two distinct sets of numerals are useful, for clock faces, for ceremonial and monumental purposes, and by publishers and film distributors who have an interest in making copyright dates difficult to read.

Roman numerals in ancient Rome

Perhaps the biggest difference between modern and Roman Roman numerals is that the Romans rarely used the subtraction principle. Nine was much more likely to be VIIII than IX.

A line drawn over a numeral meant that its value was to be multiplied by 1000. If lines were drawn on the top and both sides of a numeral, its value was multiplied by a hundred thousand.

In Rome and later elsewhere characters were used which we no longer have. Some examples:

	½ (alternate symbols)
	4½
	9½ (alternate symbols)
	5,000
	10,000

Roman numerals in Europe during the Middle Ages

This system was almost the only one used in Europe until about the 11ᵗʰ century, and was gradually supplanted during the next 500 years by Hindu-Arabic numerals.

In the Middle Ages, conventions we no longer use were common.

• To make altering the last digit in a number more difficult, the final “i” was extended. Until recently, this practice was often represented in print by j instead of i, for example viij instead of viii.
• If a number was followed by a raised, smaller number (as we would write an exponent), multiply the two numbers.

A very common use of this technique was to indicate a number of scores. For example:

“For there is a C of vi^xx thereby be sold muttons and other beasts and fishes, as for herring v^xx with the tale herring make a C: x^M make a last; and because that a M^I wyll not in a barrel, therefore xii barrels packed herring make a last.”

MS Cotton, Vesp. E. IX (15ᵗʰ century)

vi^xx = 6 times 20, i.e., 6 score, = 120; muttons were sold by a “hundred” (“C”) of 120 pieces

v^xx = 5 times 20, i.e., 5 score = 100; herring was sold by a hundred of 100 pieces

x^M = 10 times 1000 = 10,000

M^I = 1000

• The Roman numerals C and M sometimes did not mean 100 or 1000 (see hundred).

A phrase was frequently added to resolve the ambiguity. For example, from the same source as above:

“Also eels be sold by the stike, that is xxv eels, and x stikes make a gwyde, ii^cl by v^xx.”

Here the phrase “by v^xx” (five score) indicates that the “c” in the previous number, “ii^cl”, means 100, so ii^cl is 250 (2 times 100, plus 50).

Roman numerals in old manuscripts

Certain types of errors are typical in reading Roman numerals in old manuscripts, due to physical damage to the text. Kemble describes some:

This [inconsistency in dates] however generally arises from the latter date having been partially abraded by age, and so misread: the want of a light line at the bottom readily transforms a V (in the old charters U) into a II; an abrasion may convert an X into a V: hence we not uncommonly find in these copies indiction IIII for VII or XV; VII for XII; XII for XV, and the like. Nor is another error at all uncommon, where a letter or contraction has been taken to be part of the date: for instance, indictione u^o (uero) II^a, has often been read as if it were indictione VII^a. Again, indictione X^ma has become transformed into indictione XIII^a, the strokes of the written “m” having been taken to represent three units. This cause of error is so frequent as to render multiplied examples unnecessary.

Johannis M. Kemble.
Codex Diplomaticus Aevi Saxonici.
London: Sumptibus Societatis, 1839.
Reprinted in facsimile by Kraus Reprint Limited, Vaduz, 1964.
Volume 1, page lxxxix.

resources

Florian Cajori.
A History of Mathematical Notations.
La Salle, Illinois: Open Court, 1928 and 1929.

Republished in one volume by Dover in 1993.

Georges Ifrah.
Translated by David Bellos, E. F. Harding and Sophi Wood.
The Universal History of Numbers: From Prehistory to the Invention of the Computer.
John Wiley and Sons, 1999.

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Last revised: 4 October 2014.