Roman Numerals – The Roman Numeric System

All Roman numerals were formed by using a combination of seven basic symbols, each representing a specific numeric value:

The Seven Basic Numerical Symbols

Roman SymbolNameArabic ValueDerivation of Roman Figure
Iunus1A single digit.
Vquinque5Graphic representation of five fingers.
Xdecem10Symbolises two hands joined together.
Lquinquaginta50Adapted from a Chalcidic sign.
Ccentum100The first letter of the Roman word for one-hundred.
Mmille1,000The first letter of the Roman word for one-thousand.

Symbol Duplication

To express numbers which were multiples of the seven standard values, the symbols could simply be duplicated;

II= 2‘two digits’
XXX= 30‘three tens’
CC= 200‘two hundreds’

Note that the Roman symbols V, L and D are not duplicated, as their combined values can be expressed by other symbols; e.g. VV = 10 = X. Also, duplication was not carried beyond two or three symbols in practice, fourfold duplication being quite rare. The figure IIII = 4 was a common exception, though this and other instances were more usually expressed by subordinate subtraction vide infra.

Simple Addition

All numbers could be formed simply by listing down the appropriate number of required symbols, listing subordinate symbols to the right of those higher in value.

VIII= 8‘a five and three digits’
LXXV= 75‘a fifty, two tens and a five’
CLXVI= 166‘one-hundred, a fifty, a ten, a five and one digit’

Decimal numbers containing the digits 9 or 4, however, look decidedly dodgy when translated in this manner, for example 949 would be DCCCCXXXXVIIII, and 494 would be CCCCLXXXXIIII. The chances of making a mistake when copying down values using this annotation would be pretty high.

Subordinate Subtraction

To avoid duplication mistakes, the Roman accountant also formed compound numerals by the process of subordinate subtraction. In this process subordinates symbols were placed prior to the higher symbol and their numeric value subtracted from that of the higher numeral;

IV= 4‘one digit before five’
XLIX= 49‘ten before fifty and one digit before ten’
CD= 400‘one-hundred before five-hundred’

Note that subordinate subtraction was rarely performed with duplicated symbols. The numerals IIX and CCD are both syntactically correct, but would more properly be represented by VIII and CCC respectively.

Higher Roman Numbers

Numbers in the thousands were at first represented by the Romans with further adaptations of earlier Chalcidic signs. These adapted figures and their numeric values are listed on the right.

During the later Republic thousands were represented by a line inscribed above the appropriate numeral; e.g. I = 1,000 and V = 5,000. Hundreds of thousands were denoted by additional lines on each side of the numeral; e.g. |V| = 500,000 and |X| = 1,000,000 or a million, literally ‘ten hundred-thousands’. The M symbol was not used to denote thousands until the second century AD.

Putting It All Together

A Roman number may employ any or all of the above processes to further complicate matters. To translate any Roman numeral one should first break the number down into separate processes; these breaks are indicated wherever a subordinate symbol follows one of higher value. Each part of the number may then be dealt with separately, and the results combined to give the final figure. These steps are indicated in the following examples;

‘two hundreds’ (= 200),
‘ten before fifty’ (= 40),
‘five and three digits’ (= 8)
= 248
‘one-hundred before five-hundred’ (= 400),
‘two tens’ (= 20),
‘one digit before ten’ (= 9)
= 429
‘one-thousand’ (= 1,000),
‘one-hundred before five-hundred’ (= 400),
‘fifty and two tens’ (= 70)
‘one digit before five’ (= 4)
= 1,474

It is quite a complicated task to add two Roman numerals together or even to subtract one from another, and multiplication and division are both extremely difficult. Mental arithmetic using the Roman numeric system would be beyond the capabilities of most mortal men. The dreary toil of the Roman accountant is reflected in the Latin name for the occupation, Tabularius ‘the worker of tables’. The tabularius was a highly skilled – and thus highly valued – member of any senators household, though usually of the slave class, the lowest order of Roman society, he would be well looked-after for his services. The Roman accountant’s task would be simplified to some extent by the use of an abacus, or by utilising a complicated system of counting using both the digits and finger joints of each hand.

Was 3,999 the Highest That the Romans Could Count?

The highest number that can be expressed in pure Roman numeral form is 3,999 which is written as MMMCMXCIX. This is because the number 4,000 would have to be written as MMMM, which goes against the principle of not having four consecutive letters of the same type together.