Numerālia

Latin numerals fall into four classes, each answering a different question about quantity:

(1) Cardinals (cardinālia) — answering how many?. The basic counting numbers: ūnus, duo, trēs, quattuor, quīnque, sex, septem, octō, novem, decem... Most cardinals are indeclinable adjectives; only the numbers 1, 2, 3, and the multiples of 100 (ducentī, trecentī...) decline. Mīlle (1000) is indeclinable in the singular but declines in the plural (mīlia, mīlium, mīlibus).

(2) Ordinals (ōrdinālia) — answering which in order?. The order numbers: prīmus (first), secundus (second), tertius (third), quārtus (fourth)... All ordinals decline like bonus, -a, -um (1·2-decl. adjectives).

(3) Distributives (distribūtīva) — answering how many each? / how many at a time?. The "by-counts": singulī, -ae, -a (one each), bīnī (two each), ternī (three each), etc. Decline like 1·2 plurals (always plural). Used with collective and naturally-plural nouns and in certain idioms.

(4) Numeral adverbs (adverbia numerālia) — answering how many times?. The "times": semel (once), bis (twice), ter (thrice), quater (four times), quīnquiēs (five times)... Indeclinable.

The four classes systematize different ways of quantifying — by amount, order, distribution, frequency. Latin tightly distinguishes them; English often uses single forms (e.g., "two" for both cardinal and distributive in some contexts). Reading classical Latin requires recognizing which class is intended.

Note that compound ordinals are written as two separate words: tertius decimus (13th), vīcēsimus quīntus (25th). The order can be either way: quīntus et vīcēsimus is also possible.

Distinguish: secundus "second" (in order, sequential) from alter "the other (of two), the second of two": alter exercitus "the second of two armies"; secundus exercitus "the second army (in a series)".

Secundus also means "favorable" (as adj.) — secunda fortūna "favorable fortune". The dual sense reflects the verb root sequor "follow" — "the one following / the favorable one".

Ordinals are used in Latin where English sometimes uses cardinal: dates, regnal years, anniversaries (quīnquāgēsimus annus = "the 50th year" of a reign).

Distributives are used:

(a) With "plurals tantum" — nouns that have only a plural sense (cāstra "camp", litterae "letter", arma "arms"): bīna cāstra "two camps" (NOT *duo cāstra); ternae litterae "three letters".

(b) Where English uses "each" or "apiece": singulī cōnsulēs "one consul each (year)"; bīnī mīlitēs "two soldiers per (group)".

(c) Idiomatically in multiplication: bis bīna sunt quattuor "twice two is four" (the multiplier is the adverb, the multiplicand the distributive); ter ternī = "thrice three each" = 9.

(d) Regularly in poetry instead of cardinals: bīnī for duo, ternī for trēs, etc. Vergil and Horace use distributives where prose would use cardinals.

The numeral adverbs end in -iēs (or -iēns) from 4 onward. They are indeclinable. The forms semel (once), bis (twice), ter (thrice) are very ancient and irregular; from quater onward, regular formation begins.

Roman numerals use seven basic symbols:

• I = 1 (originally a vertical stroke).
• V = 5 (the upper half of X).
• X = 10 (originally a stroke with a crossbar).
• L = 50 (the early form of an angle ┴).
• C = 100 (from centum).
• D = 500 (from earlier ID).
• M = 1000 (from mīlle; in Roman times often written ⅽⅼↄ or CIↃ).

Subtractive notation (IV = 4, IX = 9, XL = 40, etc.): regular by classical times. A smaller numeral immediately before a larger one is subtracted: IV = 5−1 = 4; XC = 100−10 = 90.

Additive notation (without subtraction): VI = 5+1 = 6; LX = 50+10 = 60; CCC = 100×3 = 300.

Bar above a numeral multiplies it by 1000: V̄ = 5,000; X̄ = 10,000; C̄ = 100,000; M̄ = 1,000,000. The Romans used this convention for very large numbers.

For numbers in the millions, the formula was deciēs centēna mīlia = "ten times a hundred thousand" = 1,000,000 (literally "10 × 100,000"). Even larger numbers used compounded multiplications.

Modern usage often retains Roman numerals for: chapter numbers, monarchs (Henry VIII), film sequels (Rocky II), copyright dates on films (MMXXVI = 2026), and clock faces.

Fractions are expressed in two main ways:

(a) Ordinal in numerator + plural feminine ordinal in denominator (with partēs "parts" understood or expressed): duae tertiae partēs "two thirds (parts)"; tria septima "three sevenths". The fraction is treated as a fraction of a whole.

(b) Halves and twelfths use special vocabulary (from old Roman fractional system, Roman asses):
• dīmidius, -a, -um "half" = 1/2.
• uncia = 1/12.
• sextāns = 2/12 = 1/6.
• quadrāns = 3/12 = 1/4.
• triēns = 4/12 = 1/3.
• quīncūnx = 5/12.
• sēmis = 6/12 = 1/2 (the "half-as", from old coin).
• septūnx = 7/12. bēs = 8/12 = 2/3. dōdrāns = 9/12 = 3/4. dextāns = 10/12 = 5/6. deūnx = 11/12.

Time: Roman day = 12 hours, divided sunrise to sunset; varied by season. Hours numbered: hōra prīma "first hour" (~6am winter, ~5am summer); hōra sexta "sixth hour" (noon).

Year notation: AUC (ab urbe conditā) "from the founding of the city" — Rome traditionally founded 753 BCE. Years reckoned from this anchor or by the names of the consuls (a primary dating method in formal documents): cōnsulibus M. Tulliō et C. Antōniō = "in the consulship of Marcus Tullius and Gaius Antonius" = 63 BCE.

Numeral usage: numerals govern the partitive genitive when emphasis is on division (decem mīlitum "ten of the soldiers"), but normally agree directly (decem mīlitēs "ten soldiers"). Numerals occupy positions adjacent to the noun: trēs librī or librī trēs — both standard.