Ordinal number

Formation and grammatical use

In English ordinals are usually formed by adding an ending to a cardinal numeral. The most common endings are -st (1st), -nd (2nd), -rd (3rd) and the general -th (4th, 5th, 6th, ...). A small set of exceptions governs teens: numbers ending 11, 12 and 13 take -th despite their final digit, so 11th, 12th and 13th are regular. Ordinals can act as nouns (“the third”), determiners or adjectives (“the third chapter”), and are often hyphenated when used before a noun phrase as a compound modifier (for example, "a 21st-century idea"). They are commonly written in short form (1st, 2nd, 3rd) when space is limited, and spelled out in formal prose.

Rules and common examples

• Basic rule: attach -th to most numerals; use -st, -nd, -rd for numbers ending in 1, 2, 3 respectively (except 11–13).
• Dates: ordinals are used in spoken dates (“July fourth”) though written dates may omit the indicator (“July 4, 2021”).
• Ranking and positions: ordinals label places in contests, floors in buildings, chapters, sections, and steps in procedures.
• Indexing: in computing and mathematics an index may be called first, second, etc.; some contexts use zero-based indexing, which shifts everyday linguistic ordinals by one.

Ordinals in mathematics

Mathematical ordinals generalize the everyday notion of position to infinite sequences and ordered sets. In set theory the term is precise: an ordinal describes the order type of a well-ordered set. Finite ordinals correspond to the natural counting positions (first, second, third, ...). A standard formal approach identifies each ordinal with the set of all smaller ordinals; this construction is often attributed to von Neumann. See set theory for context.

When infinite orderings are allowed, new ordinals appear. The smallest infinite ordinal is commonly denoted by the Greek letter omega. It represents the order type of the natural numbers (including zero in the usual formalization): more background on natural numbers is available at natural numbers. The ordinal ω is the first limit ordinal and is followed by ordinals such as ω+1, ω+2, and larger limit ordinals. These illustrate that ordinals record position in ways that extend beyond mere size—the distinction between being the next element (a successor ordinal) and having no immediate predecessor (a limit ordinal) is important. For comments on infinitude see infinite.

Distinctions and significance

Ordinals differ from cardinals: cardinals measure how many elements a set has (quantity), while ordinals record the place of elements within an ordering. In practical terms, ordinals are used in rankings, schedules, legal documents, navigation, and algorithm design. Historical development spans both everyday language and rigorous mathematical formalization: language influenced notation and everyday use, while mathematics provided formal definitions that handle infinite and abstract order types. For further reading and examples consult introductory resources on ordering and set theory via the links above.