Movatterモバイル変換

[0]ホーム
URL:

Jump to content
WiktionaryThe Free Dictionary
Search

ordinal number

From Wiktionary, the free dictionary

English

[edit]
EnglishWikipedia has an article on:
Wikipedia

Pronunciation

[edit]

Noun

[edit]

ordinalnumber (pluralordinal numbers)

  1. (grammar) A word that expresses therelativeposition of an item in asequence.
    First,second andthird are theordinal numbers corresponding to one, two and three.
  2. (arithmetic) Anatural number used to denoteposition in asequence.
    In the expression a3, the "3" is anordinal number.
  3. (set theory) Such a number generalised to correspond to anycardinal number (the size of someset);formally, theorder type of somewell-orderedset of somecardinalitya, which represents anequivalence class of well-ordered sets (exactly those of cardinalitya) under theequivalence relation "existence of an order-preservingbijection".
    • 1950, Frederick Bagemihl (translator),Erich Kamke,Theory of Sets, Dover (Dover Phoenix), 2006,page 137,
      For not only do the antinomies a) to e) disappear when we admit as elements of sets only such sets,ordinal numbers, and cardinal numbers as are bounded above by a fixed cardinal number, but we see also that paradoxes always arise if we collect into a set any sets, cardinal numbers, orordinal numbers which are not bounded above by a fixed cardinal number.
    • 1960 [D. Van Nostrand],Paul R. Halmos,Naive Set Theory, 2017, Dover, Republication,page 80,
      Is there a set that consists exactly of all theordinal numbers? It is easy to see that the answer must be no. If there were such a set, then we could form the supremum of allordinal numbers. That supremum would be anordinal number greater than or equal to everyordinal number. Since, however, for eachordinal number there exists a strictly greater one (for example, its successor), this is impossible; it makes no sense to speak of the "set" of all ordinals.
    • 2009, Marek Kuczma, edited by Attila Gilányi,An Introduction to the Theory of Functional Equations and Inequalities, 2nd edition, Springer (Birkhäuser),page10:
      Ifα{\displaystyle \alpha } is anordinal number, then by definition any two well-ordered sets of typeα{\displaystyle \alpha } are similar, i.e., there exists a one-to-one mapping from one set to the other. Consequently these sets have the same cardinality. Consequently to anyordinal numberα{\displaystyle \alpha } we may assign a cardinal number, the common cardinality of all well-ordered sets of typeα{\displaystyle \alpha }.

Usage notes

[edit]

On ordinal number usage:

eleventh day
  • If an ordinal is followed by a plural noun, the two-word phrase refers to a set of items described by the phrase in singular. For examplesecond homes refers to a set of homes each of which is considered a "second home."
two fifths ,25{\displaystyle \textstyle {\frac {2}{5}}}
  • Ordinal numbers are used inexponents, where generally construed asadjectives precedingpower to which a base is raised:
two to the minus twenty-first power ,2−21{\displaystyle 2^{-21}}
six to the third ,63{\displaystyle 6^{3}}
  • Ordinal numbers are generally considered to be ordered fromhigh tolow , so thatfirst place is consideredhighest , andfifth is lower thansecond.Degree is an exception.
  • Ordinal numbers corresponding to numbers higher than 20 usecardinal numbers for all the places preceding the final ordinal part:
twenty-first or21st , occasionallyXXI
one hundred fifteenth or115th , occasionallyCXV
thirty-three thousandth or33,000th

Synonyms

[edit]

Antonyms

[edit]

Coordinate terms

[edit]

Translations

[edit]
grammar: word used to denote relative position in a sequence
arithmetic: number used to denote position in a sequence
generalized kind of number to denote the length of a well-order on a set
The translations below need to be checked and inserted above into the appropriate translation tables. See instructions atWiktionary:Entry layout § Translations.
Translations to be checked

See also

[edit]

Further reading

[edit]
Retrieved from "https://en.wiktionary.org/w/index.php?title=ordinal_number&oldid=86668504"
Categories:
Hidden categories:
[8]ページ先頭
©2009-2025 Movatter.jp