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In mathematics a number is the actual quantity, and a numeral is the symbol used to write it, like '4' or 'IV'.

In linguistics however the quantity is called the numeral, so we speak of the numerals from one to ten being un, deux, trois etc.; because the term number refers to the grammatical property of being singular or plural (or dual or trial in those languages that have it).

Mathematical logic uses the term numeral to refer to an explicit representation of numbers (almost always, of natural numbers). This use is an extension of Gritchka's "mathematical" sense: "5" is the numeral (in the decimal representation called "Indian numerals" or "Arabic numerals") of the number five. However, "17" is the numeral for seventeen in this mathematical logic sense.

More formally, "numerals" for the natural numbers are any definitions for 0 ("zero") and S ("successor") which satisfy Peano's axioms. Since any 2 models of these axioms are isomorphic, all numerals are indeed representations of the natural numbers. Examples include von Neumann numerals and other definitions from set theory (see natural numbers as sets on E2; however note also that von Neumann numerals are really "numerals" for all ordinals, not just the natural numbers), and Church numerals.

Nu"mer*al (?), a. [L. numeralis, fr. numerus number: cf. F. num'eral. See Number, n.]

1.

Of or pertaining to number; consisting of number or numerals.

A long train of numeral progressions. Locke.

2.

Expressing number; representing number; as, numeral letters or characters, as X or 10 for ten.

Nu"mer*al, n.

1.

A figure or character used to express a number; as, the Arabic numerals, 1, 2, 3, etc.; the Roman numerals, I, V, X, L, etc.

2.

A word expressing a number.

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