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| '''One''' (''1'') is a ]. | |||
| ⚫ | |||
| It is the ] following ] and preceding ]. | |||
| If a ring is recognized as a ] of ]s, its multiplicative identity can be referred to as ]. | |||
| For any number <i>x</i>: | |||
| * <i>x</i>·1 = 1·<i>x</i> = <i>x</i> (see ]); | |||
| * <i>x</i>/1 = <i>x</i> (see ]); | |||
| * <i>x</i><sup>1</sup> = <i>x</i> and 1<sup><i>x</i></sup> = 1 (see ]). | |||
| One cannot be used as the base of a ] in the ordinary way. | |||
| Sometimes ]ing is referred to as "base 1", since only one mark (the tally) is needed, but this doesn't work in the same way as the usual numeral systems. | |||
| Related to this, one cannot take ]s with base 1 (much as one cannot ]). | |||
| One is not always though of as a number, although (unlike zero) it has been accepted as such since antiquity. | |||
| Reflecting this, many languages retain a distinction between ] and ] forms of a ], the former reserved for the case when only one object is being referred to. | |||
| In the ] of natural numbers, 1 is the ] {0}. | |||
| This set has ] 1 and ] 1. | |||
| In a multiplicative ] or ], the ] is sometimes called 1, but <i>e</i> is more traditional. | |||
| ⚫ | However, 1 is especially common for the multiplicative identity of a ]. | ||
| In ]s, the one is the ], which in many games is considered the highest card in the suit, rather than the lowest as one would normally expect. | |||
Revision as of 04:40, 21 February 2003
One (1) is a number. It is the natural number following zero and preceding two.
For any number x:
- x·1 = 1·x = x (see Multiplication);
- x/1 = x (see Division);
- x = x and 1 = 1 (see Exponentiation).
One cannot be used as the base of a numeral system in the ordinary way. Sometimes tallying is referred to as "base 1", since only one mark (the tally) is needed, but this doesn't work in the same way as the usual numeral systems. Related to this, one cannot take logarithms with base 1 (much as one cannot divide by zero).
One is not always though of as a number, although (unlike zero) it has been accepted as such since antiquity. Reflecting this, many languages retain a distinction between singular and plural forms of a noun, the former reserved for the case when only one object is being referred to.
In the Von Neumann representation of natural numbers, 1 is the set {0}. This set has cardinality 1 and hereditary rank 1.
In a multiplicative group or monoid, the identity element is sometimes called 1, but e is more traditional. However, 1 is especially common for the multiplicative identity of a ring.
In playing cards, the one is the ace, which in many games is considered the highest card in the suit, rather than the lowest as one would normally expect.