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The number '''zero''', denoted 0, is the smallest natural number. It is the additive identity for natural numbers, meaning that \(a+0=0+a=a\) for any \(a\in\mathbb N\). It is also the multiplicative annihilator for natural numbers, meaning that \(a\cdot 0=0\cdot a=0\) for any \(a\in\mathbb N\). It is the number immediately before [[1]]. |
The number '''zero''', denoted 0, is the smallest natural number. It is the additive identity for natural numbers, meaning that \(a+0=0+a=a\) for any \(a\in\mathbb N\). It is also the multiplicative annihilator for natural numbers, meaning that \(a\cdot 0=0\cdot a=0\) for any \(a\in\mathbb N\). It is the number immediately before [[1]]. |
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As with any other natural number, it may be identified with either an [[ordinal]] or [[cardinal]]. In both cases, it still satisfies the aforementioned properties, being the smallest element of the structure, an additive identity, and the multiplicative annihilator. |
As with any other natural number, it may be identified with either an [[ordinal]] or [[cardinal]]. In both cases, it still satisfies the aforementioned properties, being the smallest element of the structure, an additive identity, and the multiplicative annihilator. In particular, it is equal to the identity in the monoid of [[natural numbers]] under addition. |
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Zero is both the [[cardinality]] and the [[order type]] of the [[empty set]] \(\varnothing\). |
Zero is both the [[cardinality]] and the [[order type]] of the [[empty set]] \(\varnothing\). |
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Revision as of 14:01, 31 August 2023
The number zero, denoted 0, is the smallest natural number. It is the additive identity for natural numbers, meaning that \(a+0=0+a=a\) for any \(a\in\mathbb N\). It is also the multiplicative annihilator for natural numbers, meaning that \(a\cdot 0=0\cdot a=0\) for any \(a\in\mathbb N\). It is the number immediately before 1.
As with any other natural number, it may be identified with either an ordinal or cardinal. In both cases, it still satisfies the aforementioned properties, being the smallest element of the structure, an additive identity, and the multiplicative annihilator. In particular, it is equal to the identity in the monoid of natural numbers under addition.
Zero is both the cardinality and the order type of the empty set \(\varnothing\).