0

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 \(\varnothing\).