Zero is the base natural number, written 0. In the Peano axioms, zero is the starting element: it is a natural number, and it is not the successor of any natural number. Every other natural number is obtained from zero by applying the successor operation some finite number of times.
Zero is the identity element for addition: n + 0 = 0 + n = n for all n. It is the annihilator for multiplication: n × 0 = 0 for all n. In the integers, zero is the unique number that is neither positive nor negative.
In set theory, zero is represented by the empty set ∅ — the set with no elements. The cardinality of the empty set is 0, and the empty set serves as the starting point for the von Neumann construction of the natural numbers: 0 = ∅, 1 = {∅}, 2 = {∅, {∅}}, and so on.