Definition:Zero

Disambiguation

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Zero may refer to:

• Set Theory and Order Theory:
  • The zero ordinal.
  • The zero cardinal.

• Algebra:
  • The zero of the numbers:
    • The zero of the natural numbers, a concept which follows from, and can be defined from, the definition of the natural numbers as the isomorphism class of a naturally ordered semigroup. From this definition follow:
    • The zero of the integers.
    • The zero of the rational numbers.
    • The zero of the real numbers.
    • The zero of the complex numbers.
  • The zero digit of the number base representation.

• Abstract Algebra:
  • The zero of the naturally ordered semigroup.
  • A zero element of an algebraic structure $\struct {S, \circ}$: an element $z \in S$ such that $\forall s \in S: z \circ s = z = s \circ z$.
  • The zero of a ring: that element $0_R$ of a ring $\struct {R, +, \times}$ such that $\forall a \in R: 0_R \times a = 0_R = a \times 0_R$.
  • The zero of a field: that element $0_F$ of a field $\struct {F, +, \times}$ such that $\forall a \in F: 0_F \times a = 0_F = a \times 0_F$.
  • Root of Polynomial
  • Zero Mapping

• Analysis and Complex Analysis:
  • A zero of a function: given a function $f$ (which will usually be either real-valued or complex-valued), an element $x$ such that $\map f x = 0$.

• Linear Algebra:
  • A zero vector.

• Category Theory:
  • The empty category.

Also see

• Definition:Nonzero
• Definition:Trivial

• 0
• 1

Linguistic Note

The Sanskrit word used by the early Indian mathematicians for zero was sunya, which means empty, or blank.

In Arabic this was translated as sifr.

This was translated via the Latin zephirum into various European languages as zero, cifre, cifra, and into English as zero and cipher.

Note that the plural of zero is either zeros or zeroes. On $\mathsf{Pr} \infty \mathsf{fWiki}$, zeroes is preferred.

The word zero can also be used as a verb, meaning to set (a value) to zero in the context of algorithms and computer science

The word zeroize can also be seen in this context.