# Definition:Identity Arithmetic Function

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## Definition

The **identity arithmetic function** $\iota: S \to \Z$ is defined for $n \geq 1$ by:

- $\forall n \in S: \map \iota n = \delta_{n 1}$

where:

- $S$ is (in theory) any set, but in this context is usually one of the standard number sets $\Z, \Q, \R, \C$.
- $\delta$ is the Kronecker delta.

That is:

- $\forall n \in S: \map \iota n = \begin {cases} 1 & : n = 1\\ 0 & : n \ne 1 \end {cases}$

## Also see

The **identity arithmetic function** can be expressed in terms of the characteristic function $\chi_E: S \to \set {0, 1}$ where $E = \set 1$:

- $\forall n \in S: \map \iota n = \map {\chi_{\set 1} } n$

but strictly speaking $\iota$ does not equal $\chi_{\set 1}$ because the codomains are different:

- $\Cdm \iota = \Z$
- $\Cdm {\chi_{\set 1} } = \set {0, 1}$

## Note on Name

The name of this function can be confusing. It is clearly not an identity function as such.

In fact it is a function which returns an answer to the question:

- Is $n$ equal to the identity element for multiplication? If so, return $1$, otherwise return $0$.