In mathematics, the extended natural numbers is a set which contains the values ${\displaystyle 0,1,2,\dots }$ and ${\displaystyle \infty }$ (infinity). That is, it is the result of adding a maximum element ${\displaystyle \infty }$ to the natural numbers. Addition and multiplication work as normal for finite values, and are extended by the rules ${\displaystyle n+\infty =\infty +n=\infty }$ (${\displaystyle n\in \mathbb {N} \cup \{\infty \}}$), ${\displaystyle 0\times \infty =\infty \times 0=0}$ and ${\displaystyle m\times \infty =\infty \times m=\infty }$ for ${\displaystyle m\neq 0}$.

With addition and multiplication, ${\displaystyle \mathbb {N} \cup \{\infty \}}$ is a semiring but not a ring, as ${\displaystyle \infty }$ lacks an additive inverse. ^[1] The set can be denoted by ${\displaystyle {\overline {\mathbb {N} }}}$, ${\displaystyle \mathbb {N} _{\infty }}$ or ${\displaystyle \mathbb {N} ^{\infty }}$. ^{[2] [3] [4]} It is a subset of the extended real number line, which extends the real numbers by adding ${\displaystyle -\infty }$ and ${\displaystyle +\infty }$. ^[2]

In graph theory, the extended natural numbers are used to define distances in graphs, with ${\displaystyle \infty }$ being the distance between two unconnected vertices. ^[2] They can be used to show the extension of some results, such as the max-flow min-cut theorem, to infinite graphs. ^[5]

In topology, the topos of right actions on the extended natural numbers is a category PRO of projection algebras. ^[4]

In constructive mathematics, the extended natural numbers ${\displaystyle \mathbb {N} _{\infty }}$ are a one-point compactification of the natural numbers, yielding the set of non-increasing binary sequences i.e. ${\displaystyle (x_{0},x_{1},\dots )\in 2^{\mathbb {N} }}$ such that ${\displaystyle \forall i\in \mathbb {N} :x_{i}\geq x_{i+1}}$. The sequence ${\displaystyle 1^{n}0^{\omega }}$ represents ${\displaystyle n}$, while the sequence ${\displaystyle 1^{\omega }}$ represents ${\displaystyle \infty }$. It is a retract of ${\displaystyle 2^{\mathbb {N} }}$ and the claim that ${\displaystyle \mathbb {N} \cup \{\infty \}\subseteq \mathbb {N} _{\infty }}$ implies the limited principle of omniscience. ^[3]

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• Extended natural number at the nLab