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

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

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Applications

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

In topology, the topos of right actions on the extended natural numbers is a category **PRO** of projection algebras.

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