Extended natural numbers

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.^[1] The set can be denoted by ${\overline {\mathbb {N} }}$ , $\mathbb {N} _{\infty }$ or $\mathbb {N} ^{\infty }$ .^[2]^[3]^[4] It is a subset of the extended real number line, which extends the real numbers by adding $-\infty$ and $+\infty$ .^[2]

Applications

In graph theory, the extended natural numbers are used to define distances in graphs, with $\infty$ 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 $\mathbb {N} _{\infty }$ 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.^[3]