Extended natural numbers

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

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

Applications

[edit ]

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