Extended natural numbers

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

With addition and multiplication, N ∪ { ∞ } {\displaystyle \mathbb {N} \cup \{\infty \}} {\displaystyle \mathbb {N} \cup \{\infty \}} is a semiring but not a ring, as ∞ {\displaystyle \infty } {\displaystyle \infty } lacks an additive inverse.[1] The set can be denoted by N ¯ {\displaystyle {\overline {\mathbb {N} }}} {\displaystyle {\overline {\mathbb {N} }}}, N ∞ {\displaystyle \mathbb {N} _{\infty }} {\displaystyle \mathbb {N} _{\infty }} or N ∞ {\displaystyle \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 − ∞ {\displaystyle -\infty } {\displaystyle -\infty } and + ∞ {\displaystyle +\infty } {\displaystyle +\infty }.[2]

Applications

In graph theory, the extended natural numbers are used to define distances in graphs, with ∞ {\displaystyle \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 N ∞ {\displaystyle \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 , … ) ∈ 2 N {\displaystyle (x_{0},x_{1},\dots )\in 2^{\mathbb {N} }} {\displaystyle (x_{0},x_{1},\dots )\in 2^{\mathbb {N} }} such that ∀ i ∈ N : x i ≥ x i + 1 {\displaystyle \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 ω {\displaystyle 1^{n}0^{\omega }} {\displaystyle 1^{n}0^{\omega }} represents n {\displaystyle n} {\displaystyle n}, while the sequence 1 ω {\displaystyle 1^{\omega }} {\displaystyle 1^{\omega }} represents ∞ {\displaystyle \infty } {\displaystyle \infty }. It is a retract of 2 N {\displaystyle 2^{\mathbb {N} }} {\displaystyle 2^{\mathbb {N} }} and the claim that N ∪ { ∞ } ⊆ N ∞ {\displaystyle \mathbb {N} \cup \{\infty \}\subseteq \mathbb {N} _{\infty }} {\displaystyle \mathbb {N} \cup \{\infty \}\subseteq \mathbb {N} _{\infty }} implies the limited principle of omniscience.[3]

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Further reading