Extended natural numbers

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

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

Applications

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

In topology, the Grothendieck topos of right actions on the extended natural numbers is a category PRO of projection algebras.Template:Sfnp

In constructive mathematics, the extended natural numbers ${\mathbb{N}}_{\mathrm{\infty }}$ 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 $\mathrm{\forall }i\in \mathbb{N}:x_i\ge x_{i+1}$. The sequence $1^n{0}^{\omega }$ represents $n$, while the sequence $1^{\omega }$ represents $\mathrm{\infty }$. It is a retract of $2^{\mathbb{N}}$ and the claim that $\mathbb{N}\cup \{\mathrm{\infty }\}\subseteq {\mathbb{N}}_{\mathrm{\infty }}$ implies the limited principle of omniscience.Template:Sfnp

References

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Further reading

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External links

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