Extended natural numbers explained
In mathematics, the extended natural numbers is a set which contains the values
and
(infinity). That is, it is the result of adding a maximum element
to the
natural numbers. Addition and multiplication work as normal for finite values, and are extended by the rules
(
),
and
m x infty=infty x m=infty
for
.
With addition and multiplication,
is a
semiring but not a
ring, as
lacks an
additive inverse. The set can be denoted by
},
or
. It is a subset of the
extended real number line, which extends the
real numbers by adding
and
.
Applications
In graph theory, the extended natural numbers are used to define distances in graphs, with
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
are a one-point compactification of the natural numbers, yielding the set of non-increasing binary sequences i.e.
such that
. The sequence
represents
, while the sequence
represents
. It is a
retract of
and the claim that
N\cup\{infty\}\subseteqNinfty
implies the
limited principle of omniscience.
References
Further reading
- Robert . Leonel . 0711.4396 . The Cuntz semigroup of some spaces of dimension at most two . 3 September 2013 .
- The American Mathematical Monthly. 79. 3. 1972. Lightstone. A. H.. Infinitesimals.
- Southeast Asian Bulletin of Mathematics. On Projection Algebras. 43. 2. 2019. Khanjanzadeh. Zeinab. Madanshekaf. Ali.
