In mathematics , the extended natural numbers is a set which contains the values
0
,
1
,
2
,
… … -->
{\displaystyle 0,1,2,\dots }
and
∞ ∞ -->
{\displaystyle \infty }
(infinity). That is, it is the result of adding a maximum element
∞ ∞ -->
{\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 }
(
n
∈ ∈ -->
N
∪ ∪ -->
{
∞ ∞ -->
}
{\displaystyle n\in \mathbb {N} \cup \{\infty \}}
),
0
× × -->
∞ ∞ -->
=
∞ ∞ -->
× × -->
0
=
0
{\displaystyle 0\times \infty =\infty \times 0=0}
and
m
× × -->
∞ ∞ -->
=
∞ ∞ -->
× × -->
m
=
∞ ∞ -->
{\displaystyle m\times \infty =\infty \times m=\infty }
for
m
≠ ≠ -->
0
{\displaystyle m\neq 0}
.
With addition and multiplication,
N
∪ ∪ -->
{
∞ ∞ -->
}
{\displaystyle \mathbb {N} \cup \{\infty \}}
is a semiring but not a ring , as
∞ ∞ -->
{\displaystyle \infty }
lacks an additive inverse . The set can be denoted by
N
¯ ¯ -->
{\displaystyle {\overline {\mathbb {N} }}}
,
N
∞ ∞ -->
{\displaystyle \mathbb {N} _{\infty }}
or
N
∞ ∞ -->
{\displaystyle \mathbb {N} ^{\infty }}
. It is a subset of the extended real number line , which extends the real numbers by adding
− − -->
∞ ∞ -->
{\displaystyle -\infty }
and
+
∞ ∞ -->
{\displaystyle +\infty }
.
Applications
In graph theory , the extended natural numbers are used to define distances in graphs , with
∞ ∞ -->
{\displaystyle \infty }
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
N
∞ ∞ -->
{\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} }}
such that
∀ ∀ -->
i
∈ ∈ -->
N
:
x
i
≥ ≥ -->
x
i
+
1
{\displaystyle \forall i\in \mathbb {N} :x_{i}\geq x_{i+1}}
. The sequence
1
n
0
ω ω -->
{\displaystyle 1^{n}0^{\omega }}
represents
n
{\displaystyle n}
, while the sequence
1
ω ω -->
{\displaystyle 1^{\omega }}
represents
∞ ∞ -->
{\displaystyle \infty }
. It is a retract of
2
N
{\displaystyle 2^{\mathbb {N} }}
and the claim that
N
∪ ∪ -->
{
∞ ∞ -->
}
⊆ ⊆ -->
N
∞ ∞ -->
{\displaystyle \mathbb {N} \cup \{\infty \}\subseteq \mathbb {N} _{\infty }}
implies the limited principle of omniscience .
Notes
References
Further reading
Robert, Leonel (3 September 2013). "The Cuntz semigroup of some spaces of dimension at most two". arXiv :0711.4396 .
Lightstone, A. H. (1972). "Infinitesimals". The American Mathematical Monthly . 79 (3).
Khanjanzadeh, Zeinab; Madanshekaf, Ali (2019). "On Projection Algebras". Southeast Asian Bulletin of Mathematics . 43 (2).
External links