The Argand plane or complex line is the space made from considering all points that can be labelled with one complex number; this gives it a complex dimension of 1. Comtela exist as regular subsets of the complex line.
Embedding in the Euclidean plane [ ] The complex line can be embedded in the Euclidean plane by considering each complex coordinate to correspond to two real coordinates; the typical way to do this is
x
(
z
)
=
Re
z
{\displaystyle x(z) = \operatorname{Re} z}
y
(
z
)
=
Im
z
{\displaystyle y(z) = \operatorname{Im} z}
The inverse of this is simply
z
(
x
,
y
)
=
x
+
i
y
{\displaystyle z(x,y) = x + iy}
This embedding means that the complex line is often called the complex plane , though strictly that refers to the space made from pairs of complex numbers.
See Also [ ]
Real dimensionality
0
1
2
3
...
Real space
R
n
{\displaystyle \R^n}
Point
R
0
{\displaystyle \mathbb R^{0}}
Real line
R
1
{\displaystyle \mathbb R^{1}}
Real plane
R
2
{\displaystyle \mathbb R^{2}}
Real realm
R
3
{\displaystyle \mathbb R^{3}}
...
Real projective space
R
P
n
{\displaystyle \mathbb {R}\mathbb{P}^n}
Point pair
R
P
0
{\displaystyle \mathbb {R}\mathbb{P}^0}
Real projective line
R
P
1
{\displaystyle \mathbb {R}\mathbb{P}^1}
Real projective plane
R
P
2
{\displaystyle \mathbb {R}\mathbb{P}^2}
Real projective realm
R
P
3
{\displaystyle \mathbb {R}\mathbb{P}^3}
...
Complex space
C
n
{\displaystyle \C^n}
Point
C
0
{\displaystyle \mathbb {C}^0}
Complex line
C
1
{\displaystyle \mathbb {C}^1}
Complex plane
C
2
{\displaystyle \mathbb {C}^2}
Complex realm
C
3
{\displaystyle \mathbb{C}^3}
...
Complex projective space
C
P
n
{\displaystyle \mathbb {C}\mathbb{P}^n}
Point pair
C
P
0
{\displaystyle \mathbb {C}\mathbb{P}^0}
Complex projective line
C
P
1
{\displaystyle \mathbb {C}\mathbb{P}^1}
Complex projective plane
C
P
2
{\displaystyle \mathbb {C}\mathbb{P}^2}
Complex projective realm
C
P
3
{\displaystyle \mathbb {C}\mathbb{P}^3}
...
Quaternionic space
H
n
{\displaystyle \mathbb H^{n}}
Point
H
0
{\displaystyle \mathbb {H}^0}
Quaternionic line
H
1
{\displaystyle \mathbb {H}^1}
Quaternionic plane
H
2
{\displaystyle \mathbb H^{2}}
Quaternionic realm
H
3
{\displaystyle \mathbb H^{3}}
...
Quaternionic projective space
H
P
n
{\displaystyle \mathbb {H}\mathbb{P}^n}
Point pair
H
P
0
{\displaystyle \mathbb {H}\mathbb{P}^0}
Quaternionic projective line
H
P
1
{\displaystyle \mathbb {H}\mathbb{P}^1}
Quaternionic projective plane
H
P
2
{\displaystyle \mathbb {H}\mathbb{P}^2}
Quaternionic projective realm
H
P
3
{\displaystyle \mathbb {H}\mathbb{P}^3}
...
Octonionic space
O
n
{\displaystyle \mathbb {O}^n}
Point
O
0
{\displaystyle \mathbb {O}^0}
Octonionic line
O
1
{\displaystyle \mathbb {O}^1}
Octonionic plane
O
2
{\displaystyle \mathbb {O}^2}
Octonionic realm
O
3
{\displaystyle \mathbb {O}^3}
...
Octonionic projective space
O
P
n
{\displaystyle \mathbb {O}\mathbb{P}^n}
Point pair
O
P
0
{\displaystyle \mathbb {O}\mathbb{P}^0}
Octonionic projective line
O
P
1
{\displaystyle \mathbb {O}\mathbb{P}^1}
Octonionic projective plane
O
P
2
{\displaystyle \mathbb {O}\mathbb{P}^2}
Octonionic projective realm
O
P
3
{\displaystyle \mathbb {O}\mathbb{P}^3}
...
...
...
...
...
...
...
