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complex conjugation
Redirected from "star-conjugation".
Contents
Context
Algebra
algebra , higher algebra
universal algebra
monoid , semigroup , quasigroup
nonassociative algebra
associative unital algebra
commutative algebra
Lie algebra , Jordan algebra
Leibniz algebra , pre-Lie algebra
Poisson algebra , Frobenius algebra
lattice , frame , quantale
Boolean ring , Heyting algebra
commutator , center
monad , comonad
distributive law
Group theory
Ring theory
Module theory
Complex geometry
Contents
Idea
Complex conjugation is the operation on complex numbers which reverses the sign of the imaginary part , hence the function
ℂ ⟶ ( − ) * ℂ a + i b ↦ a − i b AAAAAA for a , b ∈ ℝ .
\array{
\mathbb{C}
&
\overset{
\;\;\;
(-)^\ast
\;\;\;
}{
\longrightarrow
}
&
\mathbb{C}
\\
a + \mathrm{i} b &\mapsto& a - \mathrm{i} b
}
\phantom{AAAAAA}
for\;\; a,b \in \mathbb{R}
\,.
More generally, the anti-involution on any star-algebra may be referred to as conjugation . For instance one speaks of quaternionic conjugation for the analogous operation on quaternions :
ℍ ⟶ ( − ) * ℍ a + i b + j c + k d ↦ a − i b − j c − k d AAAAAA for a , b , c , d ∈ ℝ .
\array{
\mathbb{H}
&
\overset{
\;\;\;
(-)^\ast
\;\;\;
}{
\longrightarrow
}
&
\mathbb{H}
\\
a
+ \mathrm{i} b
+ \mathrm{j} c
+ \mathrm{k} d
&\mapsto&
a
- \mathrm{i} b
- \mathrm{j} c
- \mathrm{k} d
}
\phantom{AAAAAA}
for\;\; a, b, c, d \in \mathbb{R}
\,.
For an unrelated (or vaguely related) notion with a similar name see at conjugacy class and adjoint action .
Last revised on August 21, 2024 at 02:05:06.
See the history of this page for a list of all contributions to it.