In mathematics, the complex conjugate of a complex number is the number with equal real part and imaginary part equal in magnitude but opposite in sign. For example, the complex conjugate of 3 + 4i is 3 − 4i.
Contents
In polar form, the conjugate of
Complex conjugates are important for finding roots of polynomials. According to the complex conjugate root theorem, if a complex number is a root to a polynomial in one variable with real coefficients (such as the quadratic equation or the cubic equation), so is its conjugate.
Notation
The complex conjugate of a complex number
Properties
The following properties apply for all complex numbers z and w, unless stated otherwise, and can be proven by writing z and w in the form a + ib.
A significant property of the complex conjugate is that a complex number is equal to its complex conjugate if its imaginary part is zero, that is, if the complex number is real.
For any two complex numbers w,z:
The penultimate relation is involution; i.e., the conjugate of the conjugate of a complex number z is z. The ultimate relation is the method of choice to compute the inverse of a complex number if it is given in rectangular coordinates.
If
In general, if
The map
Use as a variable
Once a complex number
Furthermore,
is a line through the origin and perpendicular to
determines the line through
These uses of the conjugate of z as a variable are illustrated in Frank Morley's book Inversive Geometry (1933), written with his son Frank Vigor Morley.
Generalizations
The other planar real algebras, dual numbers, and split-complex numbers are also explicated by use of complex conjugation.
For matrices of complex numbers
Taking the conjugate transpose (or adjoint) of complex matrices generalizes complex conjugation. Even more general is the concept of adjoint operator for operators on (possibly infinite-dimensional) complex Hilbert spaces. All this is subsumed by the *-operations of C*-algebras.
One may also define a conjugation for quaternions and coquaternions: the conjugate of
Note that all these generalizations are multiplicative only if the factors are reversed:
Since the multiplication of planar real algebras is commutative, this reversal is not needed there.
There is also an abstract notion of conjugation for vector spaces
-
ϕ 2 = id V , where ϕ 2 = ϕ ∘ ϕ andid V is the identity map on V , -
ϕ ( z v ) = z ¯ ϕ ( v ) for allv ∈ V , z ∈ C , and -
ϕ ( v 1 + v 2 ) = ϕ ( v 1 ) + ϕ ( v 2 ) for all v 1 ∈ V , v 2 ∈ V ,
is called a complex conjugation, or a real structure. As the involution