Trait ComplexField

Source

pub trait ComplexField<R>: Fieldwhere
    R: Field,{
    // Required methods
    fn real(&self) -> R;
    fn imag(&self) -> R;
    fn conjugate(&self) -> Self;
    fn norm_sqr(&self) -> R;
    fn norm(&self) -> R;
    fn arg(&self) -> R;
    fn from_re_im(re: R, im: R) -> Self;
    fn from_polar(r: R, theta: R) -> Self;
    fn i() -> Self;
    fn is_real(&self) -> bool;
    fn is_imaginary(&self) -> bool;
}

Expand description

Represents a Complex Field — a field extension of the reals with complex conjugation and component access.

A complex field is a field where elements can be decomposed into real and imaginary parts, and complex conjugation is defined.

§Mathematical Definition

A complex field K over a real field R satisfies:

K is a Field.
There exists an involution (conjugation) *: K → K such that:
- (a + b)* = a* + b*
- (a · b)* = a* · b*
- (a*)* = a
- a · a* is a non-negative real for all a
Every element z ∈ K can be written as z = x + iy where x, y ∈ R.

§Examples

Complex numbers ℂ over ℝ
Split-complex numbers (hyperbolic numbers)

§Note

Quaternions and Octonions are NOT complex fields because they are not commutative (Quaternions) or not associative (Octonions). They implement DivisionAlgebra instead.