Dedekind 0.0.0
A structuralist C++23 Module Library
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complex.cppm File Reference
module dedekind.numbers

Minimal complex wrapper for experimental reintegration. More...

#include <cmath>
#include <concepts>
#include <functional>
#include <limits>
#include <type_traits>
#include <utility>
import dedekind.numbers:quadratic;
import dedekind.sets;
import dedekind.geometry;
import dedekind.category;
import dedekind.morphologies;
import dedekind.algebra;
Include dependency graph for complex.cppm:

Classes

class  dedekind::numbers::Complex< R >
struct  dedekind::numbers::PartialAddComplex< R >
 Partial addition transform for Complex<R>. More...
struct  dedekind::numbers::PartialMulComplex< R >
 Partial multiplication transform for Complex<R>. More...
struct  dedekind::category::is_exact_total< dedekind::numbers::Complex< R >, std::plus< dedekind::numbers::Complex< R > > >
struct  dedekind::category::is_exact_total< dedekind::numbers::Complex< R >, std::multiplies< dedekind::numbers::Complex< R > > >
struct  dedekind::category::is_associative< dedekind::numbers::Complex< R >, std::plus< dedekind::numbers::Complex< R > > >
struct  dedekind::category::is_commutative< dedekind::numbers::Complex< R >, std::plus< dedekind::numbers::Complex< R > > >
struct  dedekind::category::inverse_trait< dedekind::numbers::Complex< R >, std::multiplies< dedekind::numbers::Complex< R > > >
struct  dedekind::numbers::UnitRoot< N >
 A SYMBOLIC N-th root of unity \(\zeta_N^{\,p}\), carried by its exponent p ∈ ℤ/N (Modular<N>). The unit harmonics \(\mu_N=\{z:z^N=1\}\): the torsion subgroup of the unit circle U(1), cut out of ℂ, cyclic of order N and \(\cong\mathbb{Z}/N\) via de Moivre. More...
struct  dedekind::numbers::Character< N >
 The de Moivre CHARACTER \(\chi_m : \mathbb{Z}/N \to \mu_N\), \(k \mapsto \zeta_N^{\,mk}\) — the plane wave "as such", carried SYMBOLICALLY in UnitRoot<N> (consumes the μ_N layer; resolve() collapses to exact ℂ at N=8). More...
struct  dedekind::numbers::DiscreteCos< N >
 The exact discrete COSINE \(\cos_N(k) = \cos(2\pi k/N) = \Re\,\zeta_N^k\) over the Modular<N> angle — the EVEN part of the de Moivre character (= the \(\tfrac12(I+U)\) projection's real value). More...
struct  dedekind::numbers::DiscreteSin< N >
 The exact discrete SINE \(\sin_N(k) = \sin(2\pi k/N) = \Im\,\zeta_N^k\) — the ODD part of the character (Im of character<N>(1) resolved). An IsArrow Modular<N> → ℝ, exact at N=8. More...
struct  dedekind::category::is_exact_total< dedekind::numbers::UnitRoot< N >, std::multiplies< dedekind::numbers::UnitRoot< N > > >
struct  dedekind::category::is_associative< dedekind::numbers::UnitRoot< N >, std::multiplies< dedekind::numbers::UnitRoot< N > > >
struct  dedekind::category::is_commutative< dedekind::numbers::UnitRoot< N >, std::multiplies< dedekind::numbers::UnitRoot< N > > >
struct  dedekind::category::unit_trait< dedekind::numbers::UnitRoot< N >, std::multiplies< dedekind::numbers::UnitRoot< N > > >
struct  dedekind::category::inverse_trait< dedekind::numbers::UnitRoot< N >, std::multiplies< dedekind::numbers::UnitRoot< N > > >
struct  dedekind::category::is_cyclic_group< dedekind::numbers::UnitRoot< N >, std::multiplies< dedekind::numbers::UnitRoot< N > > >
struct  dedekind::category::cyclic_order< dedekind::numbers::UnitRoot< N >, std::multiplies< dedekind::numbers::UnitRoot< N > > >
struct  dedekind::category::SpeciesTraits< dedekind::numbers::Complex< R > >
struct  dedekind::category::is_homomorphism< std::decay_t< decltype(dedekind::numbers::embed_ℝ_ℂ)>, dedekind::numbers::QuadraticReal< 2 >, std::plus< dedekind::numbers::QuadraticReal< 2 > >, std::plus< dedekind::numbers::Complex< dedekind::numbers::QuadraticReal< 2 > > > >
struct  dedekind::category::is_homomorphism< std::decay_t< decltype(dedekind::numbers::embed_ℝ_ℂ)>, dedekind::numbers::QuadraticReal< 2 >, std::multiplies< dedekind::numbers::QuadraticReal< 2 > >, std::multiplies< dedekind::numbers::Complex< dedekind::numbers::QuadraticReal< 2 > > > >
struct  dedekind::category::quotient_algebra_base< dedekind::numbers::Complex< R > >
struct  dedekind::category::is_linear_action< dedekind::numbers::Complex< R >, R, std::plus< dedekind::numbers::Complex< R > >, std::plus< R >, std::multiplies< R >, std::multiplies<> >
struct  dedekind::category::unit_trait< dedekind::numbers::Complex< R >, std::plus< dedekind::numbers::Complex< R > > >
struct  dedekind::category::unit_trait< dedekind::numbers::Complex< R >, std::multiplies< dedekind::numbers::Complex< R > > >
struct  dedekind::category::inverse_trait< dedekind::numbers::Complex< R >, std::plus< dedekind::numbers::Complex< R > > >

Namespaces

namespace  dedekind::numbers
namespace  dedekind::category

Concepts

concept  dedekind::numbers::IsComplexScalar

Functions

template<IsComplexScalar R>
constexpr Complex< R > dedekind::numbers::conj (const Complex< R > &z)
 Complex conjugate: conj(a + bi) = a − bi.
template<IsComplexScalar R>
constexpr R dedekind::numbers::dot (const Complex< R > &z, const Complex< R > &w)
 The real inner product on ℂ ≅ ℝ²: dot(z, w) = Re(z · conj w) = re_z·re_w + im_z·im_w.
template<IsComplexScalar R>
constexpr R dedekind::numbers::abs2 (const Complex< R > &z)
 Squared norm abs2(z) = <z, z> = |z|² = re² + im²: exact, √-free.
template<IsComplexScalar R>
constexpr R dedekind::numbers::euclidean_norm_squared (const Complex< R > &z)
 Squared Euclidean norm on Complex<R>: a synonym for abs2, kept so it stays reachable by ADL in dedekind::numbers for callers that do not import dedekind.geometry (e.g. the Mandelbrot escape test and the set-algebra predicates). Kept squared to avoid a square root in hot paths.
template<unsigned N>
constexpr UnitRoot< N > dedekind::numbers::unit_root ()
 The primitive N-th root \(\zeta_N=\zeta_N^{1}\) (the μ_N generator).
template<unsigned N>
constexpr Character< N > dedekind::numbers::character (dedekind::morphologies::Modular< N > m)
 The character \(\chi_m\) at frequency m.
template<unsigned N>
requires (N == 8u)
constexpr DiscreteCos< N > dedekind::numbers::cos_n ()
 \(\cos_N = \Re\,\zeta_N^{(\cdot)}\) as an arrow.
template<unsigned N>
requires (N == 8u)
constexpr DiscreteSin< N > dedekind::numbers::sin_n ()
 \(\sin_N = \Im\,\zeta_N^{(\cdot)}\) as an arrow.

Variables

template<typename R>
constexpr bool dedekind::category::is_kleene_commutative_v< dedekind::numbers::Complex< R >, dedekind::numbers::PartialAddComplex< R > >
 Kleene traits for complex arithmetic.
template<typename R>
constexpr dedekind::numbers::Complex< R > dedekind::category::partial_unit_v< dedekind::numbers::Complex< R >, dedekind::numbers::PartialAddComplex< R > >
template<typename R>
constexpr bool dedekind::category::is_kleene_commutative_v< dedekind::numbers::Complex< R >, dedekind::numbers::PartialMulComplex< R > >
template<typename R>
constexpr dedekind::numbers::Complex< R > dedekind::category::partial_unit_v< dedekind::numbers::Complex< R >, dedekind::numbers::PartialMulComplex< R > >
template<typename R>
constexpr bool dedekind::category::is_associative_v< dedekind::numbers::Complex< R >, std::multiplies< dedekind::numbers::Complex< R > > >
template<typename R>
constexpr bool dedekind::category::is_commutative_v< dedekind::numbers::Complex< R >, std::multiplies< dedekind::numbers::Complex< R > > >
template<typename R>
constexpr bool dedekind::category::is_distributive_v< dedekind::numbers::Complex< R >, std::multiplies< dedekind::numbers::Complex< R > >, std::plus< dedekind::numbers::Complex< R > > >
constexpr auto dedekind::numbers::embed_ℝ_ℂ
 The Birkhoff S leg \(\mathbb{R}\hookrightarrow\mathbb{C}\) over the coat-hanger: QuadraticReal<2> → Complex<QuadraticReal<2>>, \(r\mapsto (r,0)\).
constexpr auto dedekind::numbers::root8
 The exact de Moivre exponential \(\zeta_8 : \mathbb{Z}/8 \to \mathbb{C}^\times\) over the coat-hanger ℝ = ℚ(√2): \(k \mapsto \zeta_8^{\,k} = (\cos\tfrac{2\pi k}{8},\ \sin\tfrac{2\pi k}{8})\).
constexpr Complex< QuadraticReal< 2 > > dedekind::numbers::ζ8
 The primitive 8th root of unity ζ8 = e^{2πi/8} = ½√2 + ½√2·i, exact in ℂ(ℚ(√2)) — the generator of the de Moivre map root8.
template<unsigned N>
constexpr bool dedekind::category::is_invertible_v< dedekind::numbers::UnitRoot< N >, std::multiplies< dedekind::numbers::UnitRoot< N > > > = true
constexpr QuadraticReal< 2 > dedekind::numbers::half_root2
constexpr auto dedekind::numbers::ℂ
 The canonical complex-number universe ℂ = 𝔸<Complex<QuadraticReal<2>>, Boole, ℶ_1> — the coat-hanger ℂ = Cplx(ℝ) over the genuine ℝ = ℚ(√2) (mirroring ℝ and 𝔻). Equality on ℚ(√2)[i] is decidable, so ℂ answers in Boole; ℶ_1 is the cardinality of the ℂ it models, not a logic.
template<>
constexpr bool dedekind::category::is_monic_arrow_v< std::decay_t< decltype(dedekind::numbers::embed_ℝ_ℂ)> >
template<dedekind::numbers::IsComplexScalar R>
constexpr bool dedekind::category::is_invertible_v< dedekind::numbers::Complex< R >, std::plus< dedekind::numbers::Complex< R > > >

Detailed Description

Minimal complex wrapper for experimental reintegration.

@partition :complex

Copyright
2026 The Dedekind Authors Licensed under the Apache License, Version 2.0.
Note
"Les mathematiciens n'etudient pas des objets, mais des relations entre des objets." ("Mathematicians do not study objects, but the relations between objects.") – Henri Poincare, La Science et l'hypothese (1902)