A real of \(\mathbb{Q}(\sqrt D)\): \(a+b\sqrt D\). More...

Public Member Functions
constexpr	QuadraticReal ()=default
	\(0\).
constexpr	QuadraticReal (Q a)
	A rational is a real of the field: \(a+0\sqrt D\).
template<typename V> requires (!std::same_as<V, Q> && std::convertible_to<V, Q>)
constexpr	QuadraticReal (V v)
	Single-step lift `V` `→` `Q` `→` field (e.g. an `int`).
constexpr Q	rational_part () const
constexpr Q	radical_part () const
constexpr QuadraticReal	infimum () const
	Extrema of the singleton \(\{*this\}\) (inf = sup = the value). Supplies `HasExtrema`, the structural prerequisite for `IsDedekindComplete`.
constexpr QuadraticReal	supremum () const

Static Public Member Functions
static constexpr QuadraticReal	root ()
	The generator \(\sqrt D\) ( \(0+1\sqrt D\)).
static constexpr QuadraticReal	of (Q a, Q b)
	\(a+b\sqrt D\) from its parts.

Friends
constexpr std::strong_ordering	operator<=> (const QuadraticReal &x, const QuadraticReal &y)
	Total, decidable order: sign of \((a-a')+(b-b')\sqrt D\) by comparing squares. This is what keeps \(\mathbb{Q}(\sqrt D)\) a genuine `IsTotallyOrdered` field rather than a Ternary residue.
constexpr bool	operator== (const QuadraticReal &x, const QuadraticReal &y)

Field operations (closed, exact) @{
constexpr QuadraticReal	operator- () const
constexpr QuadraticReal	inverse () const
	\(1/(a+b\sqrt D)\) via the conjugate; the norm \(a^2-b^2D\) is nonzero for every nonzero element ( \(\sqrt D\) irrational).
constexpr QuadraticReal	operator+ (const QuadraticReal &x, const QuadraticReal &y)
constexpr QuadraticReal	operator- (const QuadraticReal &x, const QuadraticReal &y)
constexpr QuadraticReal	operator* (const QuadraticReal &x, const QuadraticReal &y)
constexpr QuadraticReal	operator/ (const QuadraticReal &x, const QuadraticReal &y)

Detailed Description

template<long D, typename Q = Rational<default_integer>>
requires (D > 1) && IsRational<Q>
class dedekind::numbers::QuadraticReal< D, Q >

A real of \(\mathbb{Q}(\sqrt D)\): \(a+b\sqrt D\).

Template Parameters

D	the (non-square, positive) radicand, carried in the type so \(\mathbb{Q}(\sqrt2)\) and \(\mathbb{Q}(\sqrt3)\) are distinct fields.
Q	the rational coefficient carrier.

The documentation for this class was generated from the following file:

Public Types