Class Reals

java.lang.Object

org.episteme.core.mathematics.sets.Reals

All Implemented Interfaces:

AbelianGroup<Real>, AbelianMonoid<Real>, Group<Real>, Magma<Real>, Monoid<Real>, Field<Real>, Ring<Real>, Semiring<Real>, InfiniteSet<Real>, Set<Real>

public final class Reals
extends Object
implements Field<Real>, InfiniteSet<Real>

The structure of real numbers (ℝ).

This class represents the structure of real numbers, not individual elements. It implements Field because ℝ forms a field under addition and multiplication.

Real numbers are an infinite, uncountable set.

Usage

Reals structure = Reals.getInstance();
Real pi = Real.of(3.14159);
Real two = Real.of(2.0);

// Use structure for operations
Real twoPi = structure.multiply(pi, two);
Real halfPi = structure.divide(pi, two);

Since:

1.0

Author:

Gemini AI (Google DeepMind)

See Also:

• * @author Silvere Martin-Michiellot

Field Summary

Fields

Modifier and Type	Field	Description
static final Reals	INSTANCE	Singleton instance

Method Summary

Modifier and Type	Method	Description
Real	add(Real a, Real b)	Returns the sum of two elements (additive notation).
int	characteristic()	Returns the characteristic of this field.
boolean	contains(Real element)	Tests whether this set contains the specified element.
String	description()	Returns a human-readable description of this set.
static Reals	getInstance()	Returns the singleton instance.
Real	inverse(Real element)	Returns the multiplicative inverse of a non-zero element.
boolean	isCountable()	Returns true if this set is countable (i.e., its elements can be put in one-to-one correspondence with the natural numbers).
boolean	isEmpty()	Returns true if this set contains no elements.
boolean	isMultiplicationCommutative()	Fields always have commutative multiplication.
Real	multiply(Real a, Real b)	Returns the product of two elements.
Real	negate(Real element)	Returns the additive inverse (negation) of an element.
Real	one()	Returns the multiplicative identity (one element).
Real	operate(Real a, Real b)	Performs the binary operation on two elements.
String	toString()
Real	zero()	Returns the additive identity (zero element).

Methods inherited from class Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, wait, wait, wait

Methods inherited from interface AbelianGroup

isCommutative, subtract

Methods inherited from interface AbelianMonoid

identity, isCommutative

Methods inherited from interface Field

divide, hasUnity

Methods inherited from interface Monoid

isAssociative

Methods inherited from interface Semiring

pow

Field Details

INSTANCE

public static final Reals INSTANCE

Singleton instance

Method Details

getInstance

public static Reals getInstance()

Returns the singleton instance.

Returns:

the Reals structure

operate

public Real operate(Real a,
 Real b)

Description copied from interface: Magma

Performs the binary operation on two elements.

This is the fundamental operation of a magma. The result must be an element of this magma (closure property).

Properties: None required (not necessarily associative or commutative)

Specified by:

operate in interface Magma<Real>

Parameters:

a - the first operand

b - the second operand

Returns:

the result of a ∗ b

See Also:

• Magma.isAssociative()
• Magma.isCommutative()

add

public Real add(Real a,
 Real b)

Description copied from interface: AbelianMonoid

Returns the sum of two elements (additive notation).

Delegates to Magma.operate(Object, Object).

Specified by:

add in interface AbelianMonoid<Real>

Parameters:

a - the first addend

b - the second addend

Returns:

a + b

characteristic

public int characteristic()

Description copied from interface: Field

Returns the characteristic of this field.

The characteristic is the smallest positive integer n such that:
1 + 1 + ... + 1 (n times) = 0
If no such n exists, the characteristic is 0.

Examples:

• char(â„š) = char(ℝ) = char(â„‚) = 0
• char(𝔽ₚ) = p (for prime p)

Specified by:

characteristic in interface Field<Real>

Returns:

the characteristic (0 for infinite fields, p for finite fields)

zero

public Real zero()

Description copied from interface: AbelianMonoid

Returns the additive identity (zero element).

Delegates to AbelianMonoid.identity().

Specified by:

zero in interface AbelianMonoid<Real>

Returns:

the zero element

negate

public Real negate(Real element)

Description copied from interface: AbelianGroup

Returns the additive inverse (negation) of an element.

Satisfies: a + (-a) = (-a) + a = 0

Specified by:

negate in interface AbelianGroup<Real>

Parameters:

element - the element to negate

Returns:

-element

See Also:

• Group.inverse(Object)

multiply

public Real multiply(Real a,
 Real b)

Description copied from interface: Semiring

Returns the product of two elements.

Multiplication must be associative and distribute over addition.

Specified by:

multiply in interface Semiring<Real>

Parameters:

a - the first factor

b - the second factor

Returns:

a × b

one

public Real one()

Description copied from interface: Semiring

Returns the multiplicative identity (one element).

Satisfies: 1 × a = a × 1 = a for all elements a.

Specified by:

one in interface Semiring<Real>

Returns:

the multiplicative identity

isMultiplicationCommutative

public boolean isMultiplicationCommutative()

Description copied from interface: Field

Fields always have commutative multiplication.

Specified by:

isMultiplicationCommutative in interface Field<Real>

Specified by:

isMultiplicationCommutative in interface Semiring<Real>

Returns:

always true

inverse

public Real inverse(Real element)

Description copied from interface: Field

Returns the multiplicative inverse of a non-zero element.

For element a ≠0, returns a⁻¹ such that: a × a⁻¹ = a⁻¹ × a = 1

Examples:

• In â„š: inverse(2/3) = 3/2
• In ℝ: inverse(5.0) = 0.2
• In â„‚: inverse(3+4i) = (3-4i)/25

Specified by:

inverse in interface Field<Real>

Specified by:

inverse in interface Group<Real>

Parameters:

element - the element to invert (must be non-zero)

Returns:

the multiplicative inverse

See Also:

• Field.divide(Object, Object)
• Semiring.one()

isCountable

public boolean isCountable()

Description copied from interface: InfiniteSet

Returns true if this set is countable (i.e., its elements can be put in one-to-one correspondence with the natural numbers).

Examples:

• Countable: â„• (Natural numbers), ℤ (Integers), â„š (Rational numbers)
• Uncountable: ℝ (Real numbers), â„‚ (Complex numbers)

Specified by:

isCountable in interface InfiniteSet<Real>

Returns:

true if this set is countable

contains

public boolean contains(Real element)

Description copied from interface: Set

Tests whether this set contains the specified element.

This is the fundamental operation of a set - membership testing.

Specified by:

contains in interface Set<Real>

Parameters:

element - the element to test for membership

Returns:

true if this set contains the element, false otherwise

See Also:

• Set.isEmpty()
• invalid reference

  #size()

isEmpty

public boolean isEmpty()

Description copied from interface: Set

Returns true if this set contains no elements.

The empty set (∅) is a fundamental concept in set theory. It is the unique set containing no elements.

Specified by:

isEmpty in interface Set<Real>

Returns:

true if this set is empty

description

public String description()

Description copied from interface: Set

Returns a human-readable description of this set.

Examples:

• "ℝ (Real Numbers)"
• "ℤ/12ℤ (Integers modulo 12)"
• "{1, 2, 3, 4, 5}"

Specified by:

description in interface Set<Real>

Returns:

a description of this set

toString

public String toString()

Overrides:

toString in class Object