Class Rational

java.lang.Object

java.lang.Number

org.episteme.core.mathematics.numbers.rationals.Rational

All Implemented Interfaces:

Serializable, Comparable<Rational>, AbelianGroup<Rational>, AbelianMonoid<Rational>, Group<Rational>, Magma<Rational>, Monoid<Rational>, Field<Rational>, FieldElement<Rational>, Ring<Rational>, RingElement<Rational>, Semiring<Rational>, Set<Rational>

public final class Rational
extends Number
implements Comparable<Rational>, Field<Rational>, FieldElement<Rational>

Represents a rational number (â„š), defined as the quotient of two integers.

Rational numbers form a Field under addition and multiplication. They are always stored in reduced form (gcd(numerator, denominator) = 1) with a positive denominator.

* @author Silvere Martin-Michiellot

Since:

1.0

Author:

Gemini AI (Google DeepMind)

See Also:

• Serialized Form

Field Summary

Fields

Modifier and Type	Field	Description
static final Rational	ONE
static final Rational	ZERO

Method Summary

Modifier and Type	Method	Description
Rational	abs()
Rational	add(Rational other)	Returns the sum of this element and another.
Rational	add(Rational a, Rational b)	Returns the sum of two elements (additive notation).
Integer	ceil()
int	characteristic()	Returns the characteristic of this field.
int	compareTo(Rational other)
boolean	contains(Rational element)	Tests whether this set contains the specified element.
String	description()	Returns a human-readable description of this set.
Rational	divide(Rational other)	Returns this element divided by another.
Rational	divide(Rational a, Rational b)	Returns the quotient of two elements (division).
double	doubleValue()
boolean	equals(Object obj)
float	floatValue()
Integer	floor()
Rational	fractionalPart()
Integer	getDenominator()
Integer	getNumerator()
int	hashCode()
int	intValue()
Rational	inverse()	Returns the multiplicative inverse of this element.
Rational	inverse(Rational a)	Returns the multiplicative inverse of a non-zero element.
boolean	isEmpty()	Returns true if this set contains no elements.
boolean	isMultiplicationCommutative()	Fields always have commutative multiplication.
long	longValue()
Rational	mod(Rational m)
Rational	multiply(Rational other)	Returns the product of this element and another.
Rational	multiply(Rational a, Rational b)	Returns the product of two elements.
Rational	negate()	Returns the additive inverse of this element.
Rational	negate(Rational a)	Returns the additive inverse (negation) of an element.
static Rational	of(int value)
static Rational	of(int numerator, int denominator)
static Rational	of(long value)
static Rational	of(long numerator, long denominator)
static Rational	of(Integer numerator, Integer denominator)
Rational	one()	Returns the multiplicative identity (one element).
Rational	operate(Rational a, Rational b)	Performs the binary operation on two elements.
Rational	subtract(Rational other)	Returns the difference of this element and another.
Rational	subtract(Rational a, Rational b)	Returns the difference of two elements.
String	toString()
Rational	zero()	Returns the additive identity (zero element).

Methods inherited from class Number

byteValue, shortValue

Methods inherited from class Object

clone, finalize, getClass, notify, notifyAll, wait, wait, wait

Methods inherited from interface AbelianGroup

isCommutative

Methods inherited from interface AbelianMonoid

identity, isCommutative

Methods inherited from interface Field

hasUnity

Methods inherited from interface Monoid

isAssociative

Methods inherited from interface RingElement

isOne, isZero

Methods inherited from interface Semiring

pow

Field Details

ZERO

public static final Rational ZERO

ONE

public static final Rational ONE

Method Details

of

public static Rational of(Integer numerator,
 Integer denominator)

of

public static Rational of(long numerator,
 long denominator)

of

public static Rational of(int numerator,
 int denominator)

of

public static Rational of(long value)

of

public static Rational of(int value)

getNumerator

public Integer getNumerator()

getDenominator

public Integer getDenominator()

operate

public Rational operate(Rational a,
 Rational 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<Rational>

Parameters:

a - the first operand

b - the second operand

Returns:

the result of a ∗ b

See Also:

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

add

public Rational add(Rational a,
 Rational 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<Rational>

Parameters:

a - the first addend

b - the second addend

Returns:

a + b

add

public Rational add(Rational other)

Description copied from interface: RingElement

Returns the sum of this element and another.

Specified by:

add in interface RingElement<Rational>

Parameters:

other - the element to add

Returns:

this + other

zero

public Rational zero()

Description copied from interface: AbelianMonoid

Returns the additive identity (zero element).

Delegates to AbelianMonoid.identity().

Specified by:

zero in interface AbelianMonoid<Rational>

Specified by:

zero in interface RingElement<Rational>

Returns:

the zero element

negate

public Rational negate(Rational a)

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<Rational>

Parameters:

a - the element to negate

Returns:

-element

See Also:

• Group.inverse(Object)

negate

public Rational negate()

Description copied from interface: RingElement

Returns the additive inverse of this element.

Specified by:

negate in interface RingElement<Rational>

Returns:

-this

subtract

public Rational subtract(Rational a,
 Rational b)

Description copied from interface: AbelianGroup

Returns the difference of two elements.

Defined as: a - b = a + (-b)

Specified by:

subtract in interface AbelianGroup<Rational>

Parameters:

a - the minuend

b - the subtrahend

Returns:

a - b

See Also:

• AbelianMonoid.add(Object, Object)
• AbelianGroup.negate(Object)

subtract

public Rational subtract(Rational other)

Description copied from interface: RingElement

Returns the difference of this element and another.

Default implementation: this.add(other.negate())

Specified by:

subtract in interface RingElement<Rational>

Parameters:

other - the element to subtract

Returns:

this - other

multiply

public Rational multiply(Rational a,
 Rational 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<Rational>

Parameters:

a - the first factor

b - the second factor

Returns:

a × b

multiply

public Rational multiply(Rational other)

Description copied from interface: RingElement

Returns the product of this element and another.

Specified by:

multiply in interface RingElement<Rational>

Parameters:

other - the element to multiply

Returns:

this * other

one

public Rational 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 RingElement<Rational>

Specified by:

one in interface Semiring<Rational>

Returns:

the multiplicative identity

isMultiplicationCommutative

public boolean isMultiplicationCommutative()

Description copied from interface: Field

Fields always have commutative multiplication.

Specified by:

isMultiplicationCommutative in interface Field<Rational>

Specified by:

isMultiplicationCommutative in interface Semiring<Rational>

Returns:

always true

inverse

public Rational inverse(Rational a)

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<Rational>

Specified by:

inverse in interface Group<Rational>

Parameters:

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

Returns:

the multiplicative inverse

See Also:

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

inverse

public Rational inverse()

Description copied from interface: FieldElement

Returns the multiplicative inverse of this element.

For any non-zero element e: e.multiply(e.inverse()) = e.one()

Specified by:

inverse in interface FieldElement<Rational>

Returns:

1/this

divide

public Rational divide(Rational a,
 Rational b)

Description copied from interface: Field

Returns the quotient of two elements (division).

Defined as: a ÷ b = a × b⁻¹

Specified by:

divide in interface Field<Rational>

Parameters:

a - the dividend (numerator)

b - the divisor (denominator, must be non-zero)

Returns:

dividend ÷ divisor

See Also:

• Field.inverse(Object)
• Semiring.multiply(Object, Object)

divide

public Rational divide(Rational other)

Description copied from interface: FieldElement

Returns this element divided by another.

Default implementation: this.multiply(other.inverse())

Specified by:

divide in interface FieldElement<Rational>

Parameters:

other - the divisor

Returns:

this / other

compareTo

public int compareTo(Rational other)

Specified by:

compareTo in interface Comparable<Rational>

intValue

public int intValue()

Specified by:

intValue in class Number

longValue

public long longValue()

Specified by:

longValue in class Number

floatValue

public float floatValue()

Specified by:

floatValue in class Number

doubleValue

public double doubleValue()

Specified by:

doubleValue in class Number

toString

public String toString()

Overrides:

toString in class Object

equals

public boolean equals(Object obj)

Overrides:

equals in class Object

hashCode

public int hashCode()

Overrides:

hashCode in class Object

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<Rational>

Returns:

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

contains

public boolean contains(Rational 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<Rational>

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()

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<Rational>

Returns:

a description of this set

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<Rational>

Returns:

true if this set is empty

abs

public Rational abs()

floor

public Integer floor()

ceil

public Integer ceil()

fractionalPart

public Rational fractionalPart()

mod

public Rational mod(Rational m)