Class Q

Represents the set of rational numbers, denoted by ℚ in mathematics.

Inheritance

object

Q

Qb

Qp

Implements

IEquatable<Q>

IComparable<Q>

Inherited Members

object.Equals(object, object)

object.GetType()

object.MemberwiseClone()

object.ReferenceEquals(object, object)

Namespace: MathLib

Assembly: MathLib.dll

Syntax

public class Q : IEquatable<Q>, IComparable<Q>

Remarks

The set of rational numbers is defined as the set of all numbers that can be expressed as the quotient of two integers, where the denominator is non-zero. Formally:

ℚ = {a / b | a ∈ ℤ, b ∈ ℤ, b ≠ 0}

Rational numbers include integers, fractions, and terminating or repeating decimal expansions. Many operations, such as addition, subtraction, multiplication, and division (excluding division by zero) are closed within ℚ.

Rational numbers can be embedded within the set of real numbers ℝ, and the set of p-adic numbers ℚₚ. They serve as the foundation for more complex number systems.

In this library, the class Q represents the mathematical structure of rational numbers, and serves as a base for more advanced number fields like Qp and Cp.

The class methods must never throw exceptions from arithmetic operations. Instead, they return NaN for undefined results (such as from divide by zero).

Constructors

Q()

Initializes a rational number for the default value zero.

Declaration

public Q()

Q(Q)

Protected constructor that creates a copy of a Q object.

Declaration

protected Q(Q q)

Parameters

Type	Name	Description
Q	q

Q(BigInteger)

Initializes a rational number with the specified integer value.

Declaration

public Q(BigInteger numerator)

Parameters

Type	Name	Description
BigInteger	numerator	The integer value.

Q(BigInteger, BigInteger)

Initializes a rational number with the specified numerator and denominator. The rational number is automatically normalized, meaning:

Q will be simplified so the numerator and denominator is coprime.

The sign of the number will always be carried by the numerator (denominator will be strictly positive).

Declaration

public Q(BigInteger numerator, BigInteger denominator)

Parameters

Type	Name	Description
BigInteger	numerator	The numerator of the rational number.
BigInteger	denominator	The denominator of the rational number.

Q(BigInteger, BigInteger, bool)

Protected constructor for a rational number with the specified numerator and denominator.

Declaration

public Q(BigInteger numerator, BigInteger denominator, bool normalize)

Parameters

Type	Name	Description
BigInteger	numerator	The numerator of the rational number.
BigInteger	denominator	The denominator of the rational number.
bool	normalize	Iff true, the rational number will be normalized to its simplest form.

Remarks

Caution: Setting normalize to false will result in a faulty object if any of the following conditions are not met:

• The numerator and denominator are coprime (i.e., their greatest common divisor is 1).
• The denominator is greater than 0.

If unsure if above conditions hold, use the public constructor Q(BigInteger, BigInteger) instead.

If you are sure above conditions hold, you can safely set normalize to false for significantly faster execution.

Properties

Abs

Gets the absolute value of the current rational number.

Declaration

public Q Abs { get; }

Property Value

Type	Description
Q	A Q representing the absolute value of the current rational number. If the rational number is negative, a new Q instance is returned with the absolute value. If the rational number is non-negative, the current instance is returned.

Denominator

The denominator of the rational number, guaranteed to be positive in normalized form.

Declaration

public BigInteger Denominator { get; }

Property Value

Type	Description
BigInteger

FractionalPart

Declaration

public Q FractionalPart { get; }

Property Value

Type	Description
Q	The fractional part of the rational number, obtained by subtracting the integer part.

IntegralPart

The integral (integer) part of a rational number

Declaration

public BigInteger IntegralPart { get; }

Property Value

Type	Description
BigInteger

Remarks

In general mathematics, the concept of "integral part" is not objectively defined, since it has two possible interpretations (or modes). These interpretations are defined by whether we regard the set of all pure fractions to include the number 1 or not. We denote these interpretations (modes) as FEO (Fractions Exclude One) and FIO (Fractions Include One), respectively.

For instance, in decimal, the number 1 could have the following two interpretations:

1 = 1.000000... (FEO)
                                                                                           1 = 0.999999... (FIO)

In FEO the number 1 is a purely integral number, with integral part=1 and fractional part=0.

In FIO the number 1 is a purely fractional number, with integral part=0 and fractional part=1.

Irrespective of the mode, any number is always the sum of its integral and fractional parts.

In FIO, all numbers will have a non-zero-terminating expansion.

In FEO, any number that can terminate (with an ultimately infinite expansion of zeros) will do so.

Only numbers with a period of 0, can have different representations in FIO and FEO.

To make IntegralPart and FractionalPart well defined, yet allow access to both FIO and FEO, we assign FEO to all positive numbers, and FIO to all negative numbers and zero. This will make all members of Q have only one correct expansion.

This whole logic is completely governed by the implementation of IntegralPart. Hence, we do not need to manage modes anywhere in the code base.

To instead swap behavior (positive numbers = FIO and negative numbers = FEO) we can change IntegralPart to the following:

public BigInteger IntegralPart =>
        IsPositiveInteger
            ? (Numerator / Denominator) - 1
            : (Numerator / Denominator);

Examples

Q	Expansion	Mode
0	.0000000000000000	FEO=FIO, in any base
1₂	1.000000000000000	FEO
-1₂	.1111111111111111	FIO
103/16₂	110.0111000000000	FEO
-103/16₂	110.0110111111111	FIO
5/24₂	.0011010101010101	FEO=FIO
-5/24₂	.0011010101010101	FEO=FIO
3/45₅	.3333333333333333	FEO=FIO
-3/45₅	.3333333333333333	FEO=FIO
537/11₃	1210.211002110021	FEO=FIO
-537/11₃	1210.211002110021	FEO=FIO

See Also

ToStringExpanded(int)

IntegralPartAlwaysFEO

This version always sets FEO mode.

Declaration

protected BigInteger IntegralPartAlwaysFEO { get; }

Property Value

Type	Description
BigInteger

IsInteger

Indicates whether the rational number is an integer.

Declaration

public bool IsInteger { get; }

Property Value

Type	Description
bool

IsNaN

Indicates whether the rational number is NaN (not a number), indicated by a zero denominator.

Declaration

public bool IsNaN { get; }

Property Value

Type	Description
bool

IsNegative

Indicates whether the rational number is negative.

Declaration

public bool IsNegative { get; }

Property Value

Type	Description
bool

IsNegativeInteger

Indicates whether the rational number is a negative integer.

Declaration

public bool IsNegativeInteger { get; }

Property Value

Type	Description
bool

IsNonZeroInteger

Indicates whether the rational number is a non-zero integer.

Declaration

public bool IsNonZeroInteger { get; }

Property Value

Type	Description
bool

IsOne

Indicates whether the rational number is equal to one.

Declaration

public bool IsOne { get; }

Property Value

Type	Description
bool

IsPositive

Indicates whether the rational number is positive.

Declaration

public bool IsPositive { get; }

Property Value

Type	Description
bool

IsPositiveInteger

Indicates whether the rational number is a positive integer.

Declaration

public bool IsPositiveInteger { get; }

Property Value

Type	Description
bool

IsUnitFraction

Indicates whether the rational number is a unit fraction.

Declaration

public bool IsUnitFraction { get; }

Property Value

Type	Description
bool

Remarks

Unit fractions are strictly positive, meaning -1/10 is not a unit fraction.

IsZero

Indicates whether the rational number is zero.

Declaration

public bool IsZero { get; }

Property Value

Type	Description
bool

NaN

Declaration

public static Q NaN { get; }

Property Value

Type	Description
Q	Represents a NaN (not a number) denoting an invalid number. NaN is return value of operations that are not defined (e.g. when dividing by zero)

Numerator

The numerator of the rational number.

Declaration

public BigInteger Numerator { get; }

Property Value

Type	Description
BigInteger

One

Declaration

public static Q One { get; }

Property Value

Type	Description
Q	Represents the rational number one.

Reciprocal

Reciprocal of this rational number.

Declaration

public Q Reciprocal { get; }

Property Value

Type	Description
Q

Remarks

The reciprocal of a rational number a/b is b/a. Returns NaN if the numerator is 0.

Sign

Sign of the current rational number.

Declaration

public int Sign { get; }

Property Value

Type	Description
int	An integer that indicates the sign of the current rational number. Returns -1 if the number is negative, 0 if the number is zero, and 1 if the number is positive.

Zero

Declaration

public static Q Zero { get; }

Property Value

Type	Description
Q	Represents the rational number zero.

Methods

Backward(int)

Obsolete: This method is obsolete and will be removed.

Declaration

[Obsolete("This method is obsolete and will be removed")]
public Q Backward(int base_)

Parameters

Type	Name	Description
int	base_

Returns

Type	Description
Q

Coeff(int)

The value of the rational number in the specified base.

Declaration

public BigInteger Coeff(int base_)

Parameters

Type	Name	Description
int	base_	The base to convert to.

Returns

Type	Description
BigInteger	The coefficient of the rational number in the specified base.

Coefficients(int)

Light version of Coefficients() that takes a rational number and a base.

Declaration

public IEnumerable<int> Coefficients(int base_)

Parameters

Type	Name	Description
int	base_	The base to convert to.

Returns

Type	Description
IEnumerable<int>	An enumerable sequence of integers representing the coefficients of the rational number in the specified base.

Remarks

This method does not require the costly creation of a Qb.

See Also

Coefficients()

ShiftedFractions()

CompareTo(Q?)

Compares the current rational number to another rational number.

Declaration

public int CompareTo(Q? other)

Parameters

Type	Name	Description
Q	other	The rational number to compare with.

Returns

Type	Description
int	An integer that indicates the relative order of the current rational number and the specified rational number.

CompareTo(int)

Compares the current rational number to an int.

Declaration

public int CompareTo(int integer)

Parameters

Type	Name	Description
int	integer	The int to compare with.

Returns

Type	Description
int	An integer that indicates the relative order of the current rational number and integer.

CompareTo(BigInteger)

Compares the current rational number to a BigInteger.

Declaration

public int CompareTo(BigInteger integer)

Parameters

Type	Name	Description
BigInteger	integer	The BigInteger to compare with.

Returns

Type	Description
int	An integer that indicates the relative order of the current rational number and the specified BigInteger.

Equals(Q?)

Indicates whether the current rational number is equal to another rational number.

Declaration

public bool Equals(Q? other)

Parameters

Type	Name	Description
Q	other	The rational number to compare with.

Returns

Type	Description
bool	true iff the current rational number is equal to the specified rational number.

Equals(object?)

Indicates whether the current instance is equal to a specified object.

Declaration

public override bool Equals(object? obj)

Parameters

Type	Name	Description
object	obj	The object to compare with the current instance.

Returns

Type	Description
bool	true iff the current instance is equal to the specified object obj.

Overrides

object.Equals(object)

Forward(int)

Obsolete: This method is obsolete and will be removed.

Declaration

[Obsolete("This method is obsolete and will be removed")]
public Q Forward(int base_)

Parameters

Type	Name	Description
int	base_

Returns

Type	Description
Q

GetHashCode()

Returns the hash code for the current instance.

Declaration

public override int GetHashCode()

Returns

Type	Description
int	A hash code for the current instance.

Overrides

object.GetHashCode()

GetPAdicPreperiodic(Q, BaseInt, int)

Declaration

public static BaseInt GetPAdicPreperiodic(Q pAdicPreperiodicQ, BaseInt pAryPreperiodic, int firstExponent)

Parameters

Type	Name	Description
Q	pAdicPreperiodicQ
BaseInt	pAryPreperiodic
int	firstExponent

Returns

Type	Description
BaseInt

InBase(int)

Extends the current rational number to a Qb with the specified base.

Declaration

public Qb InBase(int base_)

Parameters

Type	Name	Description
int	base_	The base to convert to.

Returns

Type	Description
Qb	A new Qb representing the current rational number in the specified base.

IntegralPartSelective(bool)

This version selects between FIO and FEO mode.

Declaration

protected BigInteger IntegralPartSelective(bool fio)

Parameters

Type	Name	Description
bool	fio	If true FIO mode, otherwise FEO mode

Returns

Type	Description
BigInteger

Remarks

This property needs to replace the existing version to come in to effect. Current implementation is provided only to give an understanding.

Negation(bool)

Conditionally negates the current rational number.

Declaration

public Q Negation(bool negate)

Parameters

Type	Name	Description
bool	negate	A boolean value indicating whether to negate the rational number.

Returns

Type	Description
Q	A new Q representing the negative value if negate is true; otherwise, returns the current instance.

PAdicInterpretation(BaseInt, BaseInt, int)

Declaration

public static Q PAdicInterpretation(BaseInt pAdicPreperiodic, BaseInt pAdicPeriodic, int firstExponent = 0)

Parameters

Type	Name	Description
BaseInt	pAdicPreperiodic
BaseInt	pAdicPeriodic
int	firstExponent

Returns

Type	Description
Q

PAdicPeriodic(BaseInt, int)

Declaration

public static Q PAdicPeriodic(BaseInt pAdicPeriodic, int firstPeriodicExponent)

Parameters

Type	Name	Description
BaseInt	pAdicPeriodic
int	firstPeriodicExponent

Returns

Type	Description
Q

PAdicPreperiodic(BaseInt, int)

Declaration

public static Q PAdicPreperiodic(BaseInt pAdicPreperiodic, int firstExponent = 0)

Parameters

Type	Name	Description
BaseInt	pAdicPreperiodic
int	firstExponent

Returns

Type	Description
Q

PAryInterpretation(bool, BaseInt, BaseInt, int)

Returns a Q from the given parts of a base-specific expansion.

Declaration

public static Q PAryInterpretation(bool negative, BaseInt pAryPreperiodic, BaseInt pAryPeriodic, int firstExponent = -1)

Parameters

Type	Name	Description
bool	negative	A boolean indicating if the number is negative.
BaseInt	pAryPreperiodic	The preperiodic part as an integer.
BaseInt	pAryPeriodic	The periodic part as an integer.
int	firstExponent	The exponent index of the first coefficient of the number.

Returns

Type	Description
Q	A new Q representing the rational number derived from the given parts.

Remarks

This method derives the Numerator and Denominator from the given parameters.

PAryPeriodic(bool, BaseInt, int)

Declaration

public static Q PAryPeriodic(bool negative, BaseInt pAryPeriodic, int firstPeriodicExponent)

Parameters

Type	Name	Description
bool	negative
BaseInt	pAryPeriodic
int	firstPeriodicExponent

Returns

Type	Description
Q

PAryPreperiodic(bool, BaseInt, int)

Declaration

public static Q PAryPreperiodic(bool negative, BaseInt pAryPreperiodic, int firstExponent = -1)

Parameters

Type	Name	Description
bool	negative
BaseInt	pAryPreperiodic
int	firstExponent

Returns

Type	Description
Q

Pow(Q)

Raises the current rational number to the power of the given exponent.

Declaration

public Q Pow(Q exponent)

Parameters

Type	Name	Description
Q	exponent	The exponent to which to raise the current rational number.

Returns

Type	Description
Q	A new rational number representing the result of the exponentiation.

Pow(int)

Raises the current rational number to the power of the given integer exponent.

Declaration

public Q Pow(int exponent)

Parameters

Type	Name	Description
int	exponent	The integer exponent to which to raise the current rational number.

Returns

Type	Description
Q	A new rational number representing the result of the exponentiation.

ShiftedFractions(int)

Light version of ShiftedFractions() that takes a rational number and a base.

Declaration

public IEnumerable<(BigInteger Integer, Q Fraction)> ShiftedFractions(int base_)

Parameters

Type	Name	Description
int	base_	The base to convert to.

Returns

Type	Description
IEnumerable<(BigInteger Integer, Q Fraction)>	An enumerable sequence of tuples, where each tuple contains the integer part and the fractional part of the rational number in the specified base.

Remarks

This method does not require the costly creation of a Qb.

See Also

ShiftedFractions()

Square()

Returns the square of the current rational number.

Declaration

public Q Square()

Returns

Type	Description
Q	A new rational number representing the square of the current rational number.

ToString()

Returns the default string representation of the rational number

Declaration

public override string ToString()

Returns

Type	Description
string

Overrides

object.ToString()

ToStringCanonical()

Returns the rational number as a simple canonical representation in base 10.

Declaration

public string ToStringCanonical()

Returns

Type	Description
string	A string representing the rational number (in base 10) in the format "numerator/denominator" or just "numerator" if the denominator is 1.

Remarks

An NaN rational number will yield the string NaN

Examples

Console.WriteLine(new Q(-4,5).ToStringCanonical()); 
//outputs "-4/5"

ToStringFactorization()

Converts the factorization of the rational number to a string representation.

Declaration

public string ToStringFactorization()

Returns

Type	Description
string	A string representing the factorization of the rational number.

ToStringFinite(int)

Declaration

public string ToStringFinite(int base_)

Parameters

Type	Name	Description
int	base_

Returns

Type	Description
string

TryCastToInt32(out int)

Attempts to cast the rational number to a 32-bit integer, if possible.

Declaration

public bool TryCastToInt32(out int result)

Parameters

Type	Name	Description
int	result	The resulting 32-bit integer if the cast is successful.

Returns

Type	Description
bool	true iff the cast is successful.

Operators

operator +(Q, Q)

Sum of two rational numbers.

Declaration

public static Q operator +(Q a, Q b)

Parameters

Type	Name	Description
Q	a	The first term.
Q	b	The second term.

Returns

Type	Description
Q	A new rational number representing the sum of a and b.

operator +(Q, BigInteger)

Sum of a rational number and an integer.

Declaration

public static Q operator +(Q q, BigInteger integer)

Parameters

Type	Name	Description
Q	q	A rational number denoting the first term.
BigInteger	integer	A BigInteger denoting the second term.

Returns

Type	Description
Q	A new rational number representing the sum of q and integer.

operator +(BigInteger, Q)

Sum of an integer and a rational number.

Declaration

public static Q operator +(BigInteger integer, Q q)

Parameters

Type	Name	Description
BigInteger	integer	A BigInteger denoting the first term.
Q	q	A rational number denoting the second term.

Returns

Type	Description
Q	A new rational number representing the sum of integer and q.

operator --(Q)

Decrements this rational number by one.

Declaration

public static Q operator --(Q a)

Parameters

Type	Name	Description
Q	a	The rational number to decrement.

Returns

Type	Description
Q	A new rational number representing the result of the decrement.

operator /(Q, Q)

Quotient of two rational numbers.

Declaration

public static Q operator /(Q a, Q b)

Parameters

Type	Name	Description
Q	a	The dividend.
Q	b	The divisor.

Returns

Type	Description
Q	A new rational number representing the quotient of a and b.

operator /(Q, BigInteger)

Quotient of a rational number and an integer.

Declaration

public static Q operator /(Q q, BigInteger integer)

Parameters

Type	Name	Description
Q	q	A rational number denoting the dividend.
BigInteger	integer	A BigInteger denoting the divisor.

Returns

Type	Description
Q	A new rational number representing the quotient of q and integer.

operator /(BigInteger, Q)

Quotient of am integer and rational number.

Declaration

public static Q operator /(BigInteger integer, Q q)

Parameters

Type	Name	Description
BigInteger	integer	A BigInteger denoting the dividend.
Q	q	A rational number denoting the divisor.

Returns

Type	Description
Q	A new rational number representing the quotient of integer and q.

operator ==(Q, Q)

Indicates whether two rational numbers are equal.

Declaration

public static bool operator ==(Q a, Q b)

Parameters

Type	Name	Description
Q	a	The first rational number.
Q	b	The second rational number.

Returns

Type	Description
bool	true iff the two rational numbers are equal.

operator ==(Q, int)

Indicates whether the current rational number is equal to a specified int.

Declaration

public static bool operator ==(Q q, int integer)

Parameters

Type	Name	Description
Q	q	The rational number to compare.
int	integer	The int to compare.

Returns

Type	Description
bool	true iff the current rational number is equal to the specified integer.

operator ==(Q, BigInteger)

Indicates whether the current rational number is equal to a specified BigInteger.

Declaration

public static bool operator ==(Q q, BigInteger integer)

Parameters

Type	Name	Description
Q	q	The rational number to compare.
BigInteger	integer	The BigInteger to compare.

Returns

Type	Description
bool	true iff the current rational number is equal to integer.

operator >(Q, Q)

Indicates whether the current rational number is greater than another rational number.

Declaration

public static bool operator >(Q a, Q b)

Parameters

Type	Name	Description
Q	a	The first rational number.
Q	b	The second rational number.

Returns

Type	Description
bool	true iff the current rational number is greater than the specified rational number.

operator >(Q, int)

Indicates whether the current rational number is greater than a specified int.

Declaration

public static bool operator >(Q q, int integer)

Parameters

Type	Name	Description
Q	q	The current rational number.
int	integer	The int to compare.

Returns

Type	Description
bool	true iff the current rational number is greater than the specified integer.

operator >(Q, BigInteger)

Indicates whether the current rational number is greater than a BigInteger.

Declaration

public static bool operator >(Q q, BigInteger integer)

Parameters

Type	Name	Description
Q	q	The current rational number.
BigInteger	integer	The BigInteger to compare.

Returns

Type	Description
bool	true iff the current rational number is greater than the specified integer.

operator >=(Q, Q)

Indicates whether the current rational number is greater than or equal to another rational number.

Declaration

public static bool operator >=(Q a, Q b)

Parameters

Type	Name	Description
Q	a	The first rational number.
Q	b	The second rational number.

Returns

Type	Description
bool	true iff the current rational number is greater than or equal to the specified rational number.

operator >=(Q, int)

Indicates whether the current rational number is greater than or equal to a specified int.

Declaration

public static bool operator >=(Q q, int integer)

Parameters

Type	Name	Description
Q	q	The current rational number.
int	integer	The int to compare.

Returns

Type	Description
bool	true iff the current rational number is greater than or equal to integer.

operator >=(Q, BigInteger)

Indicates whether the current rational number is greater than or equal to a BigInteger.

Declaration

public static bool operator >=(Q q, BigInteger integer)

Parameters

Type	Name	Description
Q	q	The current rational number.
BigInteger	integer	The BigInteger to compare.

Returns

Type	Description
bool	true iff the current rational number is greater than or equal to the specified integer.

operator ++(Q)

Increments this rational number by one.

Declaration

public static Q operator ++(Q a)

Parameters

Type	Name	Description
Q	a	The rational number to increment.

Returns

Type	Description
Q	A new rational number representing the result of the increment.

operator !=(Q, Q)

Indicates whether two rational numbers are not equal.

Declaration

public static bool operator !=(Q a, Q b)

Parameters

Type	Name	Description
Q	a	The first rational number.
Q	b	The second rational number.

Returns

Type	Description
bool	true iff the two rational numbers are not equal.

operator !=(Q, int)

Indicates whether the current rational number is not equal to a specified int.

Declaration

public static bool operator !=(Q q, int integer)

Parameters

Type	Name	Description
Q	q	The rational number to compare.
int	integer	The int to compare.

Returns

Type	Description
bool	true iff the current rational number is not equal to integer.

operator !=(Q, BigInteger)

Indicates whether the current rational number is not equal to a specified BigInteger.

Declaration

public static bool operator !=(Q q, BigInteger integer)

Parameters

Type	Name	Description
Q	q	The rational number to compare.
BigInteger	integer	The BigInteger to compare.

Returns

Type	Description
bool	true if the current rational number is not equal to the specified integer.

operator <<(Q, int)

Performs a left bitwise shift on the current rational number.

Declaration

public static Q operator <<(Q a, int shift)

Parameters

Type	Name	Description
Q	a	The rational number to shift.
int	shift	The number of bits to shift. A positive value shifts left, a negative value shifts right.

Returns

Type	Description
Q	A new rational number representing the result of the shift operation.

Remarks

If shift is negative, the shift direction will be reversed, making this operation equivalent to a right shift. This behavior mirrors the standard behavior of the C# operators << and >> for integral types, where negative values reverse the shift direction.

operator <(Q, Q)

Indicates whether the current rational number is less than another rational number.

Declaration

public static bool operator <(Q a, Q b)

Parameters

Type	Name	Description
Q	a	The first rational number.
Q	b	The second rational number.

Returns

Type	Description
bool	true iff the current rational number is less than the specified rational number.

operator <(Q, int)

Indicates whether the current rational number is less than an int.

Declaration

public static bool operator <(Q q, int integer)

Parameters

Type	Name	Description
Q	q	The current rational number.
int	integer	The int to compare.

Returns

Type	Description
bool	true iff the current rational number is less than integer.

operator <(Q, BigInteger)

Indicates whether the current rational number is less than a BigInteger.

Declaration

public static bool operator <(Q q, BigInteger integer)

Parameters

Type	Name	Description
Q	q	The current rational number.
BigInteger	integer	The BigInteger to compare.

Returns

Type	Description
bool	true iff the current rational number is less than the specified integer.

operator <=(Q, Q)

Indicates whether the current rational number is less than or equal to another rational number.

Declaration

public static bool operator <=(Q a, Q b)

Parameters

Type	Name	Description
Q	a	The first rational number.
Q	b	The second rational number.

Returns

Type	Description
bool	true iff the current rational number is less than or equal to the specified rational number

operator <=(Q, int)

Indicates whether the current rational number is less than or equal to an int.

Declaration

public static bool operator <=(Q q, int integer)

Parameters

Type	Name	Description
Q	q	The current rational number.
int	integer	The int to compare.

Returns

Type	Description
bool	true iff the current rational number is less than or equal to the specified integer.

operator <=(Q, BigInteger)

Indicates whether the current rational number is less than or equal to a BigInteger.

Declaration

public static bool operator <=(Q q, BigInteger integer)

Parameters

Type	Name	Description
Q	q	The current rational number.
BigInteger	integer	The BigInteger to compare.

Returns

Type	Description
bool	true iff the current rational number is less than or equal to the specified integer.

operator %(Q, Q)

Performs the modulus operation on two rational numbers.

Declaration

public static Q operator %(Q a, Q b)

Parameters

Type	Name	Description
Q	a	The left operand.
Q	b	The right operand.

Returns

Type	Description
Q	The modulus of a and b as a Q.

Remarks

The formula for the modulus operation is:

(a_n / a_d) % (b_n / b_d) = ((a_n * b_d) % (a_d * b_n)) / (a_d * b_d)

where a_n and a_d are the numerator and denominator of a respectively, and b_n and b_d are the numerator and denominator of b respectively.

operator *(Q, Q)

Product of two rational numbers.

Declaration

public static Q operator *(Q a, Q b)

Parameters

Type	Name	Description
Q	a	The first factor.
Q	b	The second factor.

Returns

Type	Description
Q	A new rational number representing the product of a and b.

operator *(Q, BigInteger)

Product of a rational number and an integer.

Declaration

public static Q operator *(Q q, BigInteger integer)

Parameters

Type	Name	Description
Q	q	A rational number denoting the first factor.
BigInteger	integer	A BigInteger denoting the second factor.

Returns

Type	Description
Q	A new rational number representing the product of q and integer.

operator *(BigInteger, Q)

Product of an integer and a rational number.

Declaration

public static Q operator *(BigInteger integer, Q q)

Parameters

Type	Name	Description
BigInteger	integer	A BigInteger denoting the first factor.
Q	q	A rational number denoting the second factor.

Returns

Type	Description
Q	A new rational number representing the product of integer and q .

operator >>(Q, int)

Performs a right bitwise shift on the current rational number.

Declaration

public static Q operator >>(Q a, int shift)

Parameters

Type	Name	Description
Q	a	The rational number to shift.
int	shift	The number of bits to shift. A positive value shifts right, a negative value shifts left.

Returns

Type	Description
Q	A new rational number representing the result of the shift operation.

Remarks

If shift is negative, the shift direction will be reversed, making this operation equivalent to a left shift. This behavior mirrors the standard behavior of the C# operators << and >> for integral types, where negative values reverse the shift direction.

operator -(Q, Q)

Difference of two rational numbers.

Declaration

public static Q operator -(Q a, Q b)

Parameters

Type	Name	Description
Q	a	The minuend.
Q	b	The subtrahend.

Returns

Type	Description
Q	A new rational number representing the difference between a and b.

operator -(Q, BigInteger)

Difference of a rational number and an integer.

Declaration

public static Q operator -(Q q, BigInteger integer)

Parameters

Type	Name	Description
Q	q	A rational number denoting the minuend.
BigInteger	integer	A BigInteger denoting the subtrahend.

Returns

Type	Description
Q	A new rational number representing the difference of q and integer.

operator -(BigInteger, Q)

Difference of an integer and a rational number.

Declaration

public static Q operator -(BigInteger integer, Q q)

Parameters

Type	Name	Description
BigInteger	integer	A BigInteger denoting the minuend.
Q	q	A rational number denoting the subtrahend.

Returns

Type	Description
Q	A new rational number representing the difference of integer and q.

operator -(Q)

Negation of this rational number.

Declaration

public static Q operator -(Q a)

Parameters

Type	Name	Description
Q	a	The rational number to negate.

Returns

Type	Description
Q	A new rational number representing the negated value.

Implements

IEquatable<T>

IComparable<T>