Class Qb

Represents a rational number with a base-dependent expansion, denoted ℚb.

Inheritance

object

Q

Qb

Implements

IEquatable<Q>

IComparable<Q>

IEquatable<Qb>

Inherited Members

Q.IsNaN

Q.IsZero

Q.IsOne

Q.IsPositive

Q.IsNegative

Q.IsInteger

Q.IsNonZeroInteger

Q.IsPositiveInteger

Q.IsNegativeInteger

Q.IsUnitFraction

Q.Equals(Q)

Q.CompareTo(Q)

Q.CompareTo(BigInteger)

Q.CompareTo(int)

Q.Equals(object)

Q.GetHashCode()

Q.Numerator

Q.Denominator

Q.IntegralPart

Q.FractionalPart

Q.Zero

Q.One

Q.PAryInterpretation(bool, BaseInt, BaseInt, int)

Q.PAdicInterpretation(BaseInt, BaseInt, int)

Q.PAryPreperiodic(bool, BaseInt, int)

Q.PAdicPreperiodic(BaseInt, int)

Q.PAryPeriodic(bool, BaseInt, int)

Q.PAdicPeriodic(BaseInt, int)

Q.GetPAdicPreperiodic(Q, BaseInt, int)

Q.InBase(int)

Q.ShiftedFractions(int)

Q.Coefficients(int)

Q.Coeff(int)

Q.Backward(int)

Q.Forward(int)

Q.Sign

Q.Reciprocal

Q.Negation(bool)

Q.Square()

Q.Abs

Q.Pow(Q)

Q.Pow(int)

Q.TryCastToInt32(out int)

Q.IntegralPartAlwaysFEO

Q.IntegralPartSelective(bool)

Q.ToStringCanonical()

Q.ToStringFactorization()

Q.ToStringFinite(int)

object.Equals(object, object)

object.GetType()

object.MemberwiseClone()

object.ReferenceEquals(object, object)

Namespace: MathLib

Assembly: MathLib.dll

Syntax

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

Remarks

The class Qb extends Q by adopting a base, and thus defining a base-specific coefficient expansion of the rational number, with a preperiodic and a periodic part. These types of expansions are commonly used in numeral systems like p-ary expansions (e.g. decimal or binary) or in p-adic systems.

Each number is expressed in terms of a base, denoted by Base, with a preperiodic part (initial terminating sequence) and a periodic part (repeating sequence).

Each coefficient is indexed according to the exponent of the base Base. The index i corresponds to the coefficient c_i associated with the term c_i · Base^i, where the value of i decreases as we move from left to right in the expansion.

The first (leftmost) coefficient has index FirstExponent, which corresponds to the largest exponent. The index decreases through the expansion, reflecting the successive exponents of the base, including negative exponents for terms after the radix point (radix point occurs between c_0 and c_-1).

This consistent indexing system aligns with the mathematical representation of the number as a series in base Base: q = Σ (c_i · Base^i) where the index i decreases from FirstExponent to negative values as the expansion proceeds.

Overview of concepts and properties, for a full (ultimately periodic) expansion:
                         |--------------------------------------------------------|
    Indexes (example):   |-- 7   6   5   4   3   2   1   0  -1  -2  -3  -4  -5  --|  
    Numeric parts:       |-- IntegralPart    --|-- FractionalPart               --|
    Length properties:   |-- IntegralLength  --|-- FractionalLength             --|
    Numeric parts:       |-- PreperiodicPart               --|-- PeriodicPart   --|         
    Length properties:   |-- PreperiodicLength             --|-- Period         --|                  
                         |-- Length                                             --|
    Indexes (Exponents): • (FirstExponent=8)                     • (FirstPeriodicExponent=-2)
                                                           • (Radix point, after c_0)

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

Constructors

Qb(Q, int)

Extends a rational number to a Qb extension with the specific base base_.

Declaration

public Qb(Q q, int base_)

Parameters

Type	Name	Description
Q	q	A rational number of type Q
int	base_	The base of the rational number extension.

Qb(bool, BaseInt, BaseInt, int)

Constructor that creates a Qb from the defining parts of a base-specific expansion.

Declaration

public Qb(bool negative, BaseInt prePeriodicPart, BaseInt periodicPart, int firstExponent = -1)

Parameters

Type	Name	Description
bool	negative	A boolean indicating if the number is negative.
BaseInt	prePeriodicPart	The preperiodic part as a BaseInt.
BaseInt	periodicPart	The periodic part as a BaseInt>.
int	firstExponent	The exponent index of the first coefficient of the number.

Remarks

This constructor will derive the Numerator and Denominator from the parts.

Qb(BigInteger, BigInteger, int)

Constructor for a Qb that takes a numerator, denominator and a base. The rational number is automatically normalized to its simplest form.

Declaration

public Qb(BigInteger numerator, BigInteger denominator, int base_)

Parameters

Type	Name	Description
BigInteger	numerator	The numerator of the rational number.
BigInteger	denominator	The denominator of the rational number. Must be non-zero.
int	base_	The base of the rational number extension.

Remarks

If you already have an object of type Q, then the constructor Qb(Q, int) will be faster.

Or alternatively, you can simply call InBase(int) on the Q instance

Properties

Base

Gets the base of the rational number extension.

Declaration

public int Base { get; }

Property Value

Type	Description
int

FirstExponent

Index (= exponent value) of the first coefficient (of the base Base expansion)

Declaration

public int FirstExponent { get; }

Property Value

Type	Description
int

Remarks

This value is simply the length of IntegralPart - 1:

The range of possible values is [-1..∞[, since a number either starts before the radix point, or directly after it.

The coefficient at index FirstExponent is the true first coefficient in the expansion, which can be zero iffFirstExponent is -1.

Formula:

FirstExponent = IntegralPart.Length(Base) - 1

FirstPeriodicExponent

Index (= exponent value) of the first coefficient of the periodic part (of the base B expansion).

Declaration

public int FirstPeriodicExponent { get; }

Property Value

Type	Description
int

FractionalLength

Length of the fractional part (of the base B expansion).

Declaration

public int FractionalLength { get; }

Property Value

Type	Description
int

FullInteger

Numeric concatenation of the preperiodic and periodic parts.

Declaration

public BaseInt FullInteger { get; }

Property Value

Type	Description
BaseInt

IntegralLength

Length of the integer part (of the base B expansion).

Declaration

public int IntegralLength { get; }

Property Value

Type	Description
int

Remarks

This value is simply the base-specific length of IntegralPart.

Formula:

IntegralLength = IntegralPart.Length(Base)

Length

Total length of the preperiodic and periodic parts (of the base B expansion).

Declaration

public int Length { get; }

Property Value

Type	Description
int

NaN

Returns a NaN Qb instance.

Declaration

public static Qb NaN { get; }

Property Value

Type	Description
Qb

Period

Length of the periodic part (of the base B expansion).

Declaration

public int Period { get; }

Property Value

Type	Description
int

PeriodicPart

The periodic part.

Declaration

public BaseInt PeriodicPart { get; }

Property Value

Type	Description
BaseInt

Remarks

Contains the periodic coefficients (including zero-padding), and the period (=length of the periodic part).

PreperiodicLength

Length of the preperiodic part (of the base B expansion).

Declaration

public int PreperiodicLength { get; }

Property Value

Type	Description
int

PreperiodicPart

The preperiodic part.

Declaration

public BaseInt PreperiodicPart { get; }

Property Value

Type	Description
BaseInt

Remarks

Contains the preperiodic coefficients (including zero-padding), and the length of the preperiodic part.

Methods

Coefficients()

Returns the coefficient (or digit) expansion of a rational number in the specified base, expressed as a sequence of integers (modulo b).

Declaration

public IEnumerable<int> Coefficients()

Returns

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

Remarks

The coefficients are obtained from the fractional expansions generated by recursively shifting the radix point of the rational number in base Base.

Specifically, the coefficients are derived by taking the ShiftedFractions() and scaling them to the range [0..Base-1].

Thus, each coefficient is a number modulo Base.

Equals(Qb?)

Indicates whether the current object is equal to another object of the same type.

Declaration

public bool Equals(Qb? other)

Parameters

Type	Name	Description
Qb	other	An object to compare with this object.

Returns

Type	Description
bool	true if the current object is equal to the other parameter; otherwise, false.

RepetendShiftLeft()

Returns a new Q with a repetend (interpreted as an integer) that is base shifted 1 step to the left compared to the original repetend (interpreted as an integer).

Declaration

public Q RepetendShiftLeft()

Returns

Type	Description
Q	A new Q instance with the base-shifted repetend.

RepetendShiftRight()

Returns a new Q with a repetend (interpreted as an integer) that is base shifted 1 step to the right compared to the original repetend (interpreted as an integer).

Declaration

public Q RepetendShiftRight()

Returns

Type	Description
Q	A new Q instance with the base-shifted repetend.

ShiftedFractions()

Returns the left-to-right expansion of the rational number in the specified base as an infinite sequence of fractions. Each fraction corresponds to the fractional part remaining after successive base-specific left shifts.

Declaration

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

Returns

Type	Description
IEnumerable<(BigInteger Integer, Q Fraction)>	An enumerable sequence of rational numbers representing the shifted fractional parts in base Base.

Remarks

This method first applies a normalization step to the rational number, effectively shifting the radix point to the right (in base Base),
such that the integer part is zero, leaving only the fractional part. The first element of the sequence is this fractional remainder.

Subsequently, at each step, the fractional part is shifted left by the base (* Base), and the next fractional part is extracted by removing the integer part.

This process generates an infinite sequence of fractional remainders, which represent the underlying structure of the rational number in the given base. These fractions can be used to derive coefficients or compute the period of the expansion.

ToString()

Returns the default string representation of the rational number

Declaration

public override string ToString()

Returns

Type	Description
string

Overrides

Q.ToString()

ToStringExpanded(int)

Converts the current instance of Qb to its expanded string representation in the given base.

Declaration

public string ToStringExpanded(int coefficientCount = 16)

Parameters

Type	Name	Description
int	coefficientCount	The number of coefficients to include, starting from the leftmost. (default=16)

Returns

Type	Description
string	The expanded string representation of the current instance of Qb.

Remarks

The sign of a number is not encoded by (contained in) the expanded representation, since some Q and -Q can have the same string representation. However, negative numbers, completes the generation, by enabling us to generate all ultimately periodic expansions, including those that are not covered by the positive rational numbers.

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

ToStringExpandedSigned(int)

Declaration

public string ToStringExpandedSigned(int coefficientCount = 16)

Parameters

Type	Name	Description
int	coefficientCount

Returns

Type	Description
string

ToStringPeriodic()

Declaration

public string ToStringPeriodic()

Returns

Type	Description
string

ToStringRepetend()

Declaration

public string ToStringRepetend()

Returns

Type	Description
string

ToStringRotations()

Declaration

public string ToStringRotations()

Returns

Type	Description
string

Implements

IEquatable<T>

IComparable<T>

IEquatable<T>