Non-integer representation
Encyclopedia
A non-integer representation uses non-integer
numbers as the radix
, or bases, of a positional numbering system
. For a non-integer radix β > 1, the value of
is
The numbers di are non-negative integers less than β. This is also known as a β-expansion, a notion introduced by and first studied in detail by . Every real number has at least one (possibly infinite) β-expansion.
There are applications of β-expansions in coding theory
and models of quasicrystal
s .
), all finite decimal expansions are unique. However, even finite β-expansions are not necessarily unique, for example φ + 1 = φ2 for β = φ, the golden ratio
. A canonical choice for the β-expansion of a given real number can be determined by the following greedy algorithm
, essentially due to and formulated as given here by .
Let be the base and x a non-negative real number. Denote by the floor function
of x, that is, the greatest integer less than or equal to x, and let {x} = x − ⌊x⌋ be the fractional part of x. There exists an integer k such that . Set
and
For , put
In other words, the canonical β-expansion of x is defined by choosing the largest dk such that , then choosing the largest dk−1 such that , etc. Thus it chooses the lexicographically
largest string representing x.
With an integer base, this defines the usual radix expansion for the number x. This construction extends the usual algorithm to possibly non-integer values of β.
the natural logarithm
behaves like the common logarithm
as ln(1e) = 0, ln(10e) = 1, ln(100e) = 2 and ln(1000e) = 3.
The base e is the most economical choice of radix β > 1 , where the radix economy
is measured as the product of the radix and the length of the string of symbols needed to express a given range of values.
can be used to more easily show the relationship between the diameter
of a circle
to its circumference
; since circumference = diameter × π, a circle with a diameter 1π will have a circumference of 10π, a circle with a diameter 10π will have a circumference of 100π, etc. Furthermore, since the area
= π × radius
2, a circle with a radius of 1π will have an area of 10π, a circle with a radius of 10π will have an area of 1000π and a circle with a radius of 100π will have an area of 100000π.
behaves in a very similar way to base 2
as all one has to do to convert a number from binary into base √2 is put a zero digit in between every binary digit; for example, 191110 = 111011101112 becomes 101010001010100010101√2 and 511810 = 10011111111102 becomes 1000001010101010101010100√2. This means that every integer can be expressed in base √2 without the need of a decimal point. The base can also be used to show the relationship between the side
of a square
to its diagonal
as a square with a side length of 1√2 will have a diagonal of 10√2 and a square with a side length of 10√2 will have a diagonal of 100√2. Another use of the base is to show the silver ratio
as its representation in base √2 is simply 11√2.
. The set of numbers with two different representations is dense
in the reals , but the question of classifying real numbers with unique β-expansions is considerably more subtle than that of integer bases .
Another problem is to classify the real numbers whose β-expansions are periodic. Let β > 1, and Q(β) be the smallest field extension
of the rationals containing β. Then any real number in [0,1) having a periodic β-expansion must lie in Q(β). On the other hand, the converse need not be true. The converse does hold if β is a Pisot number , although necessary and sufficient conditions are not known.
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
numbers as the radix
Radix
In mathematical numeral systems, the base or radix for the simplest case is the number of unique digits, including zero, that a positional numeral system uses to represent numbers. For example, for the decimal system the radix is ten, because it uses the ten digits from 0 through 9.In any numeral...
, or bases, of a positional numbering system
Positional notation
Positional notation or place-value notation is a method of representing or encoding numbers. Positional notation is distinguished from other notations for its use of the same symbol for the different orders of magnitude...
. For a non-integer radix β > 1, the value of
is
The numbers di are non-negative integers less than β. This is also known as a β-expansion, a notion introduced by and first studied in detail by . Every real number has at least one (possibly infinite) β-expansion.
There are applications of β-expansions in coding theory
Coding theory
Coding theory is the study of the properties of codes and their fitness for a specific application. Codes are used for data compression, cryptography, error-correction and more recently also for network coding...
and models of quasicrystal
Quasicrystal
A quasiperiodic crystal, or, in short, quasicrystal, is a structure that is ordered but not periodic. A quasicrystalline pattern can continuously fill all available space, but it lacks translational symmetry...
s .
Construction
β-expansions are a generalization of decimal expansions. While infinite decimal expansions are not unique (for example, 1.000... = 0.999...0.999...
In mathematics, the repeating decimal 0.999... denotes a real number that can be shown to be the number one. In other words, the symbols 0.999... and 1 represent the same number...
), all finite decimal expansions are unique. However, even finite β-expansions are not necessarily unique, for example φ + 1 = φ2 for β = φ, the golden ratio
Golden ratio
In mathematics and the arts, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. The golden ratio is an irrational mathematical constant, approximately 1.61803398874989...
. A canonical choice for the β-expansion of a given real number can be determined by the following greedy algorithm
Greedy algorithm
A greedy algorithm is any algorithm that follows the problem solving heuristic of making the locally optimal choice at each stagewith the hope of finding the global optimum....
, essentially due to and formulated as given here by .
Let be the base and x a non-negative real number. Denote by the floor function
Floor function
In mathematics and computer science, the floor and ceiling functions map a real number to the largest previous or the smallest following integer, respectively...
of x, that is, the greatest integer less than or equal to x, and let {x} = x − ⌊x⌋ be the fractional part of x. There exists an integer k such that . Set
and
For , put
In other words, the canonical β-expansion of x is defined by choosing the largest dk such that , then choosing the largest dk−1 such that , etc. Thus it chooses the lexicographically
Lexicographical order
In mathematics, the lexicographic or lexicographical order, , is a generalization of the way the alphabetical order of words is based on the alphabetical order of letters.-Definition:Given two partially ordered sets A and B, the lexicographical order on...
largest string representing x.
With an integer base, this defines the usual radix expansion for the number x. This construction extends the usual algorithm to possibly non-integer values of β.
Base e
With base eE (mathematical constant)
The mathematical constant ' is the unique real number such that the value of the derivative of the function at the point is equal to 1. The function so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base...
the natural logarithm
Natural logarithm
The natural logarithm is the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.718281828...
behaves like the common logarithm
Common logarithm
The common logarithm is the logarithm with base 10. It is also known as the decadic logarithm, named after its base. It is indicated by log10, or sometimes Log with a capital L...
as ln(1e) = 0, ln(10e) = 1, ln(100e) = 2 and ln(1000e) = 3.
The base e is the most economical choice of radix β > 1 , where the radix economy
Radix economy
Various proposals have been made to quantify the relative costs between using different radices in representing numbers, especially in computer systems.-Definition:...
is measured as the product of the radix and the length of the string of symbols needed to express a given range of values.
Base π
Base πPi
' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...
can be used to more easily show the relationship between the diameter
Diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle. The diameters are the longest chords of the circle...
of a circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
to its circumference
Circumference
The circumference is the distance around a closed curve. Circumference is a special perimeter.-Circumference of a circle:The circumference of a circle is the length around it....
; since circumference = diameter × π, a circle with a diameter 1π will have a circumference of 10π, a circle with a diameter 10π will have a circumference of 100π, etc. Furthermore, since the area
Area
Area is a quantity that expresses the extent of a two-dimensional surface or shape in the plane. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat...
= π × radius
Radius
In classical geometry, a radius of a circle or sphere is any line segment from its center to its perimeter. By extension, the radius of a circle or sphere is the length of any such segment, which is half the diameter. If the object does not have an obvious center, the term may refer to its...
2, a circle with a radius of 1π will have an area of 10π, a circle with a radius of 10π will have an area of 1000π and a circle with a radius of 100π will have an area of 100000π.
Base √2
Base √2Square root of 2
The square root of 2, often known as root 2, is the positive algebraic number that, when multiplied by itself, gives the number 2. It is more precisely called the principal square root of 2, to distinguish it from the negative number with the same property.Geometrically the square root of 2 is the...
behaves in a very similar way to base 2
Binary numeral system
The binary numeral system, or base-2 number system, represents numeric values using two symbols, 0 and 1. More specifically, the usual base-2 system is a positional notation with a radix of 2...
as all one has to do to convert a number from binary into base √2 is put a zero digit in between every binary digit; for example, 191110 = 111011101112 becomes 101010001010100010101√2 and 511810 = 10011111111102 becomes 1000001010101010101010100√2. This means that every integer can be expressed in base √2 without the need of a decimal point. The base can also be used to show the relationship between the side
Edge (geometry)
In geometry, an edge is a one-dimensional line segment joining two adjacent zero-dimensional vertices in a polygon. Thus applied, an edge is a connector for a one-dimensional line segment and two zero-dimensional objects....
of a square
Square (geometry)
In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...
to its diagonal
Diagonal
A diagonal is a line joining two nonconsecutive vertices of a polygon or polyhedron. Informally, any sloping line is called diagonal. The word "diagonal" derives from the Greek διαγώνιος , from dia- and gonia ; it was used by both Strabo and Euclid to refer to a line connecting two vertices of a...
as a square with a side length of 1√2 will have a diagonal of 10√2 and a square with a side length of 10√2 will have a diagonal of 100√2. Another use of the base is to show the silver ratio
Silver ratio
In mathematics, two quantities are in the silver ratio if the ratio between the sum of the smaller plus twice the larger of those quantities and the larger one is the same as the ratio between the larger one and the smaller. This defines the silver ratio as an irrational mathematical constant,...
as its representation in base √2 is simply 11√2.
Properties
In no positional number system can every number be expressed uniquely. For example, in base 10, the number 1 has two representations: 1.000... and 0.999...0.999...
In mathematics, the repeating decimal 0.999... denotes a real number that can be shown to be the number one. In other words, the symbols 0.999... and 1 represent the same number...
. The set of numbers with two different representations is dense
Dense set
In topology and related areas of mathematics, a subset A of a topological space X is called dense if any point x in X belongs to A or is a limit point of A...
in the reals , but the question of classifying real numbers with unique β-expansions is considerably more subtle than that of integer bases .
Another problem is to classify the real numbers whose β-expansions are periodic. Let β > 1, and Q(β) be the smallest field extension
Field extension
In abstract algebra, field extensions are the main object of study in field theory. The general idea is to start with a base field and construct in some manner a larger field which contains the base field and satisfies additional properties...
of the rationals containing β. Then any real number in [0,1) having a periodic β-expansion must lie in Q(β). On the other hand, the converse need not be true. The converse does hold if β is a Pisot number , although necessary and sufficient conditions are not known.
See also
- Beta encoderBeta encoderA beta encoder is an analog to digital conversion system in which a real number in the unit interval is represented by a finite representation of a sequence in base beta, with beta being a real number between 1 and 2...
- Non-standard positional numeral systemsNon-standard positional numeral systemsNon-standard positional numeral systems here designates numeral systems that may be denoted positional systems, but that deviate in one way or another from the following description of standard positional systems:...
- Decimal expansion
- Power series