For any given prime number p, a p-adic fraction or p-adic rational is a rational number whose denominator, when the ratio is in minimal (coprime) terms, is a power of p, i.e., a number of the form \( {\displaystyle {\frac {a}{p^{b}}}} \) where a is an integer and b is a natural number. These are precisely the numbers possessing a finite base-p positional numeral system expansion.
When p=2, they are called dyadic fractions or dyadic rationals; for example, 1/2 or 3/8, but not 1/3.
Arithmetic
The sum, product, or difference of any two p-adic rationals is itself another p-adic rational:
\( {\displaystyle {\frac {a}{p^{b}}}+{\frac {c}{p^{d}}}={\frac {p^{d-\min(b,d)}a+p^{b-\min(b,d)}c}{p^{\max(b,d)}}}} \)
\( {\displaystyle {\frac {a}{p^{b}}}-{\frac {c}{p^{d}}}={\frac {p^{d-b}a-c}{p^{d}}}\quad (d\geq b)} \)
\( {\displaystyle {\frac {a}{p^{b}}}-{\frac {c}{p^{d}}}={\frac {a-p^{b-d}c}{p^{b}}}\quad (d<b)} \)
\( {\displaystyle {\frac {a}{p^{b}}}\times {\frac {c}{p^{d}}}={\frac {a\times c}{p^{b+d}}}.} \)
However, the result of dividing one p-adic fraction by another is not necessarily a p-adic fraction.
Additional properties
Because they are closed under addition, subtraction, and multiplication, but not division, the p-adic fractions are a ring but not a field. As a ring, the p-adic fractions are a subring of the rational numbers Q, and an overring of the integers Z. Algebraically, this subring is the localization of the integers Z with respect to the set of powers of p.
The set of all p-adic fractions is dense in the real line: any real number x can be arbitrarily closely approximated by dyadic rationals of the form \( \lfloor 2^i x \rfloor / 2^i \) . Compared to other dense subsets of the real line, such as the rational numbers, the p-adic rationals are in some sense a relatively "small" dense set, which is why they sometimes occur in proofs. (See for instance Urysohn's lemma for the dyadic rationals.)
The p-adic fractions are precisely those numbers possessing finite base-p expansions. Their base-p expansions are not unique; there is one finite and one infinite representation of each p-adic rational other than 0 (ignoring terminal 0s). For example, in binary ( p=2), 0.12 = 0.0111...2 = 1/4 + 1/8 + 1/16 + … = 1/2. Also, 0.112 = 0.10111...2 = 3/4.
Addition modulo 1 forms a group; this is the Prüfer p-group. (This is the same as taking the quotient group of the p-adic rationals by the integers.)
Dual group
Considering only the addition and subtraction operations of the p-adic rationals gives them the structure of an additive abelian group. The dual group of a group consists of its characters, group homomorphisms to the multiplicative group of the complex numbers, and in the spirit of Pontryagin duality the dual group of the additive p-adic rationals can also be viewed as a topological group. It is called the p-adic solenoid and is an example of a solenoid group and of a protorus.
The p-adic rationals are the direct limit of infinite cyclic subgroups of the rational numbers,
\( {\displaystyle \varinjlim \left\{p^{-i}\mathbb {Z} \mid i=0,1,2,\dots \right\}} \)
and their dual group can be constructed as the inverse limit of the unit circle group under the repeated map
\( {\displaystyle \zeta \mapsto \zeta ^{p}.} \)
An element of the p-adic solenoid can be represented as an infinite sequence of complex numbers q0, q1, qp, ..., with the properties that each qi lies on the unit circle and that, for all i > 0, qip = qi − 1. The group operation on these elements multiplies any two sequences componentwise. Each element of the dyadic solenoid corresponds to a character of the p-adic rationals that maps a/pb to the complex number qba. Conversely, every character χ of the p-adic rationals corresponds to the element of the p-adic solenoid given by qi = χ(1/pi).
As a topological space the p-adic solenoid is a solenoid, and an indecomposable continuum.[1]
Related constructions
The surreal numbers are generated by an iterated construction principle which starts by generating all finite dyadic fractions, and then goes on to create new and strange kinds of infinite, infinitesimal and other numbers.
The binary van der Corput sequence is an equidistributed permutation of the positive dyadic rational numbers.
Applications
In metrology
The inch is customarily subdivided in dyadic rather than decimal fractions; similarly, the customary divisions of the gallon into half-gallons, quarts, and pints are dyadic. The ancient Egyptians also used dyadic fractions in measurement, with denominators up to 64.[2]
In music
Time signatures in Western musical notation traditionally consist of dyadic fractions (for example: 2/2, 4/4, 6/8...), although non-dyadic time signatures have been introduced by composers in the twentieth century (for example: 2/dotted quarter note., which would literally mean 2/3⁄8). Non-dyadic time signatures are called irrational in musical terminology, but this usage does not correspond to the irrational numbers of mathematics, because they still consist of ratios of integers. Irrational time signatures in the mathematical sense are very rare, but one example (√42/1) appears in Conlon Nancarrow's Studies for Player Piano.
In computing
As a data type used by computers, floating-point numbers are often defined as integers multiplied by positive or negative powers of two, and thus all numbers that can be represented for instance by binary IEEE floating-point datatypes are dyadic rationals. The same is true for the majority of fixed-point datatypes, which also uses powers of two implicitly in the majority of cases.
See also
Half-integer, a dyadic rational formed by dividing an odd number by two
p-adic number, a number system that extends the p-adic rationals
Decimal fractions or 10-adic rationals
References
Nadler, S. B. Jr. (1973), "The indecomposability of the dyadic solenoid", American Mathematical Monthly, 80 (6): 677–679, doi:10.2307/2319174, JSTOR 2319174.
Curtis, Lorenzo J. (1978), "Concept of the exponential law prior to 1900", American Journal of Physics, 46 (9): 896–906, doi:10.1119/1.11512.
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