Dyadic rational

Dyadic rationals in the interval from 0 to 1.

In mathematics, a dyadic fraction or dyadic rational is a rational number whose denominator is a power of two, i.e., a number of the form $\frac{a}{2^b}$ where a is an integer and b is a natural number; for example, 1/2 or 3/8, but not 1/3. These are precisely the numbers whose binary expansion is finite.

Use in measurement

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.^[1]

Arithmetic

The sum, product, or difference of any two dyadic fractions is itself another dyadic fraction:

$\frac{a}{2^b}+\frac{c}{2^d}=\frac{2^{d-b}a+c}{2^d} \quad (d\ge b)$

$\frac{a}{2^b}-\frac{c}{2^d}=\frac{2^{d-b}a-c}{2^d} \quad (d\ge b)$

$\frac{a}{2^b}-\frac{c}{2^d}=\frac{a-2^{b-d}c}{2^b} \quad (d< b)$

$\frac{a}{2^b}\times \frac{c}{2^d} = \frac{ a \times c}{2^{b+d}}.$

However, the result of dividing one dyadic fraction by another is not necessarily a dyadic fraction.

Addition modulo 1 forms a group; this is the Prüfer 2-group.

Additional properties

Because they are closed under addition, subtraction, and multiplication, but not division, the dyadic fractions form 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 two.

The set of all dyadic 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 dyadic rationals are in some sense a relatively "small" dense set, which is why they sometimes occur in proofs. (See for instance Urysohn's lemma.)

Dual group

Considering only the addition and subtraction operations of the dyadic 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 dyadic rationals can also be viewed as a topological group. It is called the dyadic solenoid and is an example of a solenoid group and of a protorus.

The dyadic rationals are the direct limit of infinite cyclic subgroups of the rational numbers,

$\varinjlim \left\{2^{-i}\mathbb{Z}\mid i = 0, 1, 2, \dots \right\}$

and their dual group can be constructed as the inverse limit of the unit circle group under the repeated squaring map

$\zeta\mapsto\zeta^2.$

An element of the dyadic solenoid can be represented as an infinite sequence of complex numbers q₀, q₁, q₂, ..., with the properties that each q_i lies on the unit circle and that, for all i > 0, q_i² = q_i − 1. The group operation on these elements multiplies any two sequences componentwise. Each element of the dyadic solenoid corresponds to a character of the dyadic rationals that maps a/2^b to the complex number q_b^a. Conversely, every character χ of the dyadic rationals corresponds to the element of the dyadic solenoid given by q_i = χ(1/2ⁱ).

As a topological space the dyadic solenoid is a solenoid, and an indecomposable continuum.^[2]

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.

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/., which would literally mean 2/⁸⁄₃, or 9/14). 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 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
• 2-adic number, a number system that extends the dyadic rationals

References

1. 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.
2. Nadler, S. B., Jr. (1973), "The indecomposability of the dyadic solenoid", American Mathematical Monthly 80 (6): 677–679, doi:10.2307/2319174, JSTOR 2319174.