Dyadic rational
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.
Contents
 Use in measurement 1
 Arithmetic 2
 Additional properties 3
 Dual group 4
 Related constructions 5
 In music 6
 See also 7
 References 8
Use in measurement
The inch is customarily subdivided in dyadic rather than decimal fractions; similarly, the customary divisions of the gallon into halfgallons, quarts, and pints are dyadic. The ancient Egyptians also used dyadic fractions in measurement, with denominators up to 1/64.^{[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^{db}a+c}{2^d} \quad (d\ge b)
 \frac{a}{2^b}\frac{c}{2^d}=\frac{2^{db}ac}{2^d} \quad (d\ge b)
 \frac{a}{2^b}\frac{c}{2^d}=\frac{a2^{bd}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.
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 q_{0}, q_{1}, q_{2}, ..., with the properties that each q_{i} lies on the unit circle and that, for all i > 0, q_{i}^{2} = 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(1/2^{i}).
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, although nondyadic time signatures have been introduced by composers in the twentieth century. (Nondyadic time signatures are often called "irrational" by musicians, but this is a misnomer because they still consist of ratios of integers.)
See also
 Halfinteger, a dyadic rational formed by dividing an odd number by two
 2adic number, a number system that extends the dyadic rationals
References

