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 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 1/64.
However, the result of dividing one dyadic fraction by another is not necessarily a dyadic fraction.
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 . 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.)
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.
An element of the dyadic solenoid can be represented as an infinite sequence of complex numbers q0, q1, q2, ..., with the properties that each qi lies on the unit circle and that, for all i > 0, qi2 = 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 dyadic rationals that maps a/2b to the complex number qba. Conversely, every character χ of the dyadic rationals corresponds to the element of the dyadic solenoid given by qi = χ(1/2i).
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.
Time signatures in Western musical notation traditionally consist of dyadic fractions, although non-dyadic time signatures have been introduced by composers in the twentieth century. (Non-dyadic time signatures are often called "irrational" by musicians, but this is a misnomer because they still consist of ratios of integers.)
- Half-integer, a dyadic rational formed by dividing an odd number by two
- 2-adic number, a number system that extends the dyadic rationals