Dyadic rational
These numbers are important in computer science because they are the only ones with finite binary representations.
Dyadic rationals also have applications in weights and measures, musical time signatures, and early mathematics education.
Many traditional systems of weights and measures are based on the idea of repeated halving, which produces dyadic rationals when measuring fractional amounts of units.
The inch is customarily subdivided in dyadic rationals rather than using a decimal subdivision.
[1] The customary divisions of the gallon into half-gallons, quarts, pints, and cups are also dyadic.
[3] Similarly, systems of weights from the Indus Valley civilisation are for the most part based on repeated halving; anthropologist Heather M.-L. Miller writes that "halving is a relatively simple operation with beam balances, which is likely why so many weight systems of this time period used binary systems".
[5] In particular, as a data type used by computers, floating-point numbers are often defined as integers multiplied by positive or negative powers of two.
[6] The same is true for fixed-point datatypes, which also use powers of two implicitly in the majority of cases.
[9][10][11] Generating a random variable from random bits, in a fixed amount of time, is possible only when the variable has finitely many outcomes whose probabilities are all dyadic rational numbers.
[12] Time signatures in Western musical notation traditionally are written in a form resembling fractions (for example: 22, 44, or 68),[13] although the horizontal line of the musical staff that separates the top and bottom number is usually omitted when writing the signature separately from its staff.
[16] This stage of development of the concept of fractions has been called "algorithmic halving".
In contrast, addition and subtraction of more general fractions involves integer multiplication and factorization to reach a common denominator.
[19] Another equivalent way of defining the dyadic rationals is that they are the real numbers that have a terminating binary representation.
Every real number can be arbitrarily closely approximated by dyadic rationals.
[22] More strongly, this set is uniformly dense, in the sense that the dyadic rationals with denominator
[9] The dyadic rationals are precisely those numbers possessing finite binary expansions.
[9] Their binary expansions are not unique; there is one finite and one infinite representation of each dyadic rational other than 0 (ignoring terminal 0s).
[9][25] The dyadic rationals are the only numbers whose binary expansions are not unique.
[9] Because they are closed under addition, subtraction, and multiplication, but not division, the dyadic rationals are a ring but not a field.
, meaning that it can be generated by evaluating polynomials with integer coefficients, at the argument 1/2.
[28] Algebraically, this ring is the localization of the integers with respect to the set of powers of two.
Embedding the dyadic rationals into the 2-adic numbers does not change the arithmetic of the dyadic rationals, but it gives them a different topological structure than they have as a subring of the real numbers.
Every 2-adic number can be decomposed into the sum of a 2-adic integer and a dyadic rational; in this sense, the dyadic rationals can represent the fractional parts of 2-adic numbers, but this decomposition is not unique.
It is called the dyadic solenoid, and is isomorphic to the topological product of the real numbers and 2-adic numbers, quotiented by the diagonal embedding of the dyadic rationals into this product.
[26] Similarly, the dyadic rationals parameterize the discontinuities in the boundary between stable and unstable points in the parameter space of the Hénon map.
[36] The set of piecewise linear homeomorphisms from the unit interval to itself that have power-of-2 slopes and dyadic-rational breakpoints forms a group under the operation of function composition.
Defining real numbers in this way allows many of the basic results of mathematical analysis to be proven within a restricted theory of second-order arithmetic called "feasible analysis" (BTFA).
[39] The surreal numbers are generated by an iterated construction principle which starts by generating all finite dyadic rationals, and then goes on to create new and strange kinds of infinite, infinitesimal and other numbers.
[40] This number system is foundational to combinatorial game theory, and dyadic rationals arise naturally in this theory as the set of values of certain combinatorial games.
[41][42][19] The fusible numbers are a subset of the dyadic rationals, the closure of the set
