Dyadic rational

Dyadic rationals in the interval from 0 to 1

In mathematics, a dyadic rational or binary rational is a number that can be expressed as a fraction whose denominator is a power of two. For example, 1/2, 3/2, and 3/8 are dyadic rationals, but 1/3 is not. 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. They can accurately approximate any real number.

The sum, difference, or product of any two dyadic rational numbers is another dyadic rational number, given by a simple formula. However, division of one dyadic rational number by another does not always produce a dyadic rational result. Mathematically, this means that the dyadic rational numbers form a ring, lying between the ring of integers and the field of rational numbers. This ring may be denoted $\mathbb {Z} [{\tfrac {1}{2}}]$.

In advanced mathematics, the dyadic rational numbers are central to the constructions of the dyadic solenoid, Minkowski's question-mark function, Daubechies wavelets, Thompson's group, Prüfer 2-group, surreal numbers, and fusible numbers. These numbers are order-isomorphic to the rational numbers; they form a subsystem of the 2-adic numbers as well as of the reals, and can represent the fractional parts of 2-adic numbers. Functions from natural numbers to dyadic rationals have been used to formalize mathematical analysis in reverse mathematics.

Applications

In measurement

Kitchen weights measuring dyadic fractions of a pound from 2 lb down to 1/64 lb (1/4 oz)

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. The customary divisions of the gallon into half-gallons, quarts, pints, and cups are also dyadic. The ancient Egyptians used dyadic rationals in measurement, with denominators up to 64. 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".

In computing

Dyadic rationals are central to computer science as a type of fractional number that many computers can manipulate directly. 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. The numbers that can be represented precisely in a floating-point format, such as the IEEE floating-point datatypes, are called its representable numbers. For most floating-point representations, the representable numbers are a subset of the dyadic rationals. The same is true for fixed-point datatypes, which also use powers of two implicitly in the majority of cases. Because of the simplicity of computing with dyadic rationals, they are also used for exact real computing using interval arithmetic, and are central to some theoretical models of computable numbers.