Dyadic rational

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.

In advanced mathematics[edit]

Algebraic structure[edit]

Because they are closed under addition, subtraction, and multiplication, but not division, the dyadic rationals are a ring but not a field.^[26] The ring of dyadic rationals may be denoted $\mathbb {Z} [{\tfrac {1}{2}}]$ , meaning that it can be generated by evaluating polynomials with integer coefficients, at the argument 1/2.^[27] As a ring, the dyadic rationals are a subring of the rational numbers, and an overring of the integers.^[28] Algebraically, this ring is the localization of the integers with respect to the set of powers of two.^[29]

As well as forming a subring of the real numbers, the dyadic rational numbers form a subring of the 2-adic numbers, a system of numbers that can be defined from binary representations that are finite to the right of the binary point but may extend infinitely far to the left. The 2-adic numbers include all rational numbers, not just the dyadic rationals. 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. As they do in the reals, the dyadic rationals form a dense subset of the 2-adic numbers,^[30] and are the set of 2-adic numbers with finite binary expansions. 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.^[31]

Addition of dyadic rationals modulo 1 (the quotient group $\mathbb {Z} [{\tfrac {1}{2}}]/\mathbb {Z}$ of the dyadic rationals by the integers) forms the Prüfer 2-group.^[32]

Dyadic solenoid[edit]

Considering only the addition and subtraction operations of the dyadic rationals gives them the structure of an additive abelian group. Pontryagin duality is a method for understanding abelian groups by constructing dual groups, whose elements are characters of the original group, group homomorphisms to the multiplicative group of the complex numbers, with pointwise multiplication as the dual group operation. The dual group of the additive dyadic rationals, constructed in this way, can also be viewed as a topological group. 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.^[30] It is an example of a protorus, a solenoid, and an indecomposable continuum.^[33]