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These are precisely the numbers whose binary expansion is finite. The set of all dyadic fractions is dense in the real line; it is a rather "small" dense set, which is why it sometimes occurs in proofs, see for instance Urysohn's Lemma. The dyadic fractions form a subring of Q.
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 ancient Egyptians used Horus-eye notation for dyadic fractions.
As an abelian group the dyadic rationals are the direct limitIn mathematics, the direct limit (also called the inductive limit is a general method of taking limits of "directed families of objects". We will first give the definition for algebraic structures like groups and modules, and then the general definition w of infinite cyclic subgroups
for n = 0, 1, 2, ... . In the spirit of Pontryagin dualityTopological groups Harmonic analysis Theorems In mathematics, in particular in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform. It places in a unified context a number of, there is a dual object, namely the inverse limitIn mathematics, the inverse limit (also called the projective limit is a construction which allows one to "glue together" several related objects, the precise matter of the gluing process being specified by morphisms between the objects. Inverse limits ca of the unit circleIllustration of a unit circle. t is an angle measure. In mathematics, a unit circle is a circle with unit radius, i. a circle whose radius is 1. Frequently, especially in trigonometry, "the" unit circle is the circle of radius 1 centered at the origin (0, group under the repeated squaring map
The resulting topological groupIn mathematics, a topological group ''G is a group that is also a topological space such that the group multiplication G × G → G and the inverse operation G → G are continuous maps. Here, G × G is viewed as a topological space by using the produ D is called the dyadic solenoid. As a topological space it is an indecomposable continuum .
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