Dyadic rational

In mathematics, a dyadic fraction or dyadic rational is a rational number that when written as a vulgar fraction has a denominator that is a power of two, i.e., a rational number of the form a/2^b where a is an integer and b is a natural number. For example, 1/2 or 3/8 but not 1/3. (Like fractions of an inch as commonly used in the US, for instance.)

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.

Dyadic solenoid

As an abelian group the dyadic rationals are the direct limit of infinite cyclic subgroups

2⁻ⁿZ

for n = 0, 1, 2, ... . In the spirit of Pontryagin duality, there is a dual object, namely the inverse limit of the unit circle group under the repeated squaring map

ζ → ζ².

The resulting topological group D is called the dyadic solenoid. As a topological space it is an indecomposable continuum.

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