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In mathematics, a dyadic fraction or dyadic rational is a rational number whose denominator is a power of two, i.e., a 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. These are precisely the numbers whose binary expansion is finite. ## Dyadic solenoid

As an additive abelian group the dyadic rationals are the direct limit of infinite cyclic subgroups of the rational numbers,
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## References

The inch is customarily subdivided in dyadic rather than decimal fractions; similarly, the customary divisions of the gallon into half-gallons, quarts, and pints are dyadic. The ancient Egyptians also used dyadic fractions in measurement, with denominators up to 1/64, using a notation based on the Eye of Horus (see, e.g., Curtis).

The set of all dyadic fractions is dense in the real line: any real number x can be arbitrarily closely approximated by dyadic rationals of the form $lfloor\; 2^i\; x\; rfloor\; /\; 2^i$. Compared to other dense subsets of the real line, such as the rational numbers, the dyadic rationals are in some sense a relatively "small" dense set, which is why they sometimes occur in proofs. (See for instance Urysohn's lemma.)

The sum, product, or difference of any two dyadic fractions is itself another dyadic fraction:

- $frac\{a\}\{2^b\}+frac\{c\}\{2^d\}=frac\{2^\{d-b\}a+c\}\{2^d\}\; quad\; (dge\; b)$

- $frac\{a\}\{2^b\}-frac\{c\}\{2^d\}=frac\{2^\{d-b\}a-c\}\{2^d\}\; quad\; (dge\; b)$

- $frac\{a\}\{2^b\}-frac\{c\}\{2^d\}=frac\{a-2^\{b-d\}c\}\{2^b\}\; quad\; (d<\; b)$

- $frac\{a\}\{2^b\}times\; frac\{c\}\{2^d\}\; =\; frac\{\; a\; times\; c\}\{2^\{b+d\}\}.$

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.

- $varinjlim\; left\{2^\{-i\}mathbb\{Z\}mid\; i\; =\; 0,\; 1,\; 2,\; dots\; right\}$

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

- $zetamapstozeta^2.$

An element of the dyadic solenoid can be represented as an infinite sequence of complex numbers q_{0}, q_{1}, q_{2}, ..., with the properties that each q_{i} lies on the unit circle and that, for all i > 0, q_{i}^{2} = q_{i-1}. The group operation on these elements multiplies any two sequences componentwise.

As a topological space the dyadic solenoid is an indecomposable continuum.

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Last updated on Tuesday June 17, 2008 at 05:19:36 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Tuesday June 17, 2008 at 05:19:36 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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