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_numeral_system 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 rational number. 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−''n''''Z'' 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. Fractions
The question about ring properties: better dealt with on the localization of a ring page? The question certainly does not belong in the article so I have deleted ''what properties does this ring have?''. --User:Henrygb 23:09, 30 Apr 2005 (UTC) ---- I have added stuff about the dyadic solenoid as fitting well here, but I'm somewhat unclear about a couple of points. On the general level, Pontryagin duality interchanging direct and inverse limits: this would be OK, but I guess there is a point here, namely that the dyadic rationals must be taken with the discrete topology, and certainly not as a subspace of the real line. Also, te inverse limit could be taken either as indexed by N, or bi-infinite. User:Charles Matthews 11:14, 13 Sep 2004 (UTC)
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Words begining with Dyadic_rational:
Dyadic_rational
Dyadic_rational
Dyadic_rational_number
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_numeral_system 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 rational number. 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−''n''''Z'' 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. Fractions
Dyadic rational
The question about ring properties: better dealt with on the localization of a ring page? The question certainly does not belong in the article so I have deleted ''what properties does this ring have?''. --User:Henrygb 23:09, 30 Apr 2005 (UTC) ---- I have added stuff about the dyadic solenoid as fitting well here, but I'm somewhat unclear about a couple of points. On the general level, Pontryagin duality interchanging direct and inverse limits: this would be OK, but I guess there is a point here, namely that the dyadic rationals must be taken with the discrete topology, and certainly not as a subspace of the real line. Also, te inverse limit could be taken either as indexed by N, or bi-infinite. User:Charles Matthews 11:14, 13 Sep 2004 (UTC)
See other meanings of words starting from letter:
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DA | DB | DC | DE | DF | DG | DH | DI | DJ | DK | DL | DM | DN | DO | DP | DR | DS | DT | DU | DW | DX | DY | DZ |Words begining with Dyadic_rational:
Dyadic_rational
Dyadic_rational
Dyadic_rational_number
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