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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/2b 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.) Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ...
In mathematics, a rational number (or informally fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ...
In arithmetic, a vulgar fraction (or common fraction) consists of one integer divided by a non-zero integer. ...
In algebra, a vulgar fraction consists of one integer divided by a non-zero integer. ...
In mathematics, a power of two is any of the nonnegative integer powers of the number two; in other words, two times itself a certain number of times. ...
In mathematics, a rational number (or informally fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ...
The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ...
In mathematics, a natural number is either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). The former definition is generally used in number theory, while the latter is preferred in set theory. ...
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 binary numeral system represents numeric values using two symbols, typically 0 and 1. ...
In mathematics, the term dense has at least three different meanings. ...
In mathematics, the real line is simply the set of real numbers. ...
Urysohns lemma in topology states that if X is a normal topological space and A and B are disjoint closed subsets of X, then there exists a continuous function from X into the unit interval [0, 1], f : X → [0, 1], such that f(a) = 0 for all a...
In abstract algebra, a branch of mathematics, a subring is a subset of a ring containing the multiplicative identity, which is itself a ring under the same binary operations. ...
In mathematics, a rational number (or informally fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ...
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. In mathematics, the surreal numbers are a field containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number, and therefore the surreals are algebraically similar to superreal numbers and hyperreal numbers. ...
The ancient Egyptians used Horus-eye notation for dyadic fractions when an exact partition of a hekat unity was desired, when the divisor was not greater than 64. This was done by writing out the quotient (Q/64) from the expression:
(64/64)/n = Q/64 + R/(n*64)
thereby creating a binary or dyadic series.
For example, let n = 3, then
(64/64)/3 = 21/64 + 1/(3*64)
= (16 + 4 + 1)/64 + 1/192 (as a modern translation) = 1/4 + 1/16 + 1/64 + (5/3)*1/320 (obtaining Ahmes' version) = 1/4 + 1/16 + 1/64 + (1 + 2/3)*ro (Ahmes' version) Ahmes' version took the remainder 1/64 term and replaced it by its equal 5/320, as the 2,000 BCE Akhmim (Cairo) Wooden Tablet clearly had set down as a Middle Kingdom practice 350 years earlier, setting ro = 1/320. In other situations ro, a common divisor, was set to a value other than 1/320.
Dyadic solenoid
As an abelian group the dyadic rationals are the direct limit of infinite cyclic subgroups In mathematics, an abelian group, also called a commutative group, is a group (G, *) such that a * b = b * a for all a and b in G. Abelian groups are named after Niels Henrik Abel. ...
In 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 which can be used in any category. ...
In mathematics, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a generator of the group) such that every element of the group is a power of a. ...
- 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 In mathematics, in particular in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform. ...
In 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. ...
Illustration of a unit circle. ...
- ζ → ζ2.
The resulting topological group D is called the dyadic solenoid (see solenoid group). As a topological space it is an indecomposable continuum. In 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. ...
In mathematics, for a given prime number p, the p-adic solenoid is the topological group defined as inverse limit of the inverse system (Si, qi) where i runs over natural numbers, and each Si is a circle, and qi wraps the circle Si+1 p times around the circle...
In point-set topology, an indecomposable continuum is a continuum that is not decomposable. ...
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