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Quadratic Irrational Numbers
In mathematics, a quadratic irrational number (also known as a quadratic irrational or quadratic surd) is an irrational number that is the solution to some quadratic equation with rational coefficients which is irreducible over the rational numbers. Since fractions in the coefficients of a quadratic equation can be cleared by multiplying both sides by their least common denominator, a quadratic irrational is an irrational root of some quadratic equation with integer coefficients. The quadratic irrational numbers, a subset of the complex numbers, are algebraic numbers of degree 2, and can therefore be expressed as :, for integers ; with , and non-zero, and with square-free. When is positive, we get real quadratic irrational numbers, while a negative gives complex quadratic irrational numbers which are not real numbers. This defines an injection from the quadratic irrationals to quadruples of integers, so their cardinality is at most countable; since on the other hand every ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, or , involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorization, factorized as a product of primes that is unique up to their order. The property of being prime is called primality. A simple but slow primality test, method of checking the primality of a given number , called trial division, tests whether is a multiple of any integer between 2 and . Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance of error ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gauss–Kuzmin–Wirsing Operator
In mathematics, the Gauss–Kuzmin–Wirsing operator is the transfer operator of the Gauss map that takes a positive number to the fractional part of its reciprocal. (This is not the same as the Gauss map in differential geometry.) It is named after Carl Gauss, Rodion Kuzmin, and Eduard Wirsing. It occurs in the study of continued fractions; it is also related to the Riemann zeta function. Relationship to the maps and continued fractions The Gauss map The Gauss function (map) ''h'' is : :h(x)=1/x-\lfloor 1/x \rfloor. where \lfloor 1/x \rfloor denotes the floor function. It has an infinite number of jump discontinuities at ''x'' = 1/''n'', for positive integers ''n''. It is hard to approximate it by a single smooth polynomial. Operator on the maps The Gauss–Kuzmin–Wirsing operator G acts on functions f as : fx) = \int_0^1 \delta(x-h(y)) f(y) \, dy = \sum_^\infty \frac f \left(\frac \right). it has the fixed point \rho(x) = \frac, unique up to scaling, whi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dyadic Transformation
The dyadic transformation (also known as the dyadic map, bit shift map, 2''x'' mod 1 map, Bernoulli map, doubling map or sawtooth map) is the mapping (i.e., recurrence relation) : T: , 1) \to [0, 1)^\infty : x \mapsto (x_0, x_1, x_2, \ldots) (where [0, 1)^\infty is the set of sequences from [0, 1)) produced by the rule : x_0 = x : \text n \ge 0,\ x_ = (2 x_n) \bmod 1. Equivalently, the dyadic transformation can also be defined as the iterated function map of the piecewise linear function : T(x)=\begin2x & 0 \le x < \frac \\2x-1 & \frac \le x < 1. \end The name ''bit shift map'' arises because, if the value of an iterate is written in binary notation, the next iterate is obtained by shifting the binary point one bit to the right, and if the bit to the left of the new binary point is a "one", replacing it with a zero. The dyadic transformation provides an example of ho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Periodic Orbit
In mathematics, in the study of iterated functions and dynamical systems, a periodic point of a function is a point which the system returns to after a certain number of function iterations or a certain amount of time. Iterated functions Given a mapping from a set into itself, :f: X \to X, a point in is called periodic point if there exists an >0 so that :\ f_n(x) = x where is the th iterate of . The smallest positive integer satisfying the above is called the ''prime period'' or ''least period'' of the point . If every point in is a periodic point with the same period , then is called ''periodic'' with period (this is not to be confused with the notion of a periodic function). If there exist distinct and such that :f_n(x) = f_m(x) then is called a preperiodic point. All periodic points are preperiodic. If is a diffeomorphism of a differentiable manifold, so that the derivative f_n^\prime is defined, then one says that a periodic point is ''hyperbolic'' if :, f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minkowski's Question Mark Function
In mathematics, Minkowski's question-mark function, denoted , is a Function (mathematics), function with unusual fractal properties, defined by Hermann Minkowski in 1904. It maps quadratic irrational numbers to rational numbers on the unit interval, via an expression relating the continued fraction expansions of the quadratics to the binary expansions of the rationals, given by Arnaud Denjoy in 1938. It also maps rational numbers to dyadic rationals, as can be seen by a recursive definition closely related to the Stern–Brocot tree. Definition and intuition One way to define the question-mark function involves the correspondence between two different ways of representing fractional numbers using finite or infinite binary sequences. Most familiarly, a string of 0s and 1s with a single point mark ".", like "11.0010010000111111..." can be interpreted as the binary representation of a number. In this case this number is 2+1+\frac18+\frac1+\cdots=\pi. There is a different way of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
