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Tetranacci Number
In mathematics, the Fibonacci numbers form a sequence defined recursively by: :F_n = \begin 0 & n = 0 \\ 1 & n = 1 \\ F_ + F_ & n > 1 \end That is, after two starting values, each number is the sum of the two preceding numbers. The Fibonacci sequence has been studied extensively and generalized in many ways, for example, by starting with other numbers than 0 and 1, by adding more than two numbers to generate the next number, or by adding objects other than numbers. Extension to negative integers Using F_ = F_n - F_, one can extend the Fibonacci numbers to negative integers. So we get: :... −8, 5, −3, 2, −1, 1, 0, 1, 1, 2, 3, 5, 8, ... and F_ = (-1)^ F_n. See also Negafibonacci coding. Extension to all real or complex numbers There are a number of possible generalizations of the Fibonacci numbers which include the real numbers (and sometimes the complex numbers) in their domain. These each involve the golden ratio , and are based on Binet's formula :F_n = \frac. The ana ... [...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] |
Abelian Group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian group may be viewed as a generalization of these examples. Abelian groups are named after the Norwegian mathematician Niels Henrik Abel. The concept of an abelian group underlies many fundamental algebraic structures, such as fields, rings, vector spaces, and algebras. The theory of abelian groups is generally simpler than that of their non-abelian counterparts, and finite abelian groups are very well understood and fully classified. Definition An abelian group is a set A, together with an operation ･ , that combines any two elements a and b of A to form another element of A, denoted a \cdot b. The sym ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Silver Ratio
In mathematics, the silver ratio is a geometrical aspect ratio, proportion with exact value the positive polynomial root, solution of the equation The name ''silver ratio'' results from analogy with the golden ratio, the positive solution of the equation Although its name is recent, the silver ratio (or silver mean) has been studied since ancient times because of its connections to the square root of 2, almost-isosceles Pythagorean triple#Special cases and related equations, Pythagorean triples, square triangular numbers, Pell numbers, the octagon, and six polyhedron, polyhedra with octahedral symmetry. Definition If the ratio of two quantities is proportionate to the sum of two and their reciprocal ratio, they are in the silver ratio: \frac =\frac The ratio \frac is here denoted Substituting a=\sigma b \, in the second fraction, \sigma =\frac. It follows that the silver ratio is the positive solution of quadratic equation \sigma^2 -2\sigma -1 =0. The quadratic for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Root Of A Polynomial
In mathematics, a zero (also sometimes called a root) of a real-, complex-, or generally vector-valued function f, is a member x of the domain of f such that f(x) ''vanishes'' at x; that is, the function f attains the value of 0 at x, or equivalently, x is a solution to the equation f(x) = 0. A "zero" of a function is thus an input value that produces an output of 0. A root of a polynomial is a zero of the corresponding polynomial function. The fundamental theorem of algebra shows that any non-zero polynomial has a number of roots at most equal to its degree, and that the number of roots and the degree are equal when one considers the complex roots (or more generally, the roots in an algebraically closed extension) counted with their multiplicities. For example, the polynomial f of degree two, defined by f(x)=x^2-5x+6=(x-2)(x-3) has the two roots (or zeros) that are 2 and 3. f(2)=2^2-5\times 2+6= 0\textf(3)=3^2-5\times 3+6=0. If the function maps real numbers to real n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metallic Mean
The metallic mean (also metallic ratio, metallic constant, or noble mean) of a natural number is a positive real number, denoted here S_n, that satisfies the following equivalent characterizations: * the unique positive real number x such that x=n+\frac 1x * the positive root of the quadratic equation x^2-nx-1=0 * the number \frac2 = \frac2 * the number whose expression as a continued fraction is *: [n;n,n,n,n,\dots] = n + \cfrac Metallic means are (successive) derivations of the golden ratio, golden (n=1) and silver ratios (n=2), and share some of their interesting properties. The term "bronze ratio" (n=3) (Cf. Golden Age and Olympic Medals) and even metals such as copper (n=4) and nickel (n=5) are occasionally found in the literature.This name appears to have originated from de Spinadel's paper. In terms of algebraic number theory, the metallic means are exactly the real quadratic integers that are greater than 1 and have -1 as their quadratic integer#Norm and conjugation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
