|
Davenport–Schmidt Theorem
In mathematics, specifically the area of Diophantine approximation, the Davenport–Schmidt theorem tells us how well a certain kind of real number can be approximated by another kind. Specifically it tells us that we can get a good approximation to irrational numbers that are not quadratic by using either quadratic irrationals or simply rational numbers. It is named after Harold Davenport and Wolfgang M. Schmidt. Statement Given a number α which is either rational or a quadratic irrational, we can find unique integers ''x'', ''y'', and ''z'' such that ''x'', ''y'', and ''z'' are not all zero, the first non-zero one among them is positive, they are relatively prime, and we have :x\alpha^2 +y\alpha +z=0. If α is a quadratic irrational we can take ''x'', ''y'', and ''z'' to be the coefficients of its minimal polynomial. If α is rational we will have ''x'' = 0. With these integers uniquely determined for each such α we can define the ''height'' of α to be :H(\a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diophantine Approximation
In number theory, the study of Diophantine approximation deals with the approximation of real numbers by rational numbers. It is named after Diophantus of Alexandria. The first problem was to know how well a real number can be approximated by rational numbers. For this problem, a rational number ''a''/''b'' is a "good" approximation of a real number ''α'' if the absolute value of the difference between ''a''/''b'' and ''α'' may not decrease if ''a''/''b'' is replaced by another rational number with a smaller denominator. This problem was solved during the 18th century by means of continued fractions. Knowing the "best" approximations of a given number, the main problem of the field is to find sharp upper and lower bounds of the above difference, expressed as a function of the denominator. It appears that these bounds depend on the nature of the real numbers to be approximated: the lower bound for the approximation of a rational number by another rational number is larger than ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and more generally in all mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives. The set of real numbers is denoted or \mathbb and is sometimes called "the reals". The adjective ''real'' in this context was introduced in the 17th century by René Descartes to distinguish real numbers, associated with physical reality, from imaginary numbers (such as the square roots of ), which seemed like a theoretical contrivance unrelated to physical reality. The real numbers include the rational numbers, such as the integer and the fraction . The rest of the real number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quadratic Irrational
In mathematics, a quadratic irrational number (also known as a quadratic irrational, a quadratic irrationality or quadratic surd) is an irrational number that is the solution to some quadratic equation with rational coefficients which is irreducible over the rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...s. 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 Algebraic number#Properties, degree 2, and can therefore be expressed as :, for integers ; with , and non-zero, and with Square-free in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rational Number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rational numbers, also referred to as "the rationals", the field of rationals or the field of rational numbers is usually denoted by boldface , or blackboard bold \mathbb. A rational number is a real number. The real numbers that are rational are those whose decimal expansion either terminates after a finite number of digits (example: ), or eventually begins to repeat the same finite sequence of digits over and over (example: ). This statement is true not only in base 10, but also in every other integer base, such as the binary and hexadecimal ones (see ). A real number that is not rational is called irrational. Irrational numbers include , , , and . Since the set of rational numbers is countable, and the set of real numbers is uncountable ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harold Davenport
Harold Davenport FRS (30 October 1907 – 9 June 1969) was an English mathematician, known for his extensive work in number theory. Early life Born on 30 October 1907 in Huncoat, Lancashire, Davenport was educated at Accrington Grammar School, the University of Manchester (graduating in 1927), and Trinity College, Cambridge. He became a research student of John Edensor Littlewood, working on the question of the distribution of quadratic residues. First steps in research The attack on the distribution question leads quickly to problems that are now seen to be special cases of those on local zeta-functions, for the particular case of some special hyperelliptic curves such as Y^2 = X(X-1)(X-2)\ldots (X-k). Bounds for the zeroes of the local zeta-function immediately imply bounds for sums \sum \chi(X(X-1)(X-2)\ldots (X-k)), where χ is the Legendre symbol '' modulo'' a prime number ''p'', and the sum is taken over a complete set of residues mod ''p''. In the light of this co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wolfgang M
Wolfgang is a German male given name traditionally popular in Germany, Austria and Switzerland. The name is a combination of the Old High German words ''wolf'', meaning "wolf", and ''gang'', meaning "path", "journey", "travel". Besides the regular "wolf", the first element also occurs in Old High German as the combining form "-olf". The earliest reference of the name being used was in the 8th century. The name was also attested as "Vulfgang" in the Reichenauer Verbrüderungsbuch in the 9th century. The earliest recorded famous bearer of the name was a tenth-century Saint Wolfgang of Regensburg. Due to the lack of conflict with the pagan reference in the name with Catholicism, it is likely a much more ancient name whose meaning had already been lost by the tenth century. Grimm (''Teutonic Mythology'' p. 1093) interpreted the name as that of a hero in front of whom walks the "wolf of victory". A Latin gloss by Arnold of St Emmeram interprets the name as ''Lupambulus''.E. Förs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minimal Polynomial (field Theory)
In field theory, a branch of mathematics, the minimal polynomial of an element of a field is, roughly speaking, the polynomial of lowest degree having coefficients in the field, such that is a root of the polynomial. If the minimal polynomial of exists, it is unique. The coefficient of the highest-degree term in the polynomial is required to be 1, and the type for the remaining coefficients could be integers, rational numbers, real numbers, or others. More formally, a minimal polynomial is defined relative to a field extension and an element of the extension field . The minimal polynomial of an element, if it exists, is a member of , the ring of polynomials in the variable with coefficients in . Given an element of , let be the set of all polynomials in such that . The element is called a root or zero of each polynomial in . The set is so named because it is an ideal of . The zero polynomial, all of whose coefficients are 0, is in every since for all and . This ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thue–Siegel–Roth Theorem
In mathematics, Roth's theorem is a fundamental result in diophantine approximation to algebraic numbers. It is of a qualitative type, stating that algebraic numbers cannot have many rational number approximations that are 'very good'. Over half a century, the meaning of ''very good'' here was refined by a number of mathematicians, starting with Joseph Liouville in 1844 and continuing with work of , , , and . Statement Roth's theorem states that every irrational algebraic number \alpha has approximation exponent equal to 2. This means that, for every \varepsilon>0, the inequality :\left, \alpha - \frac\ \frac with C(\alpha,\varepsilon) a positive number depending only on \varepsilon>0 and \alpha. Discussion The first result in this direction is Liouville's theorem on approximation of algebraic numbers, which gives an approximation exponent of ''d'' for an algebraic number α of degree ''d'' ≥ 2. This is already enough to demonstrate the existence of transcen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Effective Results In Number Theory
For historical reasons and in order to have application to the solution of Diophantine equations, results in number theory have been scrutinised more than in other branches of mathematics to see if their content is effectively computable. Where it is asserted that some list of integers is finite, the question is whether in principle the list could be printed out after a machine computation. Littlewood's result An early example of an ineffective result was J. E. Littlewood's theorem of 1914, that in the prime number theorem the differences of both ψ(''x'') and π(''x'') with their asymptotic estimates change sign infinitely often. In 1933 Stanley Skewes obtained an effective upper bound for the first sign change, now known as Skewes' number. In more detail, writing for a numerical sequence ''f'' (''n''), an ''effective'' result about its changing sign infinitely often would be a theorem including, for every value of ''N'', a value ''M'' > ''N'' such that ''f'' ('' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
