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Lindemann–Weierstrass Theorem
In transcendental number theory, the Lindemann–Weierstrass theorem is a result that is very useful in establishing the transcendence of numbers. It states the following: In other words, the extension field \mathbb(e^, \dots, e^) has transcendence degree over \mathbb. An equivalent formulation , is the following: This equivalence transforms a linear relation over the algebraic numbers into an algebraic relation over \mathbb by using the fact that a symmetric polynomial whose arguments are all conjugates of one another gives a rational number. The theorem is named for Ferdinand von Lindemann and Karl Weierstrass. Lindemann proved in 1882 that is transcendental for every non-zero algebraic number thereby establishing that is transcendental (see below). Weierstrass proved the above more general statement in 1885. The theorem, along with the Gelfond–Schneider theorem, is extended by Baker's theorem, and all of these would be further generalized by Schanuel's conjecture. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transcendental Number Theory
Transcendental number theory is a branch of number theory that investigates transcendental numbers (numbers that are not solutions of any polynomial equation with rational coefficients), in both qualitative and quantitative ways. Transcendence The fundamental theorem of algebra tells us that if we have a non-constant polynomial with rational coefficients (or equivalently, by clearing denominators, with integer coefficients) then that polynomial will have a root in the complex numbers. That is, for any non-constant polynomial P with rational coefficients there will be a complex number \alpha such that P(\alpha)=0. Transcendence theory is concerned with the converse question: given a complex number \alpha, is there a polynomial P with rational coefficients such that P(\alpha)=0? If no such polynomial exists then the number is called transcendental. More generally the theory deals with algebraic independence of numbers. A set of numbers is called algebraically independent ove ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rational Integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language of mathematics, the set of integers is often denoted by the boldface or blackboard bold \mathbb. The set of natural numbers \mathbb is a subset of \mathbb, which in turn is a subset of the set of all rational numbers \mathbb, itself a subset of the real numbers \mathbb. Like the natural numbers, \mathbb is countably infinite. An integer may be regarded as a real number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , and are not. The integers form the smallest group and the smallest ring containing the natural numbers. In algebraic number theory, the integers are sometimes qualified as rational integers to distinguish them from the more general algebraic integers. In ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language of mathematics, the set of integers is often denoted by the boldface or blackboard bold \mathbb. The set of natural numbers \mathbb is a subset of \mathbb, which in turn is a subset of the set of all rational numbers \mathbb, itself a subset of the real numbers \mathbb. Like the natural numbers, \mathbb is countably infinite. An integer may be regarded as a real number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , and are not. The integers form the smallest group and the smallest ring containing the natural numbers. In algebraic number theory, the integers are sometimes qualified as rational integers to distinguish them from the more general algebraic integers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unit Disc
In mathematics, the open unit disk (or disc) around ''P'' (where ''P'' is a given point in the plane), is the set of points whose distance from ''P'' is less than 1: :D_1(P) = \.\, The closed unit disk around ''P'' is the set of points whose distance from ''P'' is less than or equal to one: :\bar D_1(P)=\.\, Unit disks are special cases of disks and unit balls; as such, they contain the interior of the unit circle and, in the case of the closed unit disk, the unit circle itself. Without further specifications, the term ''unit disk'' is used for the open unit disk about the origin, D_1(0), with respect to the standard Euclidean metric. It is the interior of a circle of radius 1, centered at the origin. This set can be identified with the set of all complex numbers of absolute value less than one. When viewed as a subset of the complex plane (C), the unit disk is often denoted \mathbb. The open unit disk, the plane, and the upper half-plane The function :f(z)=\frac is an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nome (mathematics)
In mathematics, specifically the theory of elliptic functions, the nome is a special function that belongs to the non-elementary functions. This function is of great importance in the description of the elliptic functions, especially in the description of the modular identity of the Jacobi theta function, the Hermite elliptic transcendents and the Weber modular functions, that are used for solving equations of higher degrees. Definition The nome function is given by :q =\mathrm^ =\mathrm^ =\mathrm^ \, where ''K'' and iK' are the quarter periods, and \omega_1 and \omega_2 are the fundamental pair of periods, and \tau=\frac=\frac is the half-period ratio. The nome can be taken to be a function of any one of these quantities; conversely, any one of these quantities can be taken as functions of the nome. Each of them uniquely determines the others when 0 |
J-invariant
In mathematics, Felix Klein's -invariant or function, regarded as a function of a Complex analysis, complex variable , is a modular function of weight zero for defined on the upper half-plane of complex numbers. It is the unique such function which is Holomorphic function, holomorphic away from a simple pole at the Cusp (singularity), cusp such that :j\left(e^\right) = 0, \quad j(i) = 1728 = 12^3. Rational functions of are modular, and in fact give all modular functions. Classically, the -invariant was studied as a parameterization of elliptic curves over , but it also has surprising connections to the symmetries of the Monster group (this connection is referred to as monstrous moonshine). Definition The -invariant can be defined as a function on the upper half-plane :j(\tau) = 1728 \frac = 1728 \frac = 1728 \frac with the third definition implying j(\tau) can be expressed as a Cube (algebra), cube, also since 1728 (number), 1728 = 12^3. The given functions are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modular Function
In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a growth condition. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections with number theory. Modular forms appear in other areas, such as algebraic topology, sphere packing, and string theory. A modular function is a function that is invariant with respect to the modular group, but without the condition that be holomorphic in the upper half-plane (among other requirements). Instead, modular functions are meromorphic (that is, they are holomorphic on the complement of a set of isolated points, which are poles of the function). Modular form theory is a special case of the more general theory of automorphic forms which are functions defined on Lie groups which transform nicely with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
P-adic Exponential Function
In mathematics, the -adic number system for any prime number extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extension is achieved by an alternative interpretation of the concept of "closeness" or absolute value. In particular, two -adic numbers are considered to be close when their difference is divisible by a high power of : the higher the power, the closer they are. This property enables -adic numbers to encode congruence information in a way that turns out to have powerful applications in number theory – including, for example, in the famous proof of Fermat's Last Theorem by Andrew Wiles. These numbers were first described by Kurt Hensel in 1897, though, with hindsight, some of Ernst Kummer's earlier work can be interpreted as implicitly using -adic numbers.Translator's introductionpage 35 "Indeed, with hindsight it becomes apparent that a discret ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
P-adic Numbers
In mathematics, the -adic number system for any prime number extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extension is achieved by an alternative interpretation of the concept of "closeness" or absolute value. In particular, two -adic numbers are considered to be close when their difference is divisible by a high power of : the higher the power, the closer they are. This property enables -adic numbers to encode congruence information in a way that turns out to have powerful applications in number theory – including, for example, in the famous proof of Fermat's Last Theorem by Andrew Wiles. These numbers were first described by Kurt Hensel in 1897, though, with hindsight, some of Ernst Kummer's earlier work can be interpreted as implicitly using -adic numbers.Translator's introductionpage 35 "Indeed, with hindsight it becomes apparent that a discret ... [...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 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 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 method of checking the primality of a given number n, called trial division, tests whether n is a multiple of any integer between 2 and \sqrt. Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance of error, and the AKS primality test, which always pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperbolic Function
In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the unit hyperbola. Also, similarly to how the derivatives of and are and respectively, the derivatives of and are and respectively. Hyperbolic functions occur in the calculations of angles and distances in hyperbolic geometry. They also occur in the solutions of many linear differential equations (such as the equation defining a catenary), cubic equations, and Laplace's equation in Cartesian coordinates. Laplace's equations are important in many areas of physics, including electromagnetic theory, heat transfer, fluid dynamics, and special relativity. The basic hyperbolic functions are: * hyperbolic sine "" (), * hyperbolic cosine "" (),''Collins Concise Dictionary'', p. 328 from which are derived: * hyperbolic tangent "" (), * hyp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler's Identity
In mathematics, Euler's identity (also known as Euler's equation) is the equality e^ + 1 = 0 where : is Euler's number, the base of natural logarithms, : is the imaginary unit, which by definition satisfies , and : is pi, the ratio of the circumference of a circle to its diameter. Euler's identity is named after the Swiss mathematician Leonhard Euler. It is a special case of Euler's formula e^ = \cos x + i\sin x when evaluated for . Euler's identity is considered to be an exemplar of mathematical beauty as it shows a profound connection between the most fundamental numbers in mathematics. In addition, it is directly used in a proof that is transcendental, which implies the impossibility of squaring the circle. Mathematical beauty Euler's identity is often cited as an example of deep mathematical beauty. Three of the basic arithmetic operations occur exactly once each: addition, multiplication, and exponentiation. The identity also links five fundamental mathematical constants ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
