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Imaginary Unit
The imaginary unit or unit imaginary number (i) is a solution to the quadratic equation x2 + 1 = 0. Although there is no real number with this property, i can be used to extend the real numbers to what are called complex numbers, using addition and multiplication
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Gamma Function
In mathematics, the gamma function (represented by Γ, the capital Greek alphabet
Greek alphabet
letter gamma) is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers. If n is a positive integer, Γ ( n ) = ( n − 1 ) ! displaystyle Gamma
Gamma
(n)=(n-1)! The gamma function is defined for all complex numbers except the non-positive integers
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Unit Circle
In mathematics, a unit circle is a circle with a radius of one. Frequently, especially in trigonometry, the unit circle is the circle of radius one centered at the origin (0, 0) in the Cartesian coordinate system in the Euclidean plane. The unit circle is often denoted S1; the generalization to higher dimensions is the unit sphere. If (x, y) is a point on the unit circle's circumference, then x and y are the lengths of the legs of a right triangle whose hypotenuse has length 1. Thus, by the Pythagorean theorem, x and y satisfy the equation x 2 + y 2 = 1. displaystyle x^ 2 +y^ 2 =1
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Additive Inverse
In mathematics, the additive inverse of a number a is the number that, when added to a, yields zero. This number is also known as the opposite (number),[1] sign change, and negation.[2] For a real number, it reverses its sign: the opposite to a positive number is negative, and the opposite to a negative number is positive. Zero is the additive inverse of itself. The additive inverse of a is denoted by unary minus: −a (see the discussion below). For example, the additive inverse of 7 is −7, because 7 + (−7) = 0, and the additive inverse of −0.3 is 0.3, because −0.3 + 0.3 = 0 . The additive inverse is defined as its inverse element under the binary operation of addition (see the discussion below), which allows a broad generalization to mathematical objects other than numbers
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Well-defined
In mathematics, an expression is called well-defined or unambiguous if its definition assigns it a unique interpretation or value. Otherwise, the expression is said to be not well-defined or ambiguous.[1] A function is well-defined if it gives the same result when the representation of the input is changed without changing the value of the input. For instance if f takes real numbers as input, and if f(0.5) does not equal f(1/2) then f is not well-defined (and thus: not a function).[2] The term well-defined is also used to indicate whether a logical statement is unambiguous. A function that is not well-defined is not the same as a function that is undefined. For example, if f(x) = 1/x, then f(0) is undefined, but this has nothing to do with the question of whether f(x) = 1/x is well-defined
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Unique (mathematics)
In mathematics and logic, the phrase "there is one and only one" is used to indicate that exactly one object with a certain property exists. In mathematical logic, this sort of quantification is known as uniqueness quantification or unique existential quantification. Uniqueness quantification is often denoted with the symbols "∃!" or ∃=1"
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Up To
In mathematics, the phrase up to appears in discussions about the elements of a set (say S), and the conditions under which subsets of those elements may be considered equivalent. The statement "elements a and b of set S are equivalent up to X" means that a and b are equivalent if criterion X (such as rotation or permutation) is ignored. That is, a and b can be transformed into one another if a transform corresponding to X (rotation, permutation etc.) is applied. Looking at the entire set S, when X is ignored the elements can be arranged in subsets whose elements are equivalent ("equivalent up to X"). Such subsets are called "equivalence classes". If X is some property or process, the phrase "up to X" means "disregarding a possible difference in X". For instance the statement "an integer's prime factorization is unique up to ordering", means that the prime factorization is unique if we disregard the order of the factors
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Isomorphism
In mathematics, an isomorphism (from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape") is a homomorphism or morphism (i.e. a mathematical mapping) that admits an inverse.[note 1] Two mathematical objects are isomorphic if an isomorphism exists between them. An automorphism is an isomorphism whose source and target coincide
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Automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism group. It is, loosely speaking, the symmetry group of the object.Contents1 Definition 2 Automorphism group 3 Examples 4 History 5 Inner and outer automorphisms 6 See also 7 References 8 External linksDefinition[edit] In the context of abstract algebra, a mathematical object is an algebraic structure such as a group, ring, or vector space. An automorphism is simply a bijective homomorphism of an object with itself. (The definition of a homomorphism depends on the type of algebraic structure; see, for example, group homomorphism, ring homomorphism, and linear operator). The identity morphism (identity mapping) is called the trivial automorphism in some contexts
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Complex Conjugate
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign.[1][2] For example, the complex conjugate of 3 + 4i is 3 − 4i. In polar form, the conjugate of r e i φ displaystyle re^ ivarphi is r e − i φ displaystyle re^ -ivarphi
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Galois Group
In mathematics, more specifically in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the polynomials that give rise to them via Galois groups is called Galois theory, so named in honor of Évariste Galois
Évariste Galois
who first discovered them. For a more elementary discussion of Galois groups in terms of permutation groups, see the article on Galois theory.Contents1 Definition 2 Examples 3 Properties 4 See also 5 Notes 6 References 7 External linksDefinition[edit] Suppose that E is an extension of the field F (written as E/F and read "E over F"). An automorphism of E/F is defined to be an automorphism of E that fixes F pointwise
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Automorphism Group
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism group. It is, loosely speaking, the symmetry group of the object.Contents1 Definition 2 Automorphism group 3 Examples 4 History 5 Inner and outer automorphisms 6 See also 7 References 8 External linksDefinition[edit] In the context of abstract algebra, a mathematical object is an algebraic structure such as a group, ring, or vector space. An automorphism is simply a bijective homomorphism of an object with itself. (The definition of a homomorphism depends on the type of algebraic structure; see, for example, group homomorphism, ring homomorphism, and linear operator). The identity morphism (identity mapping) is called the trivial automorphism in some contexts
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Absolute Value
In mathematics, the absolute value or modulus x of a real number x is the non-negative value of x without regard to its sign. Namely, x = x for a positive x, x = −x for a negative x (in which case −x is positive), and 0 = 0. For example, the absolute value of 3 is 3, and the absolute value of −3 is also 3. The absolute value of a number may be thought of as its distance from zero. Generalisations of the absolute value for real numbers occur in a wide variety of mathematical settings. For example, an absolute value is also defined for the complex numbers, the quaternions, ordered rings, fields and vector spaces
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Orthogonal Group
In mathematics, the orthogonal group in dimension n, denoted O(n), is the group of distance-preserving transformations of a Euclidean space of dimension n that preserve a fixed point, where the group operation is given by composing transformations. Equivalently, it is the group of n×n orthogonal matrices, where the group operation is given by matrix multiplication; an orthogonal matrix is a real matrix whose inverse equals its transpose. An important subgroup of O(n) is the special orthogonal group, denoted SO(n), of the orthogonal matrices of determinant 1. This group is also called the rotation group, because, in dimensions 2 and 3, its elements are the usual rotations around a point (in dimension 2) or a line (in dimension 3)
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Identity Matrix
In linear algebra, the identity matrix, or sometimes ambiguously called a unit matrix, of size n is the n × n square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by In, or simply by I if the size is immaterial or can be trivially determined by the context
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Branch Point
In the mathematical field of complex analysis, a branch point of a multi-valued function (usually referred to as a "multifunction" in the context of complex analysis) is a point such that the function is discontinuous when going around an arbitrarily small circuit around this point.[1] Multi-valued functions are rigorously studied using Riemann surfaces, and the formal definition of branch points employs this concept. Branch points fall into three broad categories: algebraic branch points, transcendental branch points, and logarithmic branch points. Algebraic branch points most commonly arise from functions in which there is an ambiguity in the extraction of a root, such as solving the equation w2  = z for w as a function of z. Here the branch point is the origin, because the analytic continuation of any solution around a closed loop containing the origin will result in a different function: there is non-trivial monodromy
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