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Imaginary Unit The IMAGINARY UNIT or UNIT IMAGINARY NUMBER (I) is a solution to the quadratic equation x2 + 1 = 0. Since there is no real number with this property, it extends the real numbers, and under the assumption that the familiar properties of addition and multiplication (namely closure , associativity , commutativity and distributivity ) continue to hold for this extension, the complex numbers are generated by including it. Imaginary numbers are an important mathematical concept, which extends the real number system ℝ to the complex number system ℂ, which in turn provides at least one root for every nonconstant polynomial P(x). (See Algebraic closure and Fundamental theorem of algebra .) The term "imaginary " is used because there is no real number having a negative square . There are two complex square roots of −1, namely i and −i, just as there are two complex square roots of every real number other than zero , which has one double square root [...More...]  "Imaginary Unit" on: Wikipedia Yahoo 

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. CONTENTS * 1 Definition * 2 Automorphism group * 3 Examples * 4 History * 5 Inner and outer automorphisms * 6 See also * 7 References * 8 External links DEFINITIONThe exact definition of an automorphism depends on the type of "mathematical object" in question and what, precisely, constitutes an "isomorphism" of that object. The most general setting in which these words have meaning is an abstract branch of mathematics called category theory . Category theory Category theory deals with abstract objects and morphisms between those objects [...More...]  "Automorphism Group" on: Wikipedia Yahoo 

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.} Since x2 = (−x)2 for all x, and since the reflection of any point on the unit circle about the x or yaxis is also on the unit circle, the above equation holds for all points (x, y) on the unit circle, not only those in the first quadrant [...More...]  "Unit Circle" on: Wikipedia Yahoo 

Absolute Value In mathematics , the ABSOLUTE VALUE or MODULUS x of a real number x is the nonnegative 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 . The absolute value is closely related to the notions of magnitude , distance , and norm in various mathematical and physical contexts [...More...]  "Absolute Value" on: Wikipedia Yahoo 

Orthogonal Group In mathematics , the ORTHOGONAL GROUP in dimension n, denoted O(n), is the group of distancepreserving 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). In low dimension, these groups have been widely studied, see SO(2) , SO(3) and SO(4) [...More...]  "Orthogonal Group" on: Wikipedia Yahoo 

Electric Current An ELECTRIC CURRENT is a flow of electric charge . In electric circuits this charge is often carried by moving electrons in a wire . It can also be carried by ions in an electrolyte , or by both ions and electrons such as in an ionised gas (plasma ). The SI unit for measuring an electric current is the ampere , which is the flow of electric charge across a surface at the rate of one coulomb per second. Electric current Electric current is measured using a device called an ammeter . Electric currents cause Joule heating , which creates light in incandescent light bulbs . They also create magnetic fields , which are used in motors, inductors and generators. The moving charged particles in an electric current are called charge carriers [...More...]  "Electric Current" on: Wikipedia Yahoo 

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. (In some fields, such as quantum mechanics , the identity matrix is denoted by a boldface one, 1; otherwise it is identical to I.) Less frequently, some mathematics books use U or E to represent the identity matrix, meaning "unit matrix" and the German word "Einheitsmatrix", respectively [...More...]  "Identity Matrix" on: Wikipedia Yahoo 

Electrical Engineering ELECTRICAL ENGINEERING is a professional engineering discipline that generally deals with the study and application of electricity , electronics , and electromagnetism . This field first became an identifiable occupation in the later half of the 19th century after commercialization of the electric telegraph , the telephone , and electric power distribution and use. Subsequently, broadcasting and recording media made electronics part of daily life. The invention of the transistor , and later the integrated circuit , brought down the cost of electronics to the point they can be used in almost any household object. Electrical engineering Electrical engineering has now subdivided into a wide range of subfields including electronics , digital computers , computer engineering , power engineering , telecommunications , control systems , radiofrequency engineering , signal processing , instrumentation , and microelectronics [...More...]  "Electrical Engineering" on: Wikipedia Yahoo 

Control Systems Engineering CONTROL ENGINEERING or CONTROL SYSTEMS ENGINEERING is an engineering discipline that applies automatic control theory to design systems with desired behaviors in control environments. The discipline of controls overlaps and is usually taught along with electrical engineering at many institutions around the world. The practice uses sensors and detectors to measure the output performance of the process being controlled; these measurements are used to provide corrective feedback helping to achieve the desired performance. Systems designed to perform without requiring human input are called automatic control systems (such as cruise control for regulating the speed of a car). Multidisciplinary in nature, control systems engineering activities focus on implementation of control systems mainly derived by mathematical modeling of a diverse range of systems [...More...]  "Control Systems Engineering" on: Wikipedia Yahoo 

Argument (complex Analysis) In mathematics , ARG is a function operating on complex numbers (visualized in a complex plane ). It gives the angle between the positive real axis to the line joining the point to the origin, shown as φ in figure 1, known as an ARGUMENT of the point. CONTENTS * 1 Definition * 2 Principal value * 2.1 Notation * 3 Covering space * 4 Computation * 5 Identities * 5.1 Example * 6 References * 6.1 Notes * 6.2 Bibliography * 7 External links DEFINITION Figure 2. Two choices for the argument φ An ARGUMENT of the complex number z = x + iy, denoted arg(z), is defined in two equivalent ways: * Geometrically, in the complex plane , as the angle φ from the positive real axis to the vector representing z [...More...]  "Argument (complex Analysis)" on: Wikipedia Yahoo 

Quadratic Polynomial In algebra , a QUADRATIC FUNCTION, a QUADRATIC POLYNOMIAL, a POLYNOMIAL OF DEGREE 2, or simply a QUADRATIC, is a polynomial function in one or more variables in which the highestdegree term is of the second degree. For example, a quadratic function in three variables x, y, and z contains exclusively terms x2, y2, z2, xy, xz, yz, x, y, z, and a constant: f ( x , y , z ) = a x 2 + b y 2 + c z 2 + d x y + e x z + f y z + g x + h y + i z + j , {displaystyle f(x,y,z)=ax^{2}+by^{2}+cz^{2}+dxy+exz+fyz+gx+hy+iz+j,} with at least one of the coefficients a, b, c, d, e, or f of the seconddegree terms being nonzero. A quadratic polynomial with two real roots (crossings of the x axis) and hence no complex roots. Some other quadratic polynomials have their minimum above the x axis, in which case there are no real roots and two complex roots [...More...]  "Quadratic Polynomial" on: Wikipedia Yahoo 

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 [...More...]  "Up To" on: Wikipedia Yahoo 

Isomorphism In mathematics , an ISOMORPHISM (from the Ancient Greek 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. Two mathematical objects are ISOMORPHIC if an isomorphism exists between them. An automorphism is an isomorphism whose source and target coincide. The interest of isomorphisms lies in the fact that two isomorphic objects cannot be distinguished by using only the properties used to define morphisms; thus isomorphic objects may be considered the same as long as one considers only these properties and their consequences. For most algebraic structures , including groups and rings , a homomorphism is an isomorphism if and only if it is bijective [...More...]  "Isomorphism" on: Wikipedia Yahoo 

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. CONTENTS * 1 Definition * 2 Automorphism group * 3 Examples * 4 History * 5 Inner and outer automorphisms * 6 See also * 7 References * 8 External links DEFINITIONThe exact definition of an automorphism depends on the type of "mathematical object" in question and what, precisely, constitutes an "isomorphism" of that object. The most general setting in which these words have meaning is an abstract branch of mathematics called category theory . Category theory deals with abstract objects and morphisms between those objects [...More...]  "Automorphism" on: Wikipedia Yahoo 

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". For example, the formal statement ! n N ( n 2 = 4 ) {displaystyle exists !nin mathbb {N} ,(n2=4)} may be read aloud as "there is exactly one natural number n such that n − 2 = 4". CONTENTS * 1 Proving uniqueness * 2 Reduction to ordinary existential and universal quantification * 3 Generalizations * 4 See also * 5 References PROVING UNIQUENESSThe most common technique to proving unique existence is to first prove existence of entity with the desired condition; then, to assume there exist two entities (say, a and b) that both satisfy the condition, and logically deduce their equality, i.e [...More...]  "Unique (mathematics)" on: Wikipedia Yahoo 

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 who first discovered them. For a more elementary discussion of Galois groups in terms of permutation groups , see the article on Galois theory . CONTENTS * 1 Definition * 2 Examples * 3 Properties * 4 See also * 5 Notes * 6 References * 7 External links DEFINITIONSuppose 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. In other words, an automorphism of E/F is an isomorphism α from E to E such that α(x) = x for each x in F [...More...]  "Galois Group" on: Wikipedia Yahoo 