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Complex Line
In mathematics, a complex line is a one-dimensional affine subspace of a vector space over the complex numbers. A common point of confusion is that while a complex line has complex dimension one over C (hence the term " line"), it has ordinary dimension two over the real numbers R, and is topologically equivalent to a real plane, not a real line.. The "complex plane" commonly refers to the graphical representation of the complex line on the real plane, and is thus generally synonymous with the complex line, not the complex coordinate plane. See also * Algebraic geometry * Complex vector * Riemann sphere In mathematics, the Riemann sphere, named after Bernhard Riemann, is a Mathematical model, model of the extended complex plane (also called the closed complex plane): the complex plane plus one point at infinity. This extended plane represents ... References Complex geometry Complex analysis {{geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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] |
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Real Numbers
In mathematics, a real number is a number that can be used to measurement, measure a continuous variable, continuous one-dimensional quantity such as a time, duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and in many other branches of mathematics), in particular by their role in the classical definitions of limit (mathematics), limits, continuous function, continuity and derivatives. The set of real numbers, sometimes called "the reals", is traditionally mathematical notation, denoted by a bold , often using blackboard bold, . The adjective ''real'', used in the 17th century by René Descartes, distinguishes real numbers from imaginary numbers such as the square roots of . The real numbers include the rational numbers, such as the integer and the fraction (mathematics), fraction . ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Riemann Sphere
In mathematics, the Riemann sphere, named after Bernhard Riemann, is a Mathematical model, model of the extended complex plane (also called the closed complex plane): the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers plus a value \infty for infinity. With the Riemann model, the point \infty is near to very large numbers, just as the point 0 is near to very small numbers. The extended complex numbers are useful in complex analysis because they allow for division by zero in some circumstances, in a way that makes expressions such as 1/0=\infty well-behaved. For example, any rational function on the complex plane can be extended to a holomorphic function on the Riemann sphere, with the Pole (complex analysis), poles of the rational function mapping to infinity. More generally, any meromorphic function can be thought of as a holomorphic function whose codomain is the Riemann sphere. In geometr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Complex Vector
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', can be added together and multiplied ("scaled") by numbers called ''scalars''. The operations of vector addition and scalar multiplication must satisfy certain requirements, called ''vector axioms''. Real vector spaces and complex vector spaces are kinds of vector spaces based on different kinds of scalars: real numbers and complex numbers. Scalars can also be, more generally, elements of any field. Vector spaces generalize Euclidean vectors, which allow modeling of physical quantities (such as forces and velocity) that have not only a magnitude, but also a direction. The concept of vector spaces is fundamental for linear algebra, together with the concept of matrices, which allows computing in vector spaces. This provides a concise and synthetic way for manipulating and studying systems of linear equations. Vector spaces are characterized by the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Algebraic Geometry
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; the modern approach generalizes this in a few different aspects. The fundamental objects of study in algebraic geometry are algebraic variety, algebraic varieties, which are geometric manifestations of solution set, solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are line (geometry), lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscate of Bernoulli, lemniscates and Cassini ovals. These are plane algebraic curves. A point of the plane lies on an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of points of special interest like singular point of a curve, singular p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Complex Coordinate Plane
In mathematics, the ''n''-dimensional complex coordinate space (or complex ''n''-space) is the set of all ordered ''n''-tuples of complex numbers, also known as ''complex vectors''. The space is denoted \Complex^n, and is the ''n''-fold Cartesian product of the complex line \Complex with itself. Symbolically, \Complex^n = \left\ or \Complex^n = \underbrace_. The variables z_i are the (complex) coordinates on the complex ''n''-space. The special case \Complex^2, called the ''complex coordinate plane'', is not to be confused with the complex plane, a graphical representation of the complex line. Complex coordinate space is a vector space over the complex numbers, with componentwise addition and scalar multiplication. The real and imaginary parts of the coordinates set up a bijection of \Complex^n with the 2''n''-dimensional real coordinate space, \mathbb R^. With the standard Euclidean topology, \Complex^n is a topological vector space over the complex numbers. A function ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Complex Plane
In mathematics, the complex plane is the plane (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, called the imaginary axis, is formed by the imaginary numbers. The complex plane allows for a geometric interpretation of complex numbers. Under addition, they add like vector (geometry), vectors. The multiplication of two complex numbers can be expressed more easily in polar coordinates: the magnitude or ' of the product is the product of the two absolute values, or moduli, and the angle or ' of the product is the sum of the two angles, or arguments. In particular, multiplication by a complex number of modulus 1 acts as a rotation. The complex plane is sometimes called the Argand plane or Gauss plane. Notational conventions Complex numbers In complex analysis, the complex numbers are customarily represented by the symbol , which can be sepa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Real Plane
In mathematics, the real coordinate space or real coordinate ''n''-space, of dimension , denoted or , is the set of all ordered -tuples of real numbers, that is the set of all sequences of real numbers, also known as ''coordinate vectors''. Special cases are called the ''real line'' , the ''real coordinate plane'' , and the ''real coordinate three-dimensional space'' . With component-wise addition and scalar multiplication, it is a real vector space. The coordinates over any basis of the elements of a real vector space form a ''real coordinate space'' of the same dimension as that of the vector space. Similarly, the Cartesian coordinates of the points of a Euclidean space of dimension , (Euclidean line, ; Euclidean plane, ; Euclidean three-dimensional space, ) form a ''real coordinate space'' of dimension . These one to one correspondences between vectors, points and coordinate vectors explain the names of ''coordinate space'' and ''coordinate vector''. It allows using geom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordinate is needed to specify a point on itfor example, the point at 5 on a number line. A surface, such as the boundary of a cylinder or sphere, has a dimension of two (2D) because two coordinates are needed to specify a point on itfor example, both a latitude and longitude are required to locate a point on the surface of a sphere. A two-dimensional Euclidean space is a two-dimensional space on the plane. The inside of a cube, a cylinder or a sphere is three-dimensional (3D) because three coordinates are needed to locate a point within these spaces. In classical mechanics, space and time are different categories and refer to absolute space and time. That conception of the world is a four-dimensional space but not the one that w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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One-dimensional
A one-dimensional space (1D space) is a mathematical space in which location can be specified with a single coordinate. An example is the number line, each point of which is described by a single real number. Any straight line or smooth curve is a one-dimensional space, regardless of the dimension of the ambient space in which the line or curve is embedded. Examples include the circle on a plane, or a parametric space curve. In physical space, a 1D subspace is called a "linear dimension" ( rectilinear or curvilinear), with units of length (e.g., metre). In algebraic geometry there are several structures that are one-dimensional spaces but are usually referred to by more specific terms. Any field K is a one-dimensional vector space over itself. The projective line over K, denoted \mathbf P^1(K), is a one-dimensional space. In particular, if the field is the complex numbers \mathbb, then the complex projective line \mathbf P^1(\mathbb) is one-dimensional with respect to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Line (mathematics)
In geometry, a straight line, usually abbreviated line, is an infinitely long object with no width, depth, or curvature, an idealization of such physical objects as a straightedge, a taut string, or a ray of light. Lines are spaces of dimension one, which may be embedded in spaces of dimension two, three, or higher. The word ''line'' may also refer, in everyday life, to a line segment, which is a part of a line delimited by two points (its ''endpoints''). Euclid's ''Elements'' defines a straight line as a "breadthless length" that "lies evenly with respect to the points on itself", and introduced several postulates as basic unprovable properties on which the rest of geometry was established. ''Euclidean line'' and ''Euclidean geometry'' are terms introduced to avoid confusion with generalizations introduced since the end of the 19th century, such as non-Euclidean, projective, and affine geometry. Properties In the Greek deductive geometry of Euclid's ''Elements'', ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Complex Dimension
In mathematics, complex dimension usually refers to the dimension of a complex manifold or a complex algebraic variety. These are spaces in which the local neighborhoods of points (or of non-singular points in the case of a variety) are modeled on a Cartesian product of the form \mathbb^d for some d, and the complex dimension is the exponent d in this product. Because \mathbb can in turn be modeled by \mathbb^2, a space with complex dimension d will have real dimension 2d. That is, a smooth manifold of complex dimension d has real dimension 2d; and a complex algebraic variety of complex dimension d, away from any singular point, will also be a smooth manifold of real dimension 2d. However, for a real algebraic variety (that is a variety defined by equations with real coefficients), its dimension refers commonly to its complex dimension, and its real dimension refers to the maximum of the dimensions of the manifolds contained in the set of its real points. The real dimension is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |