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Real Part
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a + bi, where and are real numbers. Because no real number satisfies the above equation, was called an imaginary number by René Descartes. For the complex number is called the , and is called the . The set of complex numbers is denoted by either of the symbols \mathbb C or . Despite the historical nomenclature, "imaginary" complex numbers have a mathematical existence as firm as that of the real numbers, and they are fundamental tools in the scientific description of the natural world. Complex numbers allow solutions to all polynomial equations, even those that have no solutions in real numbers. More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with real or complex coefficients h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
A Plus Bi
A, or a, is the first Letter (alphabet), letter and the first vowel letter of the Latin alphabet, used in the modern English alphabet, and others worldwide. Its name in English is ''English alphabet#Letter names, a'' (pronounced ), plural ''aes''. It is similar in shape to the Ancient Greek letter alpha, from which it derives. The uppercase version consists of the two slanting sides of a triangle, crossed in the middle by a horizontal bar. The lowercase version is often written in one of two forms: the double-storey and single-storey . The latter is commonly used in handwriting and fonts based on it, especially fonts intended to be read by children, and is also found in italic type. In English, ''English articles, a'' is the indefinite article, with the alternative form ''an''. Name In English, the name of the letter is the ''long A'' sound, pronounced . Its name in most other languages matches the letter's pronunciation in open syllables. History The earliest know ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Line
A number line is a graphical representation of a straight line that serves as spatial representation of numbers, usually graduated like a ruler with a particular origin (geometry), origin point representing the number zero and evenly spaced marks in either direction representing integers, imagined to extend infinitely. The association between numbers and point (geometry), points on the line links elementary arithmetic, arithmetical operations on numbers to geometry, geometric relations between points, and provides a conceptual framework for learning mathematics. In elementary mathematics, the number line is initially used to teach addition and subtraction of integers, especially involving negative numbers. As students progress, more kinds of numbers can be placed on the line, including fractions, decimal fractions, square roots, and transcendental numbers such as the pi, circle constant : Every point of the number line corresponds to a unique real number, and every real number to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Ordered Pair
In mathematics, an ordered pair, denoted (''a'', ''b''), is a pair of objects in which their order is significant. The ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a''), unless ''a'' = ''b''. In contrast, the '' unordered pair'', denoted , always equals the unordered pair . Ordered pairs are also called 2-tuples, or sequences (sometimes, lists in a computer science context) of length 2. Ordered pairs of scalars are sometimes called 2-dimensional vectors. (Technically, this is an abuse of terminology since an ordered pair need not be an element of a vector space.) The entries of an ordered pair can be other ordered pairs, enabling the recursive definition of ordered ''n''-tuples (ordered lists of ''n'' objects). For example, the ordered triple (''a'',''b'',''c'') can be defined as (''a'', (''b'',''c'')), i.e., as one pair nested in another. In the ordered pair (''a'', ''b''), the object ''a'' is called the ''first entry'', and the object ''b'' the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Numbers Intheplane
Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each other * Complex (psychology), a core pattern of emotions etc. in the personal unconscious organized around a common theme such as power or status Complex may also refer to: Arts, entertainment and media * Complex (English band), formed in 1968, and their 1971 album ''Complex'' * Complex (band), a Japanese rock band * ''Complex'' (album), by Montaigne, 2019, and its title track * ''Complex'' (EP), by Rifle Sport, 1985 * "Complex" (song), by Gary Numan, 1979 * "Complex", a song by Katie Gregson-MacLeod, 2022 * "Complex" a song by Be'O and Zico, 2022 * Complex Networks, publisher of the now-only-online magazine ''Complex'' Biology * Protein–ligand complex, a complex of a protein bound with a ligand * Exosome complex, a multi-pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Vector Space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces'' of any positive integer dimension ''n'', which are called Euclidean ''n''-spaces when one wants to specify their dimension. For ''n'' equal to one or two, they are commonly called respectively Euclidean lines and Euclidean planes. The qualifier "Euclidean" is used to distinguish Euclidean spaces from other spaces that were later considered in physics and modern mathematics. Ancient Greek geometers introduced Euclidean space for modeling the physical space. Their work was collected by the ancient Greek mathematician Euclid in his ''Elements'', with the great innovation of '' proving'' all properties of the space as theorems, by starting from a few fundamental properties, called '' postulates'', which either were considered as evident (f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commutative Algebra (structure)
In mathematics, an associative algebra ''A'' over a commutative ring (often a field) ''K'' is a ring ''A'' together with a ring homomorphism from ''K'' into the center of ''A''. This is thus an algebraic structure with an addition, a multiplication, and a scalar multiplication (the multiplication by the image of the ring homomorphism of an element of ''K''). The addition and multiplication operations together give ''A'' the structure of a ring; the addition and scalar multiplication operations together give ''A'' the structure of a module or vector space over ''K''. In this article we will also use the term ''K''-algebra to mean an associative algebra over ''K''. A standard first example of a ''K''-algebra is a ring of square matrices over a commutative ring ''K'', with the usual matrix multiplication. A commutative algebra is an associative algebra for which the multiplication is commutative, or, equivalently, an associative algebra that is also a commutative ring ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraically Closed Field
In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . In other words, a field is algebraically closed if the fundamental theorem of algebra holds for it. Every field K is contained in an algebraically closed field C, and the roots in C of the polynomials with coefficients in K form an algebraically closed field called an algebraic closure of K. Given two algebraic closures of K there are isomorphisms between them that fix the elements of K. Algebraically closed fields appear in the following chain of class inclusions: Examples As an example, the field of real numbers is not algebraically closed, because the polynomial equation x^2+1=0 has no solution in real numbers, even though all its coefficients (1 and 0) are real. The same argument proves that no subfield of the real field is algebraically closed; in particular, the field of rational numbers is not algebraically cl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reflection Symmetry
In mathematics, reflection symmetry, line symmetry, mirror symmetry, or mirror-image symmetry is symmetry with respect to a Reflection (mathematics), reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry. In Two-dimensional space, two-dimensional space, there is a line/axis of symmetry, in Three-dimensional space, three-dimensional space, there is a plane (mathematics), plane of symmetry. An object or figure which is indistinguishable from its transformed image is called mirror image, mirror symmetric. Symmetric function In formal terms, a mathematical object is symmetric with respect to a given mathematical operation, operation such as reflection, Rotational symmetry, rotation, or Translational symmetry, translation, if, when applied to the object, this operation preserves some property of the object. The set of operations that preserve a given property of the object form a group (algebra), group. Two objects are symmetr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Conjugation
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. That is, if a and b are real numbers, then the complex conjugate of a + bi is a - bi. The complex conjugate of z is often denoted as \overline or z^*. In polar form, if r and \varphi are real numbers then the conjugate of r e^ is r e^. This can be shown using Euler's formula. The product of a complex number and its conjugate is a real number: a^2 + b^2 (or r^2 in polar coordinates). If a root of a univariate polynomial with real coefficients is complex, then its complex conjugate is also a root. Notation The complex conjugate of a complex number z is written as \overline z or z^*. The first notation, a vinculum, avoids confusion with the notation for the conjugate transpose of a matrix, which can be thought of as a generalization of the complex conjugate. The second is preferred in physics, where ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Similarity (geometry)
In Euclidean geometry, two objects are similar if they have the same shape, or if one has the same shape as the mirror image of the other. More precisely, one can be obtained from the other by uniformly scaling (geometry), scaling (enlarging or reducing), possibly with additional translation (geometry), translation, rotation (mathematics), rotation and reflection (mathematics), reflection. This means that either object can be rescaled, repositioned, and reflected, so as to coincide precisely with the other object. If two objects are similar, each is congruence (geometry), congruent to the result of a particular uniform scaling of the other. For example, all circles are similar to each other, all squares are similar to each other, and all equilateral triangles are similar to each other. On the other hand, ellipses are not all similar to each other, rectangles are not all similar to each other, and isosceles triangles are not all similar to each other. This is because two ellipse ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Translation (geometry)
In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure, shape or space by the same Distance geometry, distance in a given direction (geometry), direction. A translation can also be interpreted as the addition of a constant vector space, vector to every point, or as shifting the Origin (mathematics), origin of the coordinate system. In a Euclidean space, any translation is an isometry. As a function If \mathbf is a fixed vector, known as the ''translation vector'', and \mathbf is the initial position of some object, then the translation function T_ will work as T_(\mathbf)=\mathbf+\mathbf. If T is a translation, then the image (mathematics), image of a subset A under the function (mathematics), function T is the translate of A by T . The translate of A by T_ is often written as A+\mathbf . Application in classical physics In classical physics, translational motion is movement that changes the Position (geometry), positio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
