Instruct: Given a question, retrieve relevant documents that answer the question.
Query: When was algebra first developed?
Mathematical historians[77] generally agree that the use of x in algebra was introduced by René Descartes and was first published in his treatise La Géométrie (1637).[78][79] In that work, he used letters from the beginning of the alphabet (a, b, c,...) for known quantities, and letters from the end of the alphabet (z, y, x,...) for unknowns.[80] It has been suggested that he later settled on x (in place of z) for the first unknown because of its relatively greater abundance in the French and Latin typographical fonts of the time.[81]
The Casio's computer algebra system (CAS) was first developed for the CFX-9970G in 1998. The following are the functionalities of the CAS in the CFX-9970G:
In the Algebra FX series, the computer algebra system was largely improved from its predecessor. The new features include calculating limits, Taylor's series, arc lengths, and etc.
If the above theory is true, the next stage of development ought to be "triple" algebra and if truly represents a line in a given plane, it ought to be possible to find a third term which added to the above would represent a line in space. Argand and some others guessed that it was although this contradicts the truth established by Euler that . De Morgan and many others worked hard at the problem, but nothing came of it until the problem was taken up by Hamilton. We now see the reason clearly: The symbol of double algebra denotes not a length and a direction; but a multiplier and "an angle". In it the angles are confined to one plane. Hence the next stage will be a "quadruple algebra", when the axis of the plane is made variable. And this gives the answer to the first question; double algebra is nothing but analytical plane trigonometry, and this is why it has been found to be the natural analysis for alternating currents. But De Morgan never got this far. He died with the belief that “double algebra must remain as the full development of the conceptions of arithmetic, so far as those symbols are concerned which arithmetic immediately suggests”.
Also developed are the concept of vector manifold and geometric integration theory (which generalizes Cartan's differential forms).Although the connection of geometry with algebra dates as far back at least to Euclid's "Elements" in the third century B.C. (see Greek geometric algebra), GA in the sense used in this article was not developed until 1844, when it was used in a "systematic way" to describe the geometrical properties and "transformations" of a space. In that year, Hermann Grassmann introduced the idea of a geometrical algebra in full generality as a certain calculus (analogous to the propositional calculus) that encoded all of the geometrical information of a space. Grassmann's algebraic system could be applied to a number of different kinds of spaces, the chief among them being Euclidean space, affine space, and projective space. Following Grassmann, in 1878 William Kingdon Clifford examined Grassmann's algebraic system alongside the quaternions of William Rowan Hamilton in . From his point of view, the quaternions described certain "transformations" (which he called "rotors"), whereas Grassmann's algebra described certain "properties" (or "Strecken" such as length, area, and volume). His contribution was to define a new product — the "geometric product" – on an existing Grassmann algebra, which realized the quaternions as living within that algebra. Subsequently, Rudolf Lipschitz in 1886 generalized Clifford's interpretation of the quaternions and applied them to the geometry of rotations in formula_102 dimensions. Later these developments would lead other 20th-century mathematicians to formalize and explore the properties of the Clifford algebra.
If a proposition which can be expressed in the language of first-order logic is true for an algebraically closed field, then it is true for every algebraically closed field with the same characteristic. Furthermore, if such a proposition is valid for an algebraically closed field with characteristic 0, then not only is it valid for all other algebraically closed fields with characteristic 0, but there is some natural number "N" such that the proposition is valid for every algebraically closed field with characteristic "p" when "p" > "N".
C-algebras (pronounced "C-star") are subjects of research in functional analysis, a branch of mathematics. A C*-algebra is a complex algebra "A" of continuous linear operators on a complex Hilbert space with two additional properties:C*-algebras were first considered primarily for their use in quantum mechanics to model algebras of physical observables. This line of research began with Werner Heisenberg's matrix mechanics and in a more mathematically developed form with Pascual Jordan around 1933. Subsequently, John von Neumann attempted to establish a general framework for these algebras which culminated in a series of papers on rings of operators. These papers considered a special class of C*-algebras which are now known as von Neumann algebras.
The development provided below can be equally well applied to the exterior algebra, using the wedge symbol formula_9 in place of the tensor symbol formula_10; a sign must also be kept track of, when permuting elements of the exterior algebra. This correspondence also lasts through the definition of the bialgebra, and on to the definition of a Hopf algebra. That is, the exterior algebra can also be given a Hopf algebra structure.
Joachim Cuntz has made fundamental contributions to the area of C*-algebras and to the field of noncommutative geometry in the sense of Alain Connes. He initiated the analysis of the structure of simple C*-algebras and introduced new methods and examples, including the Cuntz algebras and the Cuntz semigroup. He was one of the first to apply K-theory to noncommutative operator algebras and contributed to the development of that theory. In collaboration with Daniel Quillen, he developed a new approach to cyclic cohomology and proved the excision property of periodic cyclic theory. In recent years, he has been working mainly on C*-algebras that are related to structures from number theory.
The archetypal association of this kind, one fundamental to the historical origins of algebraic logic and lying at the heart of all subsequently developed subtheories, is the association between the class of Boolean algebras and classical propositional calculus. This association was discovered by George Boole in the 1850s, and refined by others, especially Ernst Schröder in the 1890s. This work culminated in Lindenbaum–Tarski algebras, devised by Alfred Tarski and his student Adolf Lindenbaum in the 1930s. Later, Tarski and his American students (whose ranks include Don Pigozzi) went on to discover cylindric algebra, which algebraizes all of classical first-order logic, and revived relation algebra, whose models include all well-known axiomatic set theories.
Since the Shang period, the Chinese had already fully developed a decimal system. Since early times, Chinese understood basic arithmetic (which dominated far eastern history), algebra, equations, and negative numbers with counting rods. Although the Chinese were more focused on arithmetic and advanced algebra for astronomical uses, they were also the first to develop negative numbers, algebraic geometry (only Chinese geometry) and the usage of decimals.
A composition algebra is either a division algebra or a split algebra, depending on the existence of a non-zero in such that , called a null vector. When there is no non-zero null vector, the multiplicative inverse of is , so the algebra is a division algebra. When there is a non-zero null vector, is an isotropic quadratic form, and "the algebra splits".
After Grassmann, developments in multilinear algebra were made in 1872 by Victor Schlegel when he published the first part of his "System der Raumlehre", and by Elwin Bruno Christoffel. A major advance in multilinear algebra came in the work of Gregorio Ricci-Curbastro and Tullio Levi-Civita (see references). It was the "absolute differential calculus" form of multilinear algebra that Marcel Grossmann and Michele Besso introduced to Albert Einstein. The publication in 1915 by Einstein of a general relativity explanation for the precession of the perihelion of Mercury, established multilinear algebra and tensors as physically important mathematics.
Rhetorical algebra, in which equations are written in full sentences. For example, the rhetorical form of x + 1 = 2 is "The thing plus one equals two" or possibly "The thing plus 1 equals 2". Rhetorical algebra was first developed by the ancient Babylonians and remained dominant up to the 16th century. Syncopated algebra, in which some symbolism is used, but which does not contain all of the characteristics of symbolic algebra. For instance, there may be a restriction that subtraction may be used only once within one side of an equation, which is not the case with symbolic algebra. Syncopated algebraic expression first appeared in Diophantus' Arithmetica (3rd century AD), followed by Brahmagupta's Brahma Sphuta Siddhanta (7th century). Symbolic algebra, in which full symbolism is used. Early steps toward this can be seen in the work of several Islamic mathematicians such as Ibn al-Banna (13th-14th centuries) and al-Qalasadi (15th century), although fully symbolic algebra was developed by François Viète (16th century). Later, René Descartes (17th century) introduced the modern notation (for example, the use of x—see below) and showed that the problems occurring in geometry can be expressed and solved in terms of algebra (Cartesian geometry).
Because Leibniz was a mathematical novice when he first wrote about the "characteristic", at first he did not conceive it as an algebra but rather as a universal language or script. Only in 1676 did he conceive of a kind of "algebra of thought", modeled on and including conventional algebra and its notation. The resulting "characteristic" included a logical calculus, some combinatorics, algebra, his "analysis situs" (geometry of situation), a universal concept language, and more. What Leibniz actually intended by his "characteristica universalis" and calculus ratiocinator, and the extent to which modern formal logic does justice to calculus, may never be established. Leibniz's idea of reasoning through a universal language of symbols and calculations remarkably foreshadows great 20th-century developments in formal systems, such as Turing completeness, where computation was used to define equivalent universal languages (see Turing degree).
When using algebra tiles to multiply a monomial by a monomial you first set up a rectangle where the length of the rectangle is the one monomial and then the width of the rectangle is the other monomial, similar to when you multiply integers using algebra tiles. Once the sides of the rectangle are represented by the algebra tiles you would then try to figure out which algebra tiles would fill in the rectangle. For instance, if you had x×x the only algebra tile that would complete the rectangle would be x, which is the answer.
The first modern and more precise definition of a vector space was introduced by Peano in 1888; by 1900, a theory of linear transformations of finite-dimensional vector spaces had emerged. Linear algebra took its modern form in the first half of the twentieth century, when many ideas and methods of previous centuries were generalized as abstract algebra. The development of computers led to increased research in efficient algorithms for Gaussian elimination and matrix decompositions, and linear algebra became an essential tool for modelling and simulations.
The distinction between geometric "spaces" and algebraic "structures" is sometimes clear, sometimes elusive. Clearly, groups are algebraic, while Euclidean spaces are geometric. Modules over rings are as algebraic as groups. In particular, when the ring appears to be a field, the module appears to be a linear space; is it algebraic or geometric? In particular, when it is finite-dimensional, over real numbers, and endowed with inner product, it becomes Euclidean space; now geometric. The (algebraic?) field of real numbers is the same as the (geometric?) real line. Its algebraic closure, the (algebraic?) field of complex numbers, is the same as the (geometric?) complex plane. It is first of all "a place we do analysis" (rather than algebra or geometry).
The Algebra FX series was the successor of the CFX-9970G, the first Casio calculator with computer algebra system (CAS) released in 1998. The computer algebra system in the Algebra FX series had been largely improved from the previous model so that more mathematical functions were added. The Casio's CAS was mainly developed by its R&D team and Professor John Kenelly of Clemson University.
An algebra satisfies an equation when the equation holds for all possible values of its variables in that algebra when the operation symbols are interpreted as specified by that algebra. The laws of Boolean algebra are the equations in the language of Boolean algebra satisfied by the prototype. The first three of the above examples are Boolean laws, but not the fourth since 1∧0 ≠ 1.
Similar statements hold when formula_1 is replaced by a non-Abelian locally compact group or more generally a locally compact quantum group, a class of Hopf algebra related to von Neumann algebras. An analogous theory has also been developed for actions on C* algebras and their crossed products.
Let formula_25 be a compact Lie group with Lie algebra formula_26. Then the complexification formula_27 of formula_26 is "reductive", that is, the direct sum of a complex semisimple Lie algebra and a commutative algebra. If formula_25 is simply connected, then formula_30 is actually semisimple. Conversely, every complex semisimple Lie algebra formula_30 has a "compact real form" formula_26, where formula_26 is the Lie algebra of a simply connected compact Lie group. For example, the complex semisimple Lie algebra formula_34 is the complexification of formula_35, the Lie algebra of the simply connected compact group SU(n). It is possible to develop the theory of complex semisimple Lie algebras from the compact group perspective, leading to a simpler way to develop the existence and properties of Cartan subalgebras.
In mathematics, a Nichols algebra is a Hopf algebra in a braided category assigned to an object "V" in this category (e.g. a braided vector space). The Nichols algebra is a quotient of the tensor algebra of "V" enjoying a certain universal property and is typically infinite-dimensional. Nichols algebras appear naturally in any pointed Hopf algebra and enabled their classification in important cases. The most well known examples for Nichols algebras are the Borel parts formula_1 of the infinite-dimensional quantum groups when "q" is no root of unity, and the first examples of finite-dimensional Nichols algebras are the Borel parts formula_2 of the Frobenius–Lusztig kernel ("small quantum group") when "q" is a root of unity.
The Dirac matrices were first written down by Paul Dirac when he was trying to write a relativistic first-order wave equation for the electron, and give an explicit isomorphism from the Clifford algebra to the algebra of complex matrices. The result was used to define the Dirac equation and introduce the Dirac operator. The entire Clifford algebra shows up in quantum field theory in the form of Dirac field bilinears.
In mathematics, Jordan operator algebras are real or complex Jordan algebras with the compatible structure of a Banach space. When the coefficients are real numbers, the algebras are called Jordan Banach algebras. The theory has been extensively developed only for the subclass of JB algebras. The axioms for these algebras were devised by . Those that can be realised concretely as subalgebras of self-adjoint operators on a real or complex Hilbert space with the operator Jordan product and the operator norm are called JC algebras. The axioms for complex Jordan operator algebras, first suggested by Irving Kaplansky in 1976, require an involution and are called JB* algebras or Jordan C* algebras. By analogy with the abstract characterisation of von Neumann algebras as C* algebras for which the underlying Banach space is the dual of another, there is a corresponding definition of JBW algebras. Those that can be realised using ultraweakly closed Jordan algebras of self-adjoint operators with the operator Jordan product are called JW algebras. The JBW algebras with trivial center, so-called JBW factors, are classified in terms of von Neumann factors: apart from the exceptional 27 dimensional Albert algebra and the spin factors, all other JBW factors are isomorphic either to the self-adjoint part of a von Neumann factor or to its fixed point algebra under a period two *-anti-automorphism. Jordan operator algebras have been applied in quantum mechanics and in complex geometry, where Koecher's description of bounded symmetric domains using Jordan algebras has been extended to infinite dimensions.
The geometric product was first briefly mentioned by Hermann Grassmann, who was chiefly interested in developing the closely related exterior algebra. In 1878, William Kingdon Clifford greatly expanded on Grassmann's work to form what are now usually called Clifford algebras in his honor (although Clifford himself chose to call them "geometric algebras"). For several decades, geometric algebras went somewhat ignored, greatly eclipsed by the vector calculus then newly developed to describe electromagnetism. The term "geometric algebra" was repopularized in the 1960s by Hestenes, who advocated its importance to relativistic physics.