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Abstract Algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathematics), modules, vector spaces, lattice (order), lattices, and algebra over a field, algebras over a field. The term ''abstract algebra'' was coined in the early 20th century to distinguish it from older parts of algebra, and more specifically from elementary algebra, the use of variable (mathematics), variables to represent numbers in computation and reasoning. The abstract perspective on algebra has become so fundamental to advanced mathematics that it is simply called "algebra", while the term "abstract algebra" is seldom used except in mathematical education, pedagogy. Algebraic structures, with their associated homomorphisms, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category (mathematics)
In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose objects are sets and whose arrows are functions. ''Category theory'' is a branch of mathematics that seeks to generalize all of mathematics in terms of categories, independent of what their objects and arrows represent. Virtually every branch of modern mathematics can be described in terms of categories, and doing so often reveals deep insights and similarities between seemingly different areas of mathematics. As such, category theory provides an alternative foundation for mathematics to set theory and other proposed axiomatic foundations. In general, the objects and arrows may be abstract entities of any kind, and the n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
La Géométrie
''La Géométrie'' () was published in 1637 as an appendix to ''Discours de la méthode'' ('' Discourse on the Method''), written by René Descartes. In the ''Discourse'', Descartes presents his method for obtaining clarity on any subject. ''La Géométrie'' and two other appendices, also by Descartes, ''La Dioptrique'' (''Optics'') and ''Les Météores'' (''Meteorology''), were published with the ''Discourse'' to give examples of the kinds of successes he had achieved following his method (as well as, perhaps, considering the contemporary European social climate of intellectual competitiveness, to show off a bit to a wider audience). The work was the first to propose the idea of uniting algebra and geometry into a single subject and invented an algebraic geometry called analytic geometry, which involves reducing geometry to a form of arithmetic and algebra and translating geometric shapes into algebraic equations. For its time this was ground-breaking. It also contributed to t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
New Algebra
New or NEW may refer to: Music * New, singer of K-pop group The Boyz * ''New'' (album), by Paul McCartney, 2013 ** "New" (Paul McCartney song), 2013 * ''New'' (EP), by Regurgitator, 1995 * "New" (Daya song), 2017 * "New" (No Doubt song), 1999 * "new", a song by Loona from the 2017 single album '' Yves'' * "The New", a song by Interpol from the 2002 album ''Turn On the Bright Lights'' Transportation * Lakefront Airport, New Orleans, U.S., IATA airport code NEW * Newcraighall railway station, Scotland, station code NEW Other uses * ''New'' (film), a 2004 Tamil movie * New (surname), an English family name * NEW (TV station), in Australia * new and delete (C++), in the computer programming language * Net economic welfare, a proposed macroeconomic indicator * Net explosive weight, also known as net explosive quantity * Network of enlightened Women, an American organization * Newar language, ISO 639-2/3 language code new * Next Entertainment World, a South Korean media com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
François Viète
François Viète (; 1540 – 23 February 1603), known in Latin as Franciscus Vieta, was a French people, French mathematician whose work on new algebra was an important step towards modern algebra, due to his innovative use of letters as parameters in equations. He was a lawyer by trade, and served as a Conseil du Roi, privy councillor to both Henry III of France, Henry III and Henry IV of France, Henry IV of France. Biography Early life and education Viète was born at Fontenay-le-Comte in present-day Vendée. His grandfather was a merchant from La Rochelle. His father, Etienne Viète, was an attorney in Fontenay-le-Comte and a notary in Le Busseau. His mother was the aunt of Barnabé Brisson, a magistrate and the first president of parliament during the ascendancy of the Ligue, Catholic League of France. Viète went to a Franciscan school and in 1558 studied law at Poitiers, graduating as a Bachelor of Laws in 1559. A year later, he began his career as an attorney in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Al-Khwarizmi
Muhammad ibn Musa al-Khwarizmi , or simply al-Khwarizmi, was a mathematician active during the Islamic Golden Age, who produced Arabic-language works in mathematics, astronomy, and geography. Around 820, he worked at the House of Wisdom in Baghdad, the contemporary capital city of the Abbasid Caliphate. One of the most prominent scholars of the period, his works were widely influential on later authors, both in the Islamic world and Europe. His popularizing treatise on algebra, compiled between 813 and 833 as '' Al-Jabr'' (''The Compendious Book on Calculation by Completion and Balancing''),Oaks, J. (2009), "Polynomials and Equations in Arabic Algebra", ''Archive for History of Exact Sciences'', 63(2), 169–203. presented the first systematic solution of linear and quadratic equations. One of his achievements in algebra was his demonstration of how to solve quadratic equations by completing the square, for which he provided geometric justifications. Because al-Khwarizmi was ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rhetorical Algebra
Algebra can essentially be considered as doing computations similar to those of arithmetic but with non-numerical mathematical objects. However, until the 19th century, algebra consisted essentially of the theory of equations. For example, the fundamental theorem of algebra belongs to the theory of equations and is not, nowadays, considered as belonging to algebra (in fact, every proof must use the completeness of the real numbers, which is not an algebraic property). This article describes the history of the theory of equations, referred to in this article as "algebra", from the origins to the emergence of algebra as a separate area of mathematics. Etymology The word "algebra" is derived from the Arabic word , and this comes from the treatise written in the year 830 by the medieval Persian mathematician, Al-Khwārizmī, whose Arabic title, ''Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala'', can be translated as ''The Compendious Book on Calculation by Completion ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Equations
In mathematics, an algebraic equation or polynomial equation is an equation of the form P = 0, where ''P'' is a polynomial with coefficients in some field, often the field of the rational numbers. For example, x^5-3x+1=0 is an algebraic equation with integer coefficients and :y^4 + \frac - \frac + xy^2 + y^2 + \frac = 0 is a multivariate polynomial equation over the rationals. For many authors, the term ''algebraic equation'' refers only to the univariate case, that is polynomial equations that involve only one variable. On the other hand, a polynomial equation may involve several variables (the ''multivariate'' case), in which case the term ''polynomial equation'' is usually preferred. Some but not all polynomial equations with rational coefficients have a solution that is an algebraic expression that can be found using a finite number of operations that involve only those same types of coefficients (that is, can be solved algebraically). This can be done for all such equati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moderne Algebra
''Moderne Algebra'' is a two-volume German textbook on graduate abstract algebra by , originally based on lectures given by Emil Artin in 1926 and by from 1924 to 1928. The English translation of 1949–1950 had the title ''Modern algebra'', though a later, extensively revised edition in 1970 had the title ''Algebra''. The book was one of the first textbooks to use an abstract axiomatic approach to groups, rings, and fields, and was by far the most successful, becoming the standard reference for graduate algebra for several decades. It "had a tremendous impact, and is widely considered to be the major text on algebra in the twentieth century." In 1975 van der Waerden described the sources he drew upon to write the book. In 1997 Saunders Mac Lane recollected the book's influence: * Upon its publication it was soon clear that this was the way that algebra should be presented. * Its simple but austere style set the pattern for mathematical texts in other subjects, from Bana ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axiom
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or fit' or 'that which commends itself as evident'. The precise definition varies across fields of study. In classic philosophy, an axiom is a statement that is so evident or well-established, that it is accepted without controversy or question. In modern logic, an axiom is a premise or starting point for reasoning. In mathematics, an ''axiom'' may be a " logical axiom" or a " non-logical axiom". Logical axioms are taken to be true within the system of logic they define and are often shown in symbolic form (e.g., (''A'' and ''B'') implies ''A''), while non-logical axioms are substantive assertions about the elements of the domain of a specific mathematical theory, for example ''a'' + 0 = ''a'' in integer arithmetic. N ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Equation
In mathematics, an algebraic equation or polynomial equation is an equation of the form P = 0, where ''P'' is a polynomial with coefficients in some field, often the field of the rational numbers. For example, x^5-3x+1=0 is an algebraic equation with integer coefficients and :y^4 + \frac - \frac + xy^2 + y^2 + \frac = 0 is a multivariate polynomial equation over the rationals. For many authors, the term ''algebraic equation'' refers only to the univariate case, that is polynomial equations that involve only one variable. On the other hand, a polynomial equation may involve several variables (the ''multivariate'' case), in which case the term ''polynomial equation'' is usually preferred. Some but not all polynomial equations with rational coefficients have a solution that is an algebraic expression that can be found using a finite number of operations that involve only those same types of coefficients (that is, can be solved algebraically). This can be done for all such eq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial
In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addition, subtraction, multiplication and exponentiation to nonnegative integer powers, and has a finite number of terms. An example of a polynomial of a single indeterminate is . An example with three indeterminates is . Polynomials appear in many areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problem (mathematics education), word problems to complicated scientific problems; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; and they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
