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Brahmagupta–Fibonacci Identity
In algebra, the Brahmagupta–Fibonacci identity expresses the product of two sums of two squares as a sum of two squares in two different ways. Hence the set of all sums of two squares is closed under multiplication. Specifically, the identity says :\begin \left(a^2 + b^2\right)\left(c^2 + d^2\right) & = \left(ac-bd\right)^2 + \left(ad+bc\right)^2 & & (1) \\ & = \left(ac+bd\right)^2 + \left(ad-bc\right)^2. & & (2) \end For example, :(1^2 + 4^2)(2^2 + 7^2) = 26^2 + 15^2 = 30^2 + 1^2. The identity is also known as the Diophantus identity, Daniel Shanks, Solved and unsolved problems in number theory, p.209, American Mathematical Society, Fourth edition 1993. as it was first proved by Diophantus of Alexandria. It is a special case of Euler's four-square identity, and also of Lagrange's identity. Brahmagupta proved and used a more general identity (the Brahmagupta identity), equivalent to :\begin \left(a^2 + nb^2\right)\left(c^2 + nd^2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebra
Algebra () is one of the areas of mathematics, broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary algebra deals with the manipulation of variable (mathematics), variables (commonly represented by Roman letters) as if they were numbers and is therefore essential in all applications of mathematics. Abstract algebra is the name given, mostly in mathematical education, education, to the study of algebraic structures such as group (mathematics), groups, ring (mathematics), rings, and field (mathematics), fields (the term is no more in common use outside educational context). Linear algebra, which deals with linear equations and linear mappings, is used for modern presentations of geometry, and has many practical applications (in weather forecasting, for example). There are many areas of mathematics that belong to algebra, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Composition Algebra
In mathematics, a composition algebra over a field is a not necessarily associative algebra over together with a nondegenerate quadratic form that satisfies :N(xy) = N(x)N(y) for all and in . A composition algebra includes an involution called a conjugation: x \mapsto x^*. The quadratic form N(x) = x x^* is called the norm of the algebra. A composition algebra (''A'', ∗, ''N'') is either a division algebra or a split algebra, depending on the existence of a non-zero ''v'' in ''A'' such that ''N''(''v'') = 0, called a null vector. When ''x'' is ''not'' a null vector, the multiplicative inverse of ''x'' is When there is a non-zero null vector, ''N'' is an isotropic quadratic form, and "the algebra splits". Structure theorem Every unital composition algebra over a field can be obtained by repeated application of the Cayley–Dickson construction starting from (if the characteristic of is different from ) or a 2-dimensional composition subalgebra (if ). ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hurwitz's Theorem (composition Algebras)
In mathematics, Hurwitz's theorem is a theorem of Adolf Hurwitz (1859–1919), published posthumously in 1923, solving the Hurwitz problem for finite-dimensional unital real non-associative algebras endowed with a positive-definite quadratic form. The theorem states that if the quadratic form defines a homomorphism into the positive real numbers on the non-zero part of the algebra, then the algebra must be isomorphic to the real numbers, the complex numbers, the quaternions, or the octonions. Such algebras, sometimes called Hurwitz algebras, are examples of composition algebras. The theory of composition algebras has subsequently been generalized to arbitrary quadratic forms and arbitrary fields. Hurwitz's theorem implies that multiplicative formulas for sums of squares can only occur in 1, 2, 4 and 8 dimensions, a result originally proved by Hurwitz in 1898. It is a special case of the Hurwitz problem, solved also in . Subsequent proofs of the restrictions on the dimension h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pfister's Sixteen-square Identity
In algebra, Pfister's sixteen-square identity is a non- bilinear identity of form \left(x_1^2+x_2^2+x_3^2+\cdots+x_^2\right)\left(y_1^2+y_2^2+y_3^2+\cdots+y_^2\right) = z_1^2+z_2^2+z_3^2+\cdots+z_^2 It was first proven to exist by H. Zassenhaus and W. Eichhorn in the 1960s, and independently by Albrecht Pfister around the same time. There are several versions, a concise one of which is \begin &\scriptstyle \\ &\scriptstyle \\ &\scriptstyle \\ &\scriptstyle \\ &\scriptstyle \\ &\scriptstyle \\ &\scriptstyle \\ &\scriptstyle \\ &\scriptstyle \\ &\scriptstyle \\ &\scriptstyle \\ &\scriptstyle \\ &\scriptstyle \\ &\scriptstyle \\ &\scriptstyle \\ &\scriptstyle \end If all x_i and y_i with i>8 are set equal to zero, then it reduces to Degen's eight-square identity (in blue). The u_i are \begin &u_1 = \tfrac \\ &u_2 = \tfrac \\ &u_3 = \tfrac \\ &u_4 = \tfrac \\ &u_5 = \tfrac \\ &u_6 = \tfrac \\ &u_7 = \tfrac \\ &u_8 = \tfrac \end and, a=-1,\;\;b=0,\;\;c=x_1^2+x_2^2+x_3^2+x_4^2+x_5^ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bott Periodicity
In mathematics, the Bott periodicity theorem describes a periodicity in the homotopy groups of classical groups, discovered by , which proved to be of foundational significance for much further research, in particular in K-theory of stable complex vector bundles, as well as the stable homotopy groups of spheres. Bott periodicity can be formulated in numerous ways, with the periodicity in question always appearing as a period-2 phenomenon, with respect to dimension, for the theory associated to the unitary group. See for example topological K-theory. There are corresponding period-8 phenomena for the matching theories, ( real) KO-theory and (quaternionic) KSp-theory, associated to the real orthogonal group and the quaternionic symplectic group, respectively. The J-homomorphism is a homomorphism from the homotopy groups of orthogonal groups to stable homotopy groups of spheres, which causes the period 8 Bott periodicity to be visible in the stable homotopy groups of spheres. St ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Octonions
In mathematics, the octonions are a normed division algebra over the real numbers, a kind of Hypercomplex number, hypercomplex Number#Classification, number system. The octonions are usually represented by the capital letter O, using boldface or blackboard bold \mathbb O. Octonions have eight dimension (vector space), dimensions; twice the number of dimensions of the quaternions, of which they are an extension. They are commutative property, noncommutative and associative property, nonassociative, but satisfy a weaker form of associativity; namely, they are alternative algebra, alternative. They are also Power associativity, power associative. Octonions are not as well known as the quaternions and complex numbers, which are much more widely studied and used. Octonions are related to exceptional structures in mathematics, among them the Simple Lie group#Exceptional cases, exceptional Lie groups. Octonions have applications in fields such as string theory, special relativity and qua ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Degen's Eight-square Identity
In mathematics, Degen's eight-square identity establishes that the product of two numbers, each of which is a sum of eight squares, is itself the sum of eight squares. Namely: \begin & \left(a_1^2+a_2^2+a_3^2+a_4^2+a_5^2+a_6^2+a_7^2+a_8^2\right)\left(b_1^2+b_2^2+b_3^2+b_4^2+b_5^2+b_6^2+b_7^2+b_8^2\right) = \\ ex & \quad \left(a_1 b_1 - a_2 b_2 - a_3 b_3 - a_4 b_4 - a_5 b_5 - a_6 b_6 - a_7 b_7 - a_8 b_8\right)^2+ \\ & \quad \left(a_1 b_2 + a_2 b_1 + a_3 b_4 - a_4 b_3 + a_5 b_6 - a_6 b_5 - a_7 b_8 + a_8 b_7\right)^2+ \\ & \quad \left(a_1 b_3 - a_2 b_4 + a_3 b_1 + a_4 b_2 + a_5 b_7 + a_6 b_8 - a_7 b_5 - a_8 b_6\right)^2+ \\ & \quad \left(a_1 b_4 + a_2 b_3 - a_3 b_2 + a_4 b_1 + a_5 b_8 - a_6 b_7 + a_7 b_6 - a_8 b_5\right)^2+ \\ & \quad \left(a_1 b_5 - a_2 b_6 - a_3 b_7 - a_4 b_8 + a_5 b_1 + a_6 b_2 + a_7 b_3 + a_8 b_4\right)^2+ \\ & \quad \left(a_1 b_6 + a_2 b_5 - a_3 b_8 + a_4 b_7 - a_5 b_2 + a_6 b_1 - a_7 b_4 + a_8 b_3\right)^2+ \\ & \quad \left(a_1 b_7 + a_2 b_8 + a_3 b_5 - a_4 b_6 - ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quaternions
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quaternion as the quotient of two ''directed line segment, directed lines'' in a three-dimensional space, or, equivalently, as the quotient of two vector (geometry), vectors. Multiplication of quaternions is noncommutative. Quaternions are generally represented in the form :a + b\ \mathbf i + c\ \mathbf j +d\ \mathbf k where , and are real numbers; and , and are the ''basic quaternions''. Quaternions are used in pure mathematics, but also have practical uses in applied mathematics, particularly for quaternions and spatial rotation, calculations involving three-dimensional rotations, such as in 3D computer graphics, three-dimensional computer graphics, computer vision, and Texture (crystalline), crystallographic texture analysis. They can be u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The Book Of Squares
''The Book of Squares'', ''(Liber Quadratorum'' in the original Latin) is a book on algebra by Leonardo Fibonacci, published in 1225. It was dedicated to Frederick II, Holy Roman Emperor. The ''Liber quadratorum'' has been passed down by a single 15th-century manuscript, the so-called ms. ''E 75 Sup.'' of the Biblioteca Ambrosiana (Milan, Italy), ff. 19r-39v. During the 19th century, the work has been published for the first time in a printed edition by Baldassarre Boncompagni Ludovisi, prince of Piombino. Appearing in the book is Fibonacci's identity, establishing that the set of all sums of two squares is closed under multiplication. The book anticipated the works of later mathematicians such as Fermat and Euler. The book examines several topics in number theory, among them an inductive method for finding Pythagorean triples based on the sequence of odd integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fibonacci
Fibonacci (; also , ; – ), also known as Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano ('Leonardo the Traveller from Pisa'), was an Italian mathematician from the Republic of Pisa, considered to be "the most talented Western mathematician of the Middle Ages". The name he is commonly called, ''Fibonacci'', was made up in 1838 by the Franco-Italian historian Guillaume Libri and is short for ('son of Bonacci'). However, even earlier in 1506 a notary of the Holy Roman Empire, Perizolo mentions Leonardo as "Lionardo Fibonacci". Fibonacci popularized the Indo–Arabic numeral system in the Western world primarily through his composition in 1202 of '' Liber Abaci'' (''Book of Calculation''). He also introduced Europe to the sequence of Fibonacci numbers, which he used as an example in ''Liber Abaci''. Biography Fibonacci was born around 1170 to Guglielmo, an Italian merchant and customs official. Guglielmo directed a trading post in Bugia (Béjaïa) in mode ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Latin
Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power of the Roman Republic it became the dominant language in the Italian region and subsequently throughout the Roman Empire. Even after the fall of Western Rome, Latin remained the common language of international communication, science, scholarship and academia in Europe until well into the 18th century, when other regional vernaculars (including its own descendants, the Romance languages) supplanted it in common academic and political usage, and it eventually became a dead language in the modern linguistic definition. Latin is a highly inflected language, with three distinct genders (masculine, feminine, and neuter), six or seven noun cases (nominative, accusative, genitive, dative, ablative, and vocative), five declensions, four ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
