Euler's four-square identityIn mathematics, Euler's four-square identity says that the product of two numbers, each of which is a sum of four squares, is itself a sum of four squares. For any pair of quadruples from a commutative ring, the following expressions are equal: Euler wrote about this identity in a letter dated May 4, 1748 to Goldbach (but he used a different sign convention from the above). It can be verified with elementary algebra. The identity was used by Lagrange to prove his four square theorem.
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.
Quaternion algebraIn mathematics, a quaternion algebra over a field F is a central simple algebra A over F that has dimension 4 over F. Every quaternion algebra becomes a matrix algebra by extending scalars (equivalently, tensoring with a field extension), i.e. for a suitable field extension K of F, is isomorphic to the 2 × 2 matrix algebra over K. The notion of a quaternion algebra can be seen as a generalization of Hamilton's quaternions to an arbitrary base field.
Split-complex numberIn algebra, a split complex number (or hyperbolic number, also perplex number, double number) is based on a hyperbolic unit j satisfying A split-complex number has two real number components x and y, and is written The conjugate of z is Since the product of a number z with its conjugate is an isotropic quadratic form. The collection D of all split complex numbers for x,y \in \R forms an algebra over the field of real numbers. Two split-complex numbers w and z have a product wz that satisfies This composition of N over the algebra product makes (D, +, ×, *) a composition algebra.
Split-octonionIn mathematics, the split-octonions are an 8-dimensional nonassociative algebra over the real numbers. Unlike the standard octonions, they contain non-zero elements which are non-invertible. Also the signatures of their quadratic forms differ: the split-octonions have a split signature (4,4) whereas the octonions have a positive-definite signature (8,0). Up to isomorphism, the octonions and the split-octonions are the only two 8-dimensional composition algebras over the real numbers.
Split-quaternionIn abstract algebra, the split-quaternions or coquaternions form an algebraic structure introduced by James Cockle in 1849 under the latter name. They form an associative algebra of dimension four over the real numbers. After introduction in the 20th century of coordinate-free definitions of rings and algebras, it was proved that the algebra of split-quaternions is isomorphic to the ring of the 2×2 real matrices.
ComplexificationIn mathematics, the complexification of a vector space V over the field of real numbers (a "real vector space") yields a vector space V^C over the complex number field, obtained by formally extending the scaling of vectors by real numbers to include their scaling ("multiplication") by complex numbers. Any basis for V (a space over the real numbers) may also serve as a basis for V^C over the complex numbers. Let be a real vector space.
Octonion algebraIn mathematics, an octonion algebra or Cayley algebra over a field F is a composition algebra over F that has dimension 8 over F. In other words, it is a unital non-associative algebra A over F with a non-degenerate quadratic form N (called the norm form) such that for all x and y in A. The most well-known example of an octonion algebra is the classical octonions, which are an octonion algebra over R, the field of real numbers. The split-octonions also form an octonion algebra over R.
BiquaternionIn abstract algebra, the biquaternions are the numbers w + x i + y j + z k, where w, x, y, and z are complex numbers, or variants thereof, and the elements of {1, i, j, k} multiply as in the quaternion group and commute with their coefficients. There are three types of biquaternions corresponding to complex numbers and the variations thereof: Biquaternions when the coefficients are complex numbers. Split-biquaternions when the coefficients are split-complex numbers. Dual quaternions when the coefficients are dual numbers.
Null vectorIn mathematics, given a vector space X with an associated quadratic form q, written (X, q), a null vector or isotropic vector is a non-zero element x of X for which q(x) = 0. In the theory of real bilinear forms, definite quadratic forms and isotropic quadratic forms are distinct. They are distinguished in that only for the latter does there exist a nonzero null vector. A quadratic space (X, q) which has a null vector is called a pseudo-Euclidean space.
