Hypercomplex number

The term hypercomplex number has been used in mathematics for the elements of algebras that extend or go beyond complex number arithmetic. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted Hypercomplex numbers have had a long lineage of devotees including Hermann Hankel, Georg Frobenius, Eduard Study, and Elie Cartan. Hermann Hankel ( February 14, 1839 - August 29, 1873) was a German Mathematician who was born in Halle, Ferdinand Georg Frobenius ( October 26, 1849 – August 3, 1917) was a German Mathematician, best-known for his contributions Eduard Study ( March 23, 1862 &ndash January 6, 1930) was a German Mathematician known for work on Invariant theory Élie Joseph Cartan ( 9 April 1869 &ndash 6 May 1951) was an influential French Mathematician, who did fundamental Study of particular hypercomplex systems leads to their representation with linear algebra. Linear algebra is the branch of Mathematics concerned with This article gives an overview of the key systems, including some not originally considered by the pioneers before modern insight from linear algebra. For details, references, and sources, please follow the particular number type link.

1 Numbers with dimensionality
2 References
3 External links

Numbers with dimensionality

Arguably the most common use of the term hypercomplex number refers to algebraic systems with dimensionality (axes), as contained in the following list. For others (like transfinite number, superreal number, hyperreal number, surreal number) see also under number. Transfinite numbers are Cardinal numbers or Ordinal numbers that are larger than all finite numbers yet not necessarily absolutely infinite. The superreal numbers are an extension of the Real numbers, similar to the Surreal numbers or Hyperreal numbers but comprising a more inclusive category In Mathematics, the surreal number system is an arithmetic continuum containing the real numbers as well as infinite and infinitesimal numbers A number is an Abstract object, tokens of which are Symbols used in Counting and measuring.

Despite their different algebraic properties, it is noted that none of these extensions form a field, because the field of complex numbers is algebraically closed — see fundamental theorem of algebra. In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Mathematics, a field F is said to be algebraically closed if every Polynomial in one Variable of degree at least 1 with Coefficients In Mathematics, the Fundamental theorem of algebra states that every non-constant single-variable Polynomial with complex coefficients has at

Distributive numbers with one real and n non-real axes

A comprehensive modern definition of hypercomplex number is given by Kantor and Solodovnikov ^[1] as unital, distributive number systems that contain at least one non-real axis and are closed under addition and multiplication. In Mathematics, an algebra is unital (some authors say unitary) if it contains a multiplicative Identity element (or unit) i In Mathematics, and in particular in Abstract algebra, distributivity is a property of Binary operations that generalises the distributive law Axes are generated through real number coefficients $(a_0 ,~... , a_n)$ to bases $\{ 1,~i_1, \dots, i_n \}$ ( $n \in \{ 1, 2, 3,\dots \}$ ). The coefficients distribute, associate, and commute with the real (1) and non-real( $~i_n$ ) bases. Three types of $~i_n$ are possible: $i_n^2 \in \{ -1, 0, +1 \}$ .

From a geometric viewpoint, these numbers form a finite-dimensional algebras over the real numbers. In Mathematics, the dimension of a Vector space V is the cardinality (i In Mathematics, an algebra over a field K, or a K -algebra, is a Vector space A over K equipped with In Mathematics, the real numbers may be described informally in several different ways

The following classifications fall under this category. At times, the term 'hypernumber' is used synonymously to 'hypercomplex number' as defined by Kantor and Solodovnikov (but see below for Musean hypernumbers, some of which are not distributive or don't include a real number axis).

One non-real axis

Split-complex numbers

Split-complex numbers are constructed from the bases $\{ 1 , ~j \}$ with $j 2 = + 1$ a non-real root of 1. In Linear algebra, a split-complex number is of the form z = x + y j where j2 = +1, and x and y are Real

Algebras that include such non-real roots of 1 contain idempotents $\tfrac{1}{2} (1 \pm j)$ and zero divisors $(1 + j)(1 - j) = 0$ , so such algebras cannot be division algebras. Idempotence ˌaɪdɨmˈpoʊtəns describes the property of operations in Mathematics and Computer science which means that multiple applications of the operation In Abstract algebra, a nonzero element a of a ring is a left zero divisor if there exists a nonzero b such that ab = 0 In the field of Mathematics called Abstract algebra, a division algebra is roughly speaking an Algebra over a field in which division is possible However, these properties can turn out to be very meaningful, for instance in describing the Lorentz transformations of special relativity. In Physics, the Lorentz transformation converts between two different observers' measurements of space and time where one observer is in constant motion with respect to Special relativity (SR (also known as the special theory of relativity or STR) is the Physical theory of Measurement in Inertial

Dual numbers

Dual numbers have bases ${1,ε}$ with nilpotent $ε 2 = 0$ . A variety of dualities in mathematics are listed at Duality (mathematics. In Mathematics, an element x of a ring R is called nilpotent if there exists some positive Integer n such that

More than one non-real axis

Clifford algebras

Clifford algebra is the unital associative algebra generated over an underlying vector space equipped with a quadratic form. In Mathematics, Clifford algebras are a type of Associative algebra. In Mathematics, a quadratic form is a Homogeneous polynomial of degree two in a number of variables This is equivalent^[2] to being able to define a symmetric scalar product, u. v = ½(uv + vu) that can be used to orthogonalise the quadratic form, to give a set of bases {e₁. In Linear algebra, orthogonalization is the process of finding a set of Orthogonal vectors that span a particular subspace. . . e_k} such that:

$\tfrac{1}{2} (e_i e_j + e_j e_i) = \Bigg\{ \begin{matrix} -1, 0, +1 & i=j, \\ 0 & i \not = j \end{matrix}$

Imposing closure under multiplication now generates a multivector space spanned by 2^k bases, {1, e₁, e₂, e₃, . . . , e₁e₂, . . . , e₁e₂e₃, . . . }. These can be interpreted as the bases of a hypercomplex number system. Unlike the bases {e₁. . . e_k}, the remaining bases may or may not anti-commute, depending on how many simple exchanges must be carried out to swap the two factors. So e₁e₂ = - e₂e₁; but e₁(e₂e₃) = + (e₂e₃)e₁.

Putting aside the bases for which e_i² = 0 (ie directions in the original space over which the quadratic form was degenerate), the remaining Clifford algebras can be identified by the label Cℓ_p,q(R) indicating that the algebra is constructed from p simple bases with e_i² = +1, q with e_i² = -1, and where R indicates that this is to be a Clifford algebra over the reals - ie coefficients of elements of the algebra are to be real numbers. In Mathematics, specifically Linear algebra, a degenerate Bilinear form f(xy on a Vector space V is one such that

These algebras, called geometric algebras, form a systematic set which turn out to be very useful in physics problems which involve rotations, phases, or spins, notably in classical and quantum mechanics, electromagnetic theory and relativity. In Mathematical physics, a geometric algebra is a Multilinear algebra described technically as a Clifford algebra over a real vector space equipped A rotation is a movement of an object in a circular motion A two- Dimensional object rotates around a center (or point) of rotation The phase of an oscillation or wave is the fraction of a complete cycle corresponding to an offset in the displacement from a specified reference point at time t = 0 In Quantum mechanics, spin is a fundamental property of atomic nuclei, Hadrons and Elementary particles For particles with non-zero spin Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons In Classical electromagnetism, Maxwell's equations are a set of four Partial differential equations that describe the properties of the electric This page is about the scientific concept of relativity for philosophical or sociological theories about relativity see Relativism.

Examples include: the complex numbers Cℓ_0,1(R); split-complex numbers Cℓ_1,0(R); quaternions Cℓ_0,2(R); split-biquaternions Cℓ_0,3(R); coquaternions Cℓ_1,1(R) ≈ Cℓ_2,0(R) (the natural algebra of 2d space); Cℓ_3,0(R) (the natural algebra of 3d space, and the algebra of the Pauli matrices); and Cℓ_1,3(R) the space-time algebra. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Linear algebra, a split-complex number is of the form z = x + y j where j2 = +1, and x and y are Real Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician In Mathematics, a split-biquaternion is a member of the Clifford algebra C �( R) In Abstract algebra, the split-quaternions or coquaternions are elements of an Associative algebra introduced by James Cockle in 1849 under the The Pauli matrices are a set of 2 × 2 complex Hermitian and unitary matrices. In Mathematical physics, spacetime algebra is a name for the Clifford algebra C �( R) which can be particularly closely associated

The elements of the algebra Cℓ_p,q(R) form an even subalgebra Cℓ⁰_q+1,p(R) of the algebra Cℓ_q+1,p(R), which can be used to parametrise rotations in the larger algebra. There is thus a close connection between complex numbers and rotations in 2D space; between quaternions and rotations in 3D space; between split-complex numbers and (hyperbolic) rotations (Lorentz transformations) in 1+1 D space, and so on. In Physics, the Lorentz transformation converts between two different observers' measurements of space and time where one observer is in constant motion with respect to

Whereas Cayley-Dickson and split-complex constructs with eight or more dimensions are not associative anymore with respect to multiplication, Clifford algebras retain associativity at any dimensionality.

Quaternion, octonion, and beyond: Cayley-Dickson construction

All of the Clifford algebras Cℓ_p,q(R) apart from the complex numbers and the quaternions contain non-real elements j that square to 1; and so cannot be division algebras. A different approach to extending the complex numbers is taken by the Cayley-Dickson construction. This generates number systems of dimension 2ⁿ, n in {2, 3, 4, . . . }, with bases $\{1, i_1, ..., i_{2^n-1}\}$ , where all the non-real bases anti-commute and satisfy $i_m^2 = -1$ .

The first algebras in this sequence are the four-dimensional quaternions, eight-dimensional octonions, and 16-dimensional sedenions. Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician In Mathematics, the octonions are a nonassociative extension of the Quaternions Their 8-dimensional Normed division algebra over the Real In Abstract algebra, sedenions form a 16- dimensional algebra over the reals. However, satisfying these requirements comes at a price: Each increase in dimensionality introduces new algebraic complications. Quaternion multiplication is not commutative anymore, octonion multiplication additionally is non-associative, and sedenions do not form a normed space with multiplicative norm. In Mathematics, commutativity is the ability to change the order of something without changing the end result In Mathematics, associativity is a property that a Binary operation can have In Abstract algebra, sedenions form a 16- dimensional algebra over the reals. In Linear algebra, Functional analysis and related areas of Mathematics, a norm is a function that assigns a strictly positive length

Because quaternions and octonions offer a (multiplicative) norm similar to lengths in four and eight dimensional Euclidean vector space respectively, these numbers can be referred to as points in some higher-dimensional Euclidean space. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added Beyond octonions, however, this analogy fails since these constructs are not normed anymore.

Modified Cayley-Dickson construction

The Cayley-Dickson construction can be modified by starting with the split-complex numbers rather than the complex numbers. This leads to coquaternions (split-quaternions; e. In Abstract algebra, the split-quaternions or coquaternions are elements of an Associative algebra introduced by James Cockle in 1849 under the g. to bases $\{ 1,~i_1, i_2, i_3 \}$ with $i_1^2 = -1, i_2^2 = i_3^2 = +1$ , ) and split-octonions (e. In Mathematics, the split-octonions are a Nonassociative extension of the Quaternions (or the Split-quaternions. g. to bases $\{ 1,~i_1, \dots , i_7 \}$ with $i_1^2 = i_2^2 = i_3^2 = -1$ , $i_4^2 = \cdots = i_7^2 = +1$ ). The coquaternions contain nilpotents, have a non-commutative multiplication, and are isomorphic to real matrices (2 x 2). In Mathematics, an element x of a ring R is called nilpotent if there exists some positive Integer n such that The 2 x 2 real matrices are the Linear mappings of the Cartesian coordinate system into itself by the rule (xy \mapsto (xy\begin{pmatrix}a & c Split-octonions are non-associative.

All non-real bases of split Cayley-Dickinson algebras are anti-commutative.

Complexified algebras: Tessarine, biquaternion, and conic sedenion

While for the Cayley-Dickson constructs and the split Cayley-Dickson constructs all non-real bases are anti-commutative, use of a commutative imaginary base leads to four-dimensional tessarines $\mathbb C\otimes\mathbb C$ , eight-dimensional biquaternions $\mathbb C\otimes\mathbb H$ , and 16-dimensional conic sedenions $\mathbb C\otimes\mathbb O$ . The tessarines are a mathematical idea introduced by James Cockle in 1848 The biquaternions are the numbers w + xi + yj + zk \ \! where w x y and z are complex numbers and the elements of {1 i j k} multiply as in the Quaternion group Musean hypernumbers are an algebraic concept envisioned by Charles A

Tessarines offer a commutative and associative multiplication, biquaternions are associative but not commutative, and conic sedenions are not associative and not commutative. They all contain idempotents and zero-divisors, are not normed, but offer a multiplicative modulus. In Mathematics, the absolute value (or modulus) of a Real number is its numerical value without regard to its sign. Biquaternions contain nilpotents, conic sedenions are also not power associative. In Abstract algebra, power associativity is a weak form of Associativity.

With the exception of their idempotents, zero-divisors, and nilpotents, the arithmetic of these numbers is closed with respect to multiplication, division, exponentiation, and logarithms (see e. In Mathematics, the logarithm of a number to a given base is the power or Exponent to which the base must be raised in order to produce g. conic quaternions, which are isomorphic to tessarines). Musean hypernumbers are an algebraic concept envisioned by Charles A

Alexander MacFarlane's hyperbolic quaternion

The hyperbolic quaternions (after Alexander MacFarlane) have a non-associative and non-commutative multiplication. In Mathematics, a hyperbolic quaternion is a mathematical concept first suggested by Alexander MacFarlane in 1891 in a speech to the American Association Alexander Macfarlane ( April 21 1851 – August 28, 1913) was a Scottish - Canadian Logician Physicist Nevertheless, they offer a ring structure somewhat richer than the Minkowski space of special relativity. In Physics and Mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einstein's theory of Special relativity Special relativity (SR (also known as the special theory of relativity or STR) is the Physical theory of Measurement in Inertial All bases are roots of 1, i. e. $i_n^2 = +1$ for $n \in \{ 1, 2, 3 \}$ . This structure is of historical and educational interest since it was a spectacle of the 1890s that presaged the spacetime revolution of the following decade. SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS

Musean hypernumber

While Kantor and Solodovnikov generalize multiplication for numbers of more than one dimension through distributive rectangular (Cartesian coordinate) products, hypernumbers after Charles A. Musès use an approach to generalization by means of absolutes and angles. Musean hypernumbers are an algebraic concept envisioned by Charles A Charles A Muses (1919&ndash2000 a figure who wrote articles and books under various pseudonyms (including Musès Musaios Kyril Demys Arthur Fontaine Kenneth Demarest and Carl Musean hypernumbers are organized in 'levels' which correspond to different algebraic properties. While arithmetics built on the first three levels (to real, imaginary $i = \sqrt{-1}$ , and counterimaginary $\varepsilon = \sqrt{+1} \ne \pm 1$ bases) are contained in the definition by Kantor and Solodovnikov (see hypernumbers for isomorphisms to numbers mentioned above), the remaining levels offer additional arithmetical properties. Musean hypernumbers are an algebraic concept envisioned by Charles A For example, they are not necessarily distributive, and not all have a real axis.

Multicomplex number

Multicomplex numbers are a commutative n-dimensional algebra generated by one element e that satisfies $~e^n = -1$ . In Mathematics, the multicomplex numbers, {\Bbb{MC}}_n form an n dimensional algebra generated by one element e which satisfies ~e^n A special case are the bicomplex numbers which are isomorphic to tessarines, conic quaternions (from Musès' hypernumbers), and are also contained in the 'hypercomplex number' definition by Kantor and Solodovnikov. In Mathematics, a bicomplex number (from the Multicomplex numbers see e

References

^ I. L. Kantor, A. S. Solodovnikov, "Hypercomplex numbers: an elementary introduction to algebras"; translated by A. Shenitzer (original in Russian). New York: Springer-Verlag, c. 1989.
^ This equivalence applies except of the very special case of vector spaces where addition is defined with characteristic m = 2; but the linear spaces in this article allow multiplication by any real scalar, so that situation does not arise. In Mathematics, the characteristic of a ring R, often denoted char( R) is defined to be the smallest number of times one must add the ring's

Eric W. Weisstein, Hypercomplex number at MathWorld. Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W
Olariu, S. (2000). "Complex Numbers in n Dimensions" arxiv:math. CV/0011044/v1.
Jeanne La Duke "The study of linear associative algebras in the United States, 1870 - 1927", see pp. 147-159 of Emmy Noether in Bryn Mawr Bhama Srinivasan & Judith Sally editors, Springer Verlag 1983.

External links

Dictionary

-noun

(mathematics) A complex number consisting of multiple square roots of negative one, creating multiple simulated dimensions.

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