NationMaster - Encyclopedia: Hypercomplex number

The term hypercomplex number has been used in mathematics for the elements of algebras that extend or go beyond complex number arithmetic. Hypercomplex numbers have had a long lineage of devotees including Hermann Hankel, Georg Frobenius, Eduard Study, and Elie Cartan. Study of particular hypercomplex systems leads to their representation with linear algebra. 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. Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = âˆ’1. ... Hermann Hankel (February 14, 1839 - August 29, 1873) was a German mathematician who was born in Halle, Germany and died in Schramberg (near Tübingen), Germany. ... Picture of Frobenius Ferdinand Georg Frobenius (October 26, 1849 - August 3, 1917) was a German mathematician, best-known for his contributions to the theory of differential equations and to group theory. ... Eduard Study (23 March 1862 - 6 Jan 1930) was a 19th-century German mathematician known for work on invariant theory of ternary forms (1889). ... Élie Joseph Cartan (9 April 1869 - 6 May 1951) was a French mathematician, who did fundamental work in the theory of Lie groups and their geometric applications. ... Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear transformations, and systems of linear equations. ...

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, also known as infinite numbers, are numbers that are not finite. ... The superreal numbers compose a more inclusive category than hyperreal number. ... The system of hyperreal numbers represents a rigorous method of treating the ideas about infinite and infinitesimal numbers that had been used casually by mathematicians, scientists, and engineers ever since the invention of calculus by Newton and Leibniz. ... In mathematics, the surreal numbers are a field containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number, and therefore the surreals are algebraically similar to superreal numbers and hyperreal numbers. ... A number is an abstract entity that represents a count or measurement. ...

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 (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, a field F is said to be algebraically closed if every polynomial of degree at least 1, with coefficients in F, has a zero (root) in F (i. ... In mathematics, the fundamental theorem of algebra states that every complex polynomial in one variable and of degree â‰¥ has some complex root. ...

Distributive numbers with one real and n non-real axes

A comprehensive modern definition of hypercomplex number is given by Kantor and Solodovnikov (see full reference below) as unitary, distributive number systems that contain at least one non-real axis and are closed under addition and multiplication. Axes are generated through real number coefficients $(a_0 ,~... , a_n)$ to bases ${ 1,~i_1, ..., i_n }$ ( $n in { 1, 2, 3... }$ ). The coefficients distribute, associate, and commute with the real (1) and non-real( $~i_n$ ) bases. Three types of $~i_n$ are possible: $i_n^2 in { -1, 0, +1 }$ . In government, see Unitary state In mathematics, see Unitary matrix Unitary operator Unitary group Unitary representation This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ... In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. ...

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 a compatible notion of multiplication of elements of A. A straightforward generalisation allows K to be any commutative ring. ... 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).

Quaternion, octonion, and beyond: Cayley-Dickson construction

Cayley-Dickson construction provides for the extention of complex numbers into number systems with dimensionality $~2^n$ ( $n in {2, 3, 4, ... }$ ). These include the four-dimensional quaternions, eight-dimensional octonions, and 16-dimensional sedenions. Increasing dimensionality introduces 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, the Cayley-Dickson construction produces a sequence of algebras over the field of real numbers, each with twice the dimension of the previous one. ... In mathematics, the quaternions are a non-commutative extension of the complex numbers. ... In mathematics, the octonions are a nonassociative extension of the quaternions. ... The sedenions form a 16-dimensional algebra over the reals obtained by applying the Cayley-Dickson construction to the octonions. ... In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ... In mathematics, associativity is a property that a binary operation can have. ... The sedenions form a 16-dimensional algebra over the reals obtained by applying the Cayley-Dickson construction to the octonions. ... In linear algebra, functional analysis and related areas of mathematics, a norm is a function which assigns a positive length or size to all vectors in a vector space, other than the zero vector. ...

In the definition of Kantor and Solodovnikov, these numbers correspond to anti-commutative bases of type $i_m^2 =~-1$ (with $m in {1, ...,~2^n - 1 }$ ).

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. Beyond octonions, however, this analogy fails since these constructs are not normed anymore. In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ...

Dual number

Dual numbers are to bases ${1, i}$ with nilpotent $i 2 = 0$ . 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 xn = 0. ...

Split-complex algebra

Split-complex numbers are to bases ${1, i}$ with $i 2 = + 1$ a non-real root of 1. They contain idempotents $frac{1}{2} (1~pm i)$ and zero divisors $(1 + i)(1 - i) =~0$ . In mathematics, the split-complex numbers are an extension of the real numbers defined analogously to the complex numbers. ... In mathematics, an idempotent element is an element which, intuitively, leaves something unchanged. ... In abstract algebra, a non-zero element a of a ring R is a left zero divisor if there exists a non-zero b such that ab = 0. ...

A modified Cayley-Dickson construction leads to coquaternions (split-quaternions; e.g. to bases ${ 1,~i_1, i_2, i_3 }$ with $i_1^2 = i_2^2 = +1$ , $i_3^2 = -1$ ) and split-octonions (e.g. to bases ${ 1,~i_1, ... , i_7 }$ with $i_1^2 = i_2^2 = i_3^2 = -1$ , $i_4^2 = ... = i_7^2 = +1$ ). Coquaternions contain nilpotents and have a non-commutative multiplication. Split-octonions are also non-associative. In abstract algebra, a coquaternion is an idea put forward by James Cockle in 1849. ... In mathematics, the split-octonions are a nonassociative extension of the quaternions (or the split-quaternions). ... In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n such that xn = 0. ...

All non-real bases of split-complex algebra are anti-commutative.

Clifford algebra

Clifford algebra is the unitary associative algebra over real, complex, and quaternionic vector spaces equipped with a quadratic form. 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. Clifford algebras are a type of associative algebra in mathematics. ... In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. ...

Tessarine, biquaternion, and conic sedenion

While for Cayley-Dickson constructs, split-complex algebra, and Clifford algebra all non-real bases are anti-commutative, use of a commutative imaginary base leads to four-dimensional tessarines, eight-dimensional biquaternions or Clifford biquaternions, and 16-dimensional conic sedenions. The tessarines are a mathematical idea introduced by James Cockle in 1848. ... In mathematics, a biquaternion is a numeric and geometric concept developed by William Kingdon Clifford, William Rowan Hamilton, and Alexander MacAuley in the nineteenth century. ... A Clifford biquaternion is a concept from geometric algebra. ... Musean Hypernumbers are a concept envisioned by Charles A. MusÃ¨s (1919â€“2000) to form a complete, integrated, connected, and natural number system [1][2][3][4][5]. MusÃ¨s sketched certain fundamental types of hypernumbers and arranged them in ten levels, each with its own associated arithmetic and geometry. ...

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. Biquaternions contain nilpotents, conic sedenions are also not power associative. In mathematics, the absolute value (or modulus1) of a real number is its numerical value without regard to its sign. ... 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.g. conic quaternions, which are isomorphic to tessarines). In mathematics, exponentiation (frequently known colloquially as raising a number to a power) is a process generalized from repeated (or iterated) multiplication, in much the same way that multiplication is a process generalized from repeated addition. ... Logarithms to various bases: is to base e, is to base 10, and is to base 1. ... Musean Hypernumbers are a concept envisioned by Charles A. MusÃ¨s (1919â€“2000) to form a complete, integrated, connected, and natural number system [1][2][3][4][5]. MusÃ¨s sketched certain fundamental types of hypernumbers and arranged them in ten levels, each with its own associated arithmetic and geometry. ...

A. MacFarlane's hyperbolic quaternion

The hyperbolic quaternions (after Alexander MacFarlane) have a non-associative and non-commutative multiplication. Nevertheless, they offer a ring structure somewhat richer than the Minkowski space of special relativity. 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. To meet Wikipedias quality standards, this article or section may require cleanup. ... Alexander MacFarlane (1851 - 1913) was a Scottish-Canadian logician, physicist, and mathematician. ... In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ... The special theory of relativity was proposed in 1905 by Albert Einstein in his article On the Electrodynamics of Moving Bodies. Some three centuries earlier, Galileos principle of relativity had stated that all uniform motion was relative, and that there was no absolute and well-defined state of rest... In physics, spacetime is a mathematical model that combines three-dimensional space and one-dimensional time into a single construct called the space-time continuum, in which time plays the role of the 4th dimension. ...

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 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. For example, they are not necessarily distributive, and not all have a real axis. Musean Hypernumbers are a concept envisioned by Charles A. MusÃ¨s (1919â€“2000) to form a complete, integrated, connected, and natural number system [1][2][3][4][5]. MusÃ¨s sketched certain fundamental types of hypernumbers and arranged them in ten levels, each with its own associated arithmetic and geometry. ... Charles A. MusÃ¨s (1919-2000), also known as Musaios, was the founder of the Lion Path, a shamanistic movement. ... Musean Hypernumbers are a concept envisioned by Charles A. MusÃ¨s (1919â€“2000) to form a complete, integrated, connected, and natural number system [1][2][3][4][5]. MusÃ¨s sketched certain fundamental types of hypernumbers and arranged them in ten levels, each with its own associated arithmetic and geometry. ...

Multicomplex number

Multicomplex numbers are a commutative n dimensional algebra generated by one element e that satisfies $~e^n = -1$ . 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, the multicomplex numbers form a commutative n dimensional algebra generated by one element e which satisfies . ... A bicomplex number is a number written in the form, a + bi1 + ci2 + dj, where i1, i2 and j are imaginary units. ...

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.
Weisstein, Eric W., Hypercomplex number at MathWorld.

Dr. Eric W. Weisstein Encyclopedist Dr. Eric W. Weisstein (born March 18, 1969, in Bloomington, Indiana) is a noted encyclopedist in several technical areas of science and mathematics. ... MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc. ...

External links

History of the Hypercomplexes on hyperjeff.com
Hypercomplex.ru
Clyde Davenport's Commutative Hypercomplex Math Page
The Generalized Number System
Hypercomplex Signal Processing
The Hypercomplex Kalman Filter

Category: Hypercomplex numbers

Results from FactBites:

FRACTINT hypercomplex type (191 words)
	It is not possible to fully generalize the complex numbers to four dimensions without sacrificing some of the algebraic properties shared by real and complex numbers.
	Hypercomplex numbers fail the rule that says all non-zero elements have multiplicative inverses - that is, if z is not 0, there should be a number 1/z such that (1/z)*(z) = 1.
	Hypercomplex numbers were brought to our attention by Clyde Davenport, author of "A Hypercomplex Calculus with Applications to Relativity", ISBN 0-9623837-0-8.

Hypercomplex number (104 words)
	Hypercomplex numbers are extensions of the complex numbers, such as quaternions, octonions and sedenions.
	Whereas complex numbers can be viewed as points in a plane, hypercomplex numbers can be viewed as points in some higher-dimensional Euclidean space (4 dimensions for the quaternions, 8 for the octonions, 16 for the sedenions).
	The Clifford algebras are another family of hypercomplex numbers.

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