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In [[mathematics]], a '''hypercomplex number''' is a traditional term for an [[element (mathematics)|element]] of an [[algebra over a field]] where the [[field (mathematics)|field]] is the [[real number]]s or the [[complex number]]s. In the nineteenth century [[number system]]s called [[quaternion]]s, [[tessarine]]s, [[coquaternion]]s, [[biquaternion]]s, and [[octonion]]s became established concepts in mathematical literature. The concept of a hypercomplex number covered them all, and called for a science to explain and classify them.
 
The cataloguing project began in 1872 when [[Benjamin Peirce]] first published his ''Linear Associative Algebra'', and was carried forward by his son [[Charles Sanders Peirce]].<ref>Linear Associative Algebra (1881) [[American Journal of Mathematics]] 4(1):221&ndash;6</ref>  Most significantly, they identified the [[nilpotent]] and the [[idempotent element]]s as useful hypercomplex numbers for classifications. The [[Cayley–Dickson construction]] used [[involution (mathematics)|involution]]s to generate complex numbers, quaternions, and octonions out of the real number system. Hurwitz and Frobenius proved theorems that put limits on hypercomplexity: [[Hurwitz's theorem (normed division algebras)]], and [[Frobenius theorem (real division algebras)]].
 
It was [[matrix (mathematics)|matrix algebra]] that harnessed the hypercomplex systems. First, matrices contributed new hypercomplex numbers like [[2 × 2 real matrices]]. Soon the matrix paradigm began to explain the others as they became represented by matrices and their operations.  In 1907 [[Joseph Wedderburn]] showed that associative hypercomplex systems could be represented by matrices, or direct sums of systems of matrices. From that date the preferred term for a hypercomplex system became [[associative algebra]] as seen in the title of Wedderburn’s thesis at [[University of Edinburgh]]. Note however, that non-associative systems like octonions and [[hyperbolic quaternion]]s represent another type of hypercomplex number.
 
As Hawkins (1972) explains, the hypercomplex numbers are stepping stones to learning about [[Lie group]]s and [[group representation]] theory. For instance, in 1929 [[Emmy Noether]] at [[Bryn Mawr College|Bryn Mawr]] wrote on "hypercomplex quantities and representation theory".
Review of the historic particulars gives body to the generalities of modern theory. In 1973 Kantor and Solodovnikov published a textbook on hypercomplex numbers which was translated in 1989; a reviewer says it has a "highly classical flavour". See [[Karen Parshall]] (1985) for a detailed exposition of the heyday of hypercomplex numbers, including the role of such luminaries as [[Theodor Molien]] and [[Eduard Study]]. For the transition to [[Abstract algebra|modern algebra]], [[Bartel van der Waerden]] devotes thirty pages to hypercomplex numbers in his ''History of Algebra'' (1985).
 
==Definition==
A definition of a '''hypercomplex number''' is given by {{harvtxt|Kantor|Solodovnikov|1989}} as an element of a finite dimensional algebra over the real numbers that is [[unital algebra|unital]] and [[distributive]] (but not necessarily associative). Elements are generated with real number coefficients <math>(a_0, \dots, a_n)</math> for a basis <math>\{ 1, i_1, \dots, i_n \}</math>. Where possible, it is conventional to choose the basis so that <math>i_k^2 \in \{ -1, 0, +1 \}</math>. A technical approach to hypercomplex numbers directs attention first to those of [[dimension]] two. Higher dimensions are configured as Cliffordian or algebraic sums of other algebras.
 
==Two-dimensional real algebras==
'''Theorem:'''<ref>[[Isaak Yaglom]] (1968) ''Complex Numbers in Geometry'', pages 10 to 14</ref><ref>John H. Ewing editor (1991) ''Numbers'', page 237, Springer, ISBN 3-540-97497-0 </ref><ref>Kantor & Solodovnikov (1978) 14,15</ref>
Up to isomorphism, there are exactly three 2-dimensional unital algebras over the reals: the ordinary [[complex number]]s, the [[split-complex number]]s, and the [[dual number]]s.
 
:proof: Since the algebra is [[closure (mathematics)|closed]] under squaring, and it has but two dimensions, the non-real basis element ''u'' squares to an arbitrary linear combination of 1 and ''u'':
:<math>u^2=a_0+a_1u</math>
with arbitrary real numbers a<sub>0</sub> and a<sub>1</sub>.
Using the common method of [[completing the square]] by
subtracting a<sub>1</sub>u and adding the quadratic complement a<sub>1</sub>²/4 to both sides yields
:<math>u^2-a_1u+\frac{a_1^2}{4}=a_0+\frac{a_1^2}{4} .</math>
:<math>u^2-a_1u+\frac{a_1^2}{4}=\left(u-\frac{a_1}{2}\right)^2= \tilde{u}^2</math>&nbsp; so that
:<math>\tilde{u}^2~=a_0+\frac{a_1^2}{4} .</math>
The three cases depend on this real value:
* If 4''a<sub>0</sub>'' = &minus;''{{SubSup|a|1|2}}'', the above formula yields ''ũ''<sup>2</sup> = 0. Hence, ''ũ'' can directly be identified with the [[nilpotent]] element <math>\epsilon</math> of the Dual numbers' basis <math>\{ 1, ~\epsilon \}</math>.
*If 4''a<sub>0</sub>'' > &minus;''{{SubSup|a|1|2}}'', the above formula yields ''ũ''<sup>2</sup> > 0. This leads to the split-complex numbers which have normalized basis <math>\{ 1 , ~j \}</math> with <math>j^2 = +1</math>. To obtain ''j'' from ''ũ'', the latter must be divided by the positive real number <math>a:=\sqrt{a_0+\frac{a_1^2}{4}}</math> which has the same square as ''ũ''.
*If 4''a<sub>0</sub>'' < &minus;''{{SubSup|a|1|2}}'', the above formula yields ''ũ''<sup>2</sup> < 0. This leads to the complex numbers which have normalized basis <math>\{ 1 , ~i \}</math> with <math>i^2 = -1</math>. To yield ''i'' from ''ũ'', the latter has to be divided by a positive real number <math>a:=\sqrt{\frac{a_1^2}{4}-a_0}</math> which squares to the negative of ''ũ''<sup>2</sup>.
 
The complex numbers are the only two-dimensional hypercomplex algebra that is a [[Field (mathematics)|field]].
Algebras such as the split-complex numbers that include non-real roots of 1 also contain [[idempotent element|idempotent]]s <math>\tfrac{1}{2} (1 \pm j)</math> and [[zero divisor]]s <math>(1 + j)(1 - j) = 0</math>, so such algebras cannot be [[division algebra]]s. However, these properties can turn out to be very meaningful, for instance in describing the [[Lorentz transformations]] of [[special relativity]].
 
Through the use of the concept of a [[projective line over a ring]], it has been shown that [[cross-ratio]]s can be formed in any of these planar hypercomplex number systems.<ref>Sky Brewer (2013) [http://link.springer.com/article/10.1007/s00006-012-0335-7 "Projective Cross-ratio on Hypercomplex Numbers"], [[Advances in Applied Clifford Algebras]] 23(1):1&ndash;14</ref>
 
==Higher dimensional examples (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]]. Over the real numbers this is equivalent to being able to define a symmetric scalar product, ''u''⋅''v'' = ½(''uv'' + ''vu'') that can be used to [[orthogonalization|orthogonalise]] the quadratic form, to give a set of bases {''e''<sub>1</sub>, ..., ''e''<sub>''k''</sub>} such that:
::<math>\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} </math>
Imposing closure under multiplication now generates a multivector space spanned by 2<sup>''k''</sup> bases, {1, ''e''<sub>1</sub>, ''e''<sub>2</sub>, ''e''<sub>3</sub>, ..., ''e''<sub>1</sub>''e''<sub>2</sub>, ..., ''e''<sub>1</sub>''e''<sub>2</sub>''e''<sub>3</sub>, ...}.  These can be interpreted as the bases of a hypercomplex number system. Unlike the bases {''e''<sub>1</sub>, ..., ''e''<sub>''k''</sub>}, 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 ''e''<sub>1</sub>''e''<sub>2</sub> = −''e''<sub>2</sub>''e''<sub>1</sub>; but  ''e''<sub>1</sub>(''e''<sub>2</sub>''e''<sub>3</sub>) = +(''e''<sub>2</sub>''e''<sub>3</sub>)''e''<sub>1</sub>.
 
Putting aside the bases for which ''e''<sub>''i''</sub><sup>2</sup> = 0 (i.e. directions in the original space over which the quadratic form was [[degenerate form|degenerate]]), the remaining Clifford algebras can be identified by the label ''C''ℓ<sub>''p'',''q''</sub>('''R''') indicating that the algebra is constructed from ''p'' simple bases with ''e''<sub>''i''</sub><sup>2</sup> = +1, ''q'' with ''e''<sub>''i''</sub><sup>2</sup> = −1, and where '''R''' indicates that this is to be a Clifford algebra over the reals—i.e. coefficients of elements of the algebra are to be real numbers.
 
These algebras, called [[geometric algebra]]s, form a systematic set which turn out to be very useful in physics problems which involve [[rotation]]s, [[phase (waves)|phase]]s, or [[Spin (physics)|spin]]s, notably in [[classical mechanics|classical]] and [[quantum mechanics]], [[electromagnetic theory]] and [[theory of relativity|relativity]].
 
Examples include: the [[complex number]]s ''C''ℓ<sub>0,1</sub>('''R'''); [[split-complex number]]s ''C''ℓ<sub>1,0</sub>('''R'''); [[quaternion]]s ''C''ℓ<sub>0,2</sub>('''R'''); [[split-biquaternion]]s ''C''ℓ<sub>0,3</sub>('''R'''); [[coquaternion]]s ''C''ℓ<sub>1,1</sub>('''R''') ≈ ''C''ℓ<sub>2,0</sub>('''R''') (the natural algebra of 2d space); ''C''ℓ<sub>3,0</sub>('''R''') (the natural algebra of 3d space, and the algebra of the [[Pauli matrices]]); and ''C''ℓ<sub>1,3</sub>('''R''') the [[spacetime algebra]].
 
The elements of the algebra ''C''ℓ<sub>''p'',''q''</sub>('''R''') form an even subalgebra ''C''ℓ<sup>0</sup><sub>''q''+1,''p''</sub>('''R''') of the algebra ''C''ℓ<sub>''q''+1,''p''</sub>('''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.
 
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.
 
In 1995 [[Ian R. Porteous]] wrote on "The recognition of subalgebras" in his book on Clifford algebras. His Proposition 11.4 summarizes the hypercomplex cases:<ref>[[Ian R. Porteous]] (1995) ''Clifford Algebras and the Classical Groups'', pages 88 & 89, [[Cambridge University Press]] ISBN 0-521-55177-3 </ref>
:Let ''A'' be a real associative algebra with unit element 1. Then
* 1 generates '''R''' ([[real number|algebra of real numbers]]),
* any two-dimensional subalgebra generated by an element e<sub>0</sub> of ''A'' such that e<sub>0</sub><sup>2</sup> = −1 is isomorphic to '''C''' ([[complex number|algebra of complex number]]s),
* any two-dimensional subalgebra generated by an element e<sub>0</sub> of ''A'' such that e<sub>0</sub><sup>2</sup> = 1 is isomorphic to <sup>2</sup>'''R''' ([[split-complex number|algebra of split-complex numbers]]),
* any four-dimensional subalgebra generated by a set {e<sub>0</sub>, e<sub>1</sub>} of mutually anti-commuting elements of ''A'' such that <math>e_0 ^2 = e_1 ^2 = -1</math> is isomorphic to '''H''' ([[quaternion|algebra of quaternions]]),
* any four-dimensional subalgebra generated by a set {e<sub>0</sub>, e<sub>1</sub>} of mutually anti-commuting elements of ''A'' such that <math>e_0 ^2 = e_1 ^2 = 1</math> is isomorphic to '''R'''(2) ([[2 × 2 real matrices]], [[coquaternion]]s),
* any eight-dimensional subalgebra generated by a set {e<sub>0</sub>, e<sub>1</sub>, e<sub>2</sub>} of mutually anti-commuting elements of ''A'' such that <math>e_0 ^2 = e_1 ^2 = e_2 ^2 = -1</math> is isomorphic to <sup>2</sup>'''H''' ([[split-biquaternion]]s),
* any eight-dimensional subalgebra generated by a set {e<sub>0</sub>, e<sub>1</sub>, e<sub>2</sub>} of mutually anti-commuting elements of ''A'' such that <math>e_0 ^2 = e_1 ^2 = e_2 ^2 = 1</math> is isomorphic to '''C'''(2) ([[biquaternion]]s, [[Pauli algebra]], 2 × 2 complex matrices).
 
For extension beyond the classical algebras, see [[Classification of Clifford algebras]].
 
===Cayley–Dickson construction===
{{details|Cayley–Dickson construction}}
All of the Clifford algebras ''C''ℓ<sub>''p'',''q''</sub>('''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 2<sup>''n''</sup>, ''n'' in {2, 3, 4, ...}, with bases <math>\{1, i_1, \dots, i_{2^n-1}\}</math>, where all the non-real bases anti-commute and satisfy <math>i_m^2 = -1</math>. In eight or more dimensions these algebras are non-associative.
 
The first algebras in this sequence are the four-dimensional [[quaternion]]s, eight-dimensional [[octonion]]s, and 16-dimensional [[sedenion]]s. However, satisfying these requirements comes at a price: Each increase in dimensionality involves a loss of algebraic symmetry: Quaternion multiplication is not [[commutative]], octonion multiplication is non-[[associative]], and the [[norm (mathematics)|norm]] of [[sedenion]]s is not multiplicative.
 
The Cayley–Dickson construction can be modified by inserting an extra sign at some stages. It then generates two of the "split algebras" in the collection of [[composition algebra]]s:
: [[split-quaternion]]s with basis <math>\{ 1, i_1, i_2, i_3 \}</math> satisfying <math>\ i_1^2 = -1, i_2^2 = i_3^2 = +1</math>, ) and
: [[split-octonion]]s with basis <math>\{ 1, i_1, \dots, i_7 \}</math> satisfying <math>\ i_1^2 = i_2^2 = i_3^2 = -1</math>, <math>\ i_4^2 = \cdots = i_7^2 = +1 .</math>
The split-quaternions contain [[nilpotent]]s, have a non-commutative multiplication, and are isomorphic to the [[2 × 2 real matrices]]. Split-octonions are non-associative.
 
===Tensor products===
The [[tensor product]] of any two algebras is another algebra, which can be used to produce many more examples of hypercomplex number systems.
 
In particular taking tensor products with the complex numbers (considered as algebras over the reals) leads to four-dimensional [[tessarine]]s <math>\mathbb C\otimes_\mathbb{R}\mathbb C</math>, eight-dimensional [[biquaternion]]s <math>\mathbb C\otimes_\mathbb{R}\mathbb H</math>, and 16-dimensional [[octonion|complex octonion]]s <math>\mathbb C\otimes_\mathbb{R}\mathbb O</math>.
 
===Further examples===
* [[bicomplex number]]s: a 4d vector space over the reals, or 2d over the complex numbers
* [[multicomplex number]]s: 2<sup>''n''−1</sup>-dimensional vector spaces over the complex numbers
* [[composition algebra]]: algebras with a [[quadratic form]] that composes with the product
 
==See also==
* [[Thomas Kirkman]]
* [[Georg Scheffers]]
* [[Richard Brauer]]
* [[Hypercomplex analysis]]
 
==Notes and references==
{{reflist}}
* Daniel Alfsmann (2006) [http://www.eurasip.org/proceedings/eusipco/eusipco2006/papers/1568981962.pdf On families of 2^N dimensional hypercomplex algebras suitable for digital signal processing], 14th European Signal Processing Conference, Florence, Italy.
* [[Emil Artin]] (1928) "Zur Theorie der hyperkomplexen Zahlen" and "Zur Arithmetik hyperkomplexer Zahlen", in ''The Collected Papers of'' Emil Artin, [[Serge Lang]] and [[John T. Tate]] editors, pp 301&ndash;45, [[Addison-Wesley]], 1965.
* {{Citation | last1=Baez | first1=John | author1-link=John Baez | title=The Octonions | url=http://math.ucr.edu/home/baez/octonions/octonions.html | year=2002 | journal=[[Bulletin of the American Mathematical Society]] | issn=0002-9904 | volume=39 | pages=145–205 | doi=10.1090/S0273-0979-01-00934-X}}
* Thomas Hawkins (1972) "Hypercomplex numbers, Lie groups, and the creation of group representation theory", ''[[Archive for History of Exact Sciences]]'' 8:243&ndash;87.
* Kantor, I.L., Solodownikow (1978), Hyperkomplexe Zahlen, BSB B.G. Teubner Verlagsgesellschaft, Leipzig.
*{{Citation | last1=Kantor | first1=I. L. | last2=Solodovnikov | first2=A. S. | title=Hypercomplex numbers | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-96980-0 | mr=996029  | year=1989}}
* Jeanne La Duke (1983) "The study of linear associative algebras in the United States, 1870–1927", see pp.&nbsp;147–159 of ''Emmy Noether in Bryn Mawr'' [[Bhama Srinivasan]] & Judith Sally editors, Springer Verlag.
* [[Theodor Molien]] (1893) "Über Systeme höher complexen Zahlen", ''Mathematische Annalen'' 41:83&ndash;156.
* Silviu Olariu (2002) ''Complex Numbers in N Dimensions'', North-Holland Mathematics Studies #190, [[Elsevier]] ISBN 0-444-51123-7 .
* [[Karen Parshall]] (1985) "Wedderburn and the Structure of Algebras" ''Archive for History of Exact Sciences'' 32:223&ndash;349.
* Irene Sabadini, Michael Shapiro & Frank Sommen, editors (2009) ''Hypercomplex Analysis and Applications'' Birkhauser ISBN 978-3-7643-9892-7 .
* [[Eduard Study]] (1898) "Theorie der gemeinen und höhern komplexen Grössen", [[Klein's encyclopedia|''Encyclopädie der mathematischen Wissenschaften]] I A '''4''' 147&ndash;83.
* Henry Taber (1904) "On Hypercomplex Number Systems", [[Transactions of the American Mathematical Society]] 5:509.
* [[B.L. van der Waerden]] (1985) ''A History of Algebra'', Chapter 10: The discovery of algebras, Chapter 11: Structure of algebras, Springer, ISBN 3-540-13610X .
* [[Joseph Wedderburn]] (1908) "On Hypercomplex Numbers", ''Proceedings of the London Mathematical Society'' 6:77&ndash;118.
 
==External links==
* {{springer|title=Hypercomplex number|id=p/h048390}}
* [http://history.hyperjeff.net/hypercomplex History of the Hypercomplexes on hyperjeff.com]
* [http://hypercomplex.xpsweb.com/index.php?&lang=en Hypercomplex.info]
* {{mathworld|urlname=HypercomplexNumber|title=Hypercomplex number}}
*[http://neo-classical-physics.info/uploads/3/0/6/5/3065888/study_-_complex_numbers_and_transformation_groups.pdf E. Study, "On systems of complex numbers and their application to the theory of transformation groups"] (English translation)
*[http://neo-classical-physics.info/uploads/3/0/6/5/3065888/frobenius_-_hypercomplex_i.pdf G. Frobenius, "Theory of hypercomplex quantities"] (English translation)
 
{{Number Systems}}
{{DEFAULTSORT:Hypercomplex Number}}
[[Category:Hypercomplex numbers| ]]
[[Category:History of mathematics]]

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Cons Of Anabolic Cocking

- Just like all diet program this particular one requires your commitment to realistically realize your primary goal.

- Only accessible through on the internet and info could be entry through your PC or mobile computer But they can be printed for your benefit.

Bonus deals and Assure:

- This price deal enters in 7 components. Not alone you have the Anabolic Cooking dish guide you will be getting the most beneficial resource guides on accomplish diet and preparing information, meal plans, the bodybuilding tips “Optimum Maximizer Caloric Calculator”, useful foodstuff log, “The Unpleasant Truth About Supplements” a properly-looked into reserve life span and guideline changes. Valued all at $419.

- Extremely reduced price, shocking small at $47 for the package deal.

- The 60-time money-back guarantee making sure you until this item will definitely function.

Bottom line on Anabolic Cookinganabolic cocking the best guideline

The Anabolic Cooking is ideal if you are interested in the most effective approaches to drop your excessive unwanted weight, make very good lean muscle explanations and flip your whole body in a toned trim body. The major valuation of this guide would be the number of mouthwatering foods that you can effortlessly get ready at home and sure to construct muscle tissue when deteriorating fatty acids.

Over time along with the additional add-ons this product have would be the most affordable and practical usually means in reshaping your system as an alternative to purchasing pre-crammed diet regime dishes that will be high priced and bland or prepare food relentlessly only to abide on your diet plan.

Some preparing food system manuals to choose from are not able to even come nearby the valuation on Anabolic Cooking simply because this a single has all the details you’ll have to superior appreciate how your body’s in a position to make muscle tissue through combined meals which can be basic, healthy and delightful. This is the ideal way to appearance your system taking into consideration its cost and price!

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