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| | Daron Babin-Daron is the CEO of Web-master Radio. Daron is lecturing and teaching on search engine marketing since 1997. Daron got his come from tv by doing work for NBC, but eventually found his way to the world of search engine marketing techniques and optimization. His purpose from the start has-been to generate tools that benefit the community. Based on a meeting with the Online Marketing Website, Daron is developing a pod cast se. H-e also mentions that Yahoo is his favorite search-engine, while Google is available in second on his list. <br><br>Matt Cutts-Matt Cutts is just a computer software engineer for Google. He began working for Google in January 2000, and is the top of Googles Webspam team. Thanks to the acceptance of his weblog, Matt Cutts has turned into a star within the search engine marketing community. By having an Alexa standing of 1,262 (at the time of writing), Matts blog is one of the busiest on the web. Since Matt has transformed into the unofficial ambassador/liason between Google and the search engine optimization group, he often attends search and web master seminars. Wherever he goes during these meetings, it is assured that he"ll have a flock of SEOs watching and following his every move. Identify new [http://Www.bbc.co.uk/search/?q=resources resources] on our favorite partner article directory - Navigate to this website: [http://natura.di.uminho.pt/~jj/ia-la/doku.php?id=The_Advantages_Of_Key_Phrase_Analysis_42997 per your request]. <br><br>Rand Fishkin-Rand Fishkin may be the owner of SEOmoz. SEOmoz focuses primarily on providing organizations around the globe with search engine optimization services. SEOmoz is comprised of eight different people, including Rebecca Kelley and Si Fishkin. Be taught more on our favorite related article directory - Hit this website: [http://hacklab.comumlab.org//wikka.php?wakka=The_Features_Of_Key_Term_Analysis_42997 seo for small businesses]. At the end of 2006, Rand and SEOmoz made the decision to produce their financial statements for the past year. My aunt found out about [http://www.okurisangurasubuy.com/showthread.php?tid=9187 SEO - This Is Keeps Expanding] by searching Google Books. Based on these statements (which were only rough estimates), SEOmoz gained a total of $600,000 throughout 2006. At the end of the year, they"d $64,000 in the bank. Just a year early, they had less than $4,000 in saved in-the bank..Orange County SEO Company, Inc<br>16400 Pacific Coast Hwy #218<br>Huntington Beach, CA 92649<br>(949) 494-0007<br><br>If you loved this informative article and you want to receive much more information concerning [http://www.alivenotdead.com/gaudyruin7875 health information] i implore you to visit our own web page. |
| In [[mathematics]], the '''Cayley–Dickson construction''', named after [[Arthur Cayley]] and [[Leonard Eugene Dickson]], produces a sequence of [[algebra over a field|algebras]] over the [[field (mathematics)|field]] of [[real number]]s, each with twice the [[dimension of a vector space|dimension]] of the previous one. The algebras produced by this process are known as '''Cayley–Dickson algebras'''. They are useful [[composition algebra]]s frequently applied in [[mathematical physics]].
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| The Cayley–Dickson construction defines a new algebra based on the [[direct sum]] of an algebra with itself, with multiplication defined in a specific way and an [[involution (mathematics)|involution]] known as [[Complex conjugate|conjugation]]. The product of an element and its conjugate (or sometimes the square root of this) called the [[norm (mathematics)|norm]].
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| The symmetries of the real field disappear as the Cayley–Dickson construction is repeatedly applied: first losing order, then commutativity of multiplication, and next associativity of multiplication.
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| More generally, the Cayley–Dickson construction takes any algebra with involution to another algebra with involution of twice the dimension.<ref name=Sch45>Schafer (1995) p.45</ref>
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| == Complex numbers as ordered pairs ==
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| The [[complex numbers]] can be written as [[ordered pair]]s (''a'', ''b'') of [[real number]]s ''a'' and ''b'', with the addition operator being component-by-component and with multiplication defined by
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| : <math>(a, b) (c, d) = (a c - b d, a d + b c).\,</math>
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| A complex number whose second component is zero is associated with a real number: the complex number (''a'', 0) is the real number ''a''.
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| Another important operation on complex numbers is conjugation. The conjugate (''a'', ''b'')<sup>*</sup> of (''a'', ''b'') is given by
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| : <math>(a, b)^* = (a, -b).\,</math>
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| The conjugate has the property that
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| : <math>(a, b)^* (a, b)
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| = (a a + b b, a b - b a) = (a^2 + b^2, 0),\,</math>
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| which is a non-negative real number. In this way, conjugation defines a ''[[norm (mathematics)|norm]]'', making the complex numbers a [[normed vector space]] over the real numbers: the norm of a complex number ''z'' is
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| : <math>|z| = (z^* z)^{1/2}.\,</math>
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| Furthermore, for any nonzero complex number ''z'', conjugation gives a [[inverse element|multiplicative inverse]],
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| : <math>z^{-1} = {z^* / |z|^2}.\,</math>
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| In as much as complex numbers consist of two independent real numbers, they form a 2-dimensional [[vector space]] over the real numbers.
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| Besides being of higher dimension, the complex numbers can be said to lack one algebraic property of the real numbers: a real number is its own conjugate.
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| == Quaternions ==
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| The next step in the construction is to generalize the multiplication and conjugation operations.
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| Form ordered pairs <math>(a, b)</math> of complex numbers <math>a</math> and <math>b</math>, with multiplication defined by
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| : <math>(a, b) (c, d)
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| = (a c - d^* b, d a + b c^*).\,</math>
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| Slight variations on this formula are possible; the resulting constructions will yield structures identical up to the signs of bases.
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| The order of the factors seems odd now, but will be important in the next step. Define the conjugate <math>(a, b)^*\,</math> of <math>(a, b)</math> by
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| : <math>(a, b)^* = (a^*, -b).\,</math>
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| These operators are direct extensions of their complex analogs: if <math>a</math> and <math>b</math> are taken from the real subset of complex numbers, the appearance of the conjugate in the formulas has no effect, so the operators are the same as those for the complex numbers.
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| The product of an element with its conjugate is a non-negative real number:
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| : <math>(a, b)^* (a, b)
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| = (a^*, -b) (a, b)
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| = (a^* a + b^* b, b a^* - b a^*)
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| = (|a|^2 + |b|^2, 0 ).\,</math>
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| As before, the conjugate thus yields a norm and an inverse for any such ordered pair. So in the sense we explained above, these pairs constitute an algebra something like the real numbers. They are the [[quaternions]], named by [[William Rowan Hamilton|Hamilton]] in 1843.
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| Inasmuch as quaternions consist of two independent complex numbers, they form a 4-dimensional vector space over the real numbers.
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| The multiplication of quaternions is not quite like the multiplication of real numbers, though. It is not [[commutative]], that is, if <math>p</math> and <math>q</math> are quaternions, it is not generally true that <math>p q = q p</math>.
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| == Octonions ==
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| {{main|Octonion}}
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| From now on, all the steps will look the same.
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| This time, form ordered pairs <math>(p, q)</math> of
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| quaternions <math>p</math> and <math>q</math>, with multiplication and conjugation defined exactly as for the quaternions:
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| : <math>(p, q) (r, s)
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| = (p r - s^* q, s p + q r^*).\,</math>
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| Note, however, that because the quaternions are not commutative, the order of the factors in the multiplication formula becomes important—if the last factor in the multiplication formula were <math>r^*q</math> rather than
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| <math>qr^*</math>, the formula for multiplication of an element by its conjugate would not yield a real number.
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| For exactly the same reasons as before, the conjugation operator yields a norm and a multiplicative inverse of any nonzero element.
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| This algebra was discovered by [[John T. Graves]] in 1843, and is called the [[octonions]] or the "[[Arthur Cayley|Cayley]] numbers".
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| Inasmuch as octonions consist of two quaternions, the octonions form an 8-dimensional vector space over the real numbers.
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| The multiplication of octonions is even stranger than that of quaternions. Besides being non-commutative, it is not [[associative]]: that is, if <math>p</math>, <math>q</math>, and <math>r</math> are octonions, it is generally not true that
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| :<math>(p q) r = p (q r).\ </math>
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| For the reason of this non-associativity, octonions have [[Octonion#Properties|no matrix representation]].
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| == Further algebras == | |
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| The algebra immediately following the octonions is called the [[sedenions]]. It retains an algebraic property called [[power associativity]], meaning that if <math>s</math> is a sedenion, <math>s^n s^m = s^{n + m}</math>, but loses the property of being an [[alternative algebra]] and hence cannot be a [[composition algebra]].
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| The Cayley–Dickson construction can be carried on ''[[ad infinitum]]'', at each step producing a power-associative algebra whose dimension is double that of the algebra of the preceding step. All the algebras generated in this way over a field are ''quadratic'': that is, each element satisfies a quadratic equation with coefficients from the field.<ref name=Sch50>Schafer (1995) p.50</ref>
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| == General Cayley–Dickson construction ==
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| {{harvtxt|Albert|1942|p= 171}} gave a slight generalization, defining the product and involution on ''B''=''A''⊕''A'' for ''A'' an [[*-algebra|algebra with involution]] (with (''xy'')<sup>*</sup> = ''y''<sup>*</sup>''x''<sup>*</sup>) to be
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| : <math>(p, q) (r, s)
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| = (p r - \gamma s^* q, s p + q r^*)\,</math>
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| :<math>(p, q)^* = (p^*, -q)\ </math>
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| for γ an additive map that commutes with * and left and right multiplication by any element. (Over the reals all choices of γ are equivalent to −1, 0 or 1.) In this construction, ''A'' is an algebra with involution, meaning:
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| *''A'' is an abelian group under +
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| *''A'' has a product that is left and right distributive over +
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| *''A'' has an involution *, with ''x''** = ''x'', (''x'' + ''y'')* = ''x''* + ''y''*, (''xy'')* = ''y''*''x''*.
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| The algebra ''B''=''A''⊕''A'' produced by the Cayley–Dickson construction is also an algebra with involution.
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| ''B'' inherits properties from ''A'' unchanged as follows.
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| *If ''A'' has an identity 1<sub>''A''</sub>, then ''B'' has an identity (1<sub>''A''</sub>, 0).
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| *If ''A'' has the property that ''x'' + ''x''<sup>*</sup>, ''xx''<sup>*</sup> associate and commute with all elements, then so does ''B''. This property implies that any element generates a commutative associative *-algebra, so in particular the algebra is power associative.
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|
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| Other properties of ''A'' only induce weaker properties of ''B'':
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| *If ''A'' is commutative and has trivial involution, then ''B'' is commutative.
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| *If ''A'' is commutative and associative then ''B'' is associative.
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| *If ''A'' is associative and ''x'' + ''x''<sup>*</sup>, ''xx''<sup>*</sup> associate and commute with everything, then ''B'' is an [[alternative algebra]].
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| == References ==
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| {{reflist}}
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| *{{Citation | last1=Albert | first1=A. A. | author1-link=Abraham Adrian Albert | title=Quadratic forms permitting composition | jstor=1968887 | mr=0006140 | year=1942 | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]] | issn=0003-486X | volume=43 | pages=161–177 | doi=10.2307/1968887 | issue=1}} (see p. 171)
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| * {{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 | issue=2}}. ''(See "[http://math.ucr.edu/home/baez/octonions/node5.html Section 2.2, The Cayley-Dickson Construction]")''
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| *{{Citation | last1=Dickson | first1=L. E. | author1-link=Leonard Dickson | title=On Quaternions and Their Generalization and the History of the Eight Square Theorem | jstor=1967865 | publisher=Annals of Mathematics | series=Second Series | year=1919 | journal=[[Annals of Mathematics]] | issn=0003-486X | volume=20 | issue=3 | pages=155–171 | doi=10.2307/1967865}}
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| * {{Citation | last1=Kantor | first1=I. L. | last2=Solodownikow | first2=A. S. | title=Hyperkomplexe Zahlen | publisher=B.G. Teubner | location=Leipzig | year=1978}}
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| * {{Citation | last1=Hamilton | first1=William Rowan | author1-link=William Rowan Hamilton | title=On Quaternions | url=http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Quatern2/Quatern2.html | year=1847 | journal=Proceedings of the Royal Irish Academy | issn=1393-7197 | volume=3 | pages=1–16}}
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| * Guy Roos (2008) "Exceptional symmetric domains", §1: Cayley algebras, in ''Symmetries in Complex Analysis'' by Bruce Gilligan & Guy Roos, volume 468 of ''Contemporary Mathematics'', [[American Mathematical Society]], ISBN 978-0-8218-4459-5 .
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| * {{citation | first=Richard D. | last=Schafer | year=1995 | origyear=1966 | zbl=0145.25601 | title=An introduction to non-associative algebras | publisher=[[Dover Publications]] | isbn=0-486-68813-5 }}
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| ==External links==
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| * Hyperjeff, ''[http://history.hyperjeff.net/hypercomplex.html Sketching the History of Hypercomplex Numbers]'' (1996–2006).
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| {{Number Systems}}
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| {{DEFAULTSORT:Cayley-Dickson construction}}
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| [[Category:Hypercomplex numbers]]
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Daron Babin-Daron is the CEO of Web-master Radio. Daron is lecturing and teaching on search engine marketing since 1997. Daron got his come from tv by doing work for NBC, but eventually found his way to the world of search engine marketing techniques and optimization. His purpose from the start has-been to generate tools that benefit the community. Based on a meeting with the Online Marketing Website, Daron is developing a pod cast se. H-e also mentions that Yahoo is his favorite search-engine, while Google is available in second on his list.
Matt Cutts-Matt Cutts is just a computer software engineer for Google. He began working for Google in January 2000, and is the top of Googles Webspam team. Thanks to the acceptance of his weblog, Matt Cutts has turned into a star within the search engine marketing community. By having an Alexa standing of 1,262 (at the time of writing), Matts blog is one of the busiest on the web. Since Matt has transformed into the unofficial ambassador/liason between Google and the search engine optimization group, he often attends search and web master seminars. Wherever he goes during these meetings, it is assured that he"ll have a flock of SEOs watching and following his every move. Identify new resources on our favorite partner article directory - Navigate to this website: per your request.
Rand Fishkin-Rand Fishkin may be the owner of SEOmoz. SEOmoz focuses primarily on providing organizations around the globe with search engine optimization services. SEOmoz is comprised of eight different people, including Rebecca Kelley and Si Fishkin. Be taught more on our favorite related article directory - Hit this website: seo for small businesses. At the end of 2006, Rand and SEOmoz made the decision to produce their financial statements for the past year. My aunt found out about SEO - This Is Keeps Expanding by searching Google Books. Based on these statements (which were only rough estimates), SEOmoz gained a total of $600,000 throughout 2006. At the end of the year, they"d $64,000 in the bank. Just a year early, they had less than $4,000 in saved in-the bank..Orange County SEO Company, Inc
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