Dördey

Taban ögelerinin çarpımı

denklikler

${\displaystyle i^{2}=j^{2}=k^{2}=ijk=-1}$,

burada i, j ve k H nın taban ögeleridir,i, j ve k nın tüm olası çarpanlarını belirtir.

örneğin −1 = ijk nın sağ çarpanlarının her ikisi de k ile verilir

${\displaystyle {\begin{aligned}-k&=ijkk=ij(k^{2})=ij(-1),\\k&=ij.\end{aligned}}}$

Diğer tüm olası çarpanlar benzer yöntemlerle belirlenebilir

${\displaystyle {\begin{alignedat}{2}ij&=k,&\qquad ji&=-k,\\jk&=i,&kj&=-i,\\ki&=j,&ik&=-j,\end{alignedat}}}$

olan satır çarpanı sol faktörü teşkil eder ve bir tablo olarak ifade edilebilir, bu yazının üstünde gösterildiği gibi kendilerinin sütunlari sağ faktörü teşkil eder.

Hamilton çarpımı

iki a₁ + b₁i + c₁j + d₁k elementler için ve a₂ + b₂i + c₂j + d₂k, burada çarpıma, Hamilton çarpımı (a₁ + b₁i + c₁j + d₁k) (a₂ + b₂i + c₂j + d₂k) denir, taban ögeler ve dağılımsal kanunun çarpımları ile tanımlanıyor.Dağılım kanunu onu çarpımın açılımı için olası yapar böylece bu taban ögelerin çarpımlarının bir toplamıdır. Bu aşağıdaki bağıntılarla veriliyor:

${\displaystyle a_{1}a_{2}+a_{1}b_{2}i+a_{1}c_{2}j+a_{1}d_{2}k}$

${\displaystyle {}+b_{1}a_{2}i+b_{1}b_{2}i^{2}+b_{1}c_{2}ij+b_{1}d_{2}ik}$

${\displaystyle {}+c_{1}a_{2}j+c_{1}b_{2}ji+c_{1}c_{2}j^{2}+c_{1}d_{2}jk}$

${\displaystyle {}+d_{1}a_{2}k+d_{1}b_{2}ki+d_{1}c_{2}kj+d_{1}d_{2}k^{2}.}$

Şimdi taban elemanları kullanılarak elde etmek için yukarıda verilen kuralları çoğaltılabilir:

${\displaystyle a_{1}a_{2}-b_{1}b_{2}-c_{1}c_{2}-d_{1}d_{2}}$

${\displaystyle {}+(a_{1}b_{2}+b_{1}a_{2}+c_{1}d_{2}-d_{1}c_{2})i}$

${\displaystyle {}+(a_{1}c_{2}-b_{1}d_{2}+c_{1}a_{2}+d_{1}b_{2})j}$

${\displaystyle {}+(a_{1}d_{2}+b_{1}c_{2}-c_{1}b_{2}+d_{1}a_{2})k.}$

Sıralı liste formu

H'nın 1, i, j, k tabanları kullanılıyor dört katının bir kümesi olarak H yazmak için mümkün kılar:

${\displaystyle \mathbf {H} =\{(a,b,c,d)\mid a,b,c,d\in \mathbf {R} \}.}$

ise taban ögeleri:

${\displaystyle {\begin{aligned}1&=(1,0,0,0),\\i&=(0,1,0,0),\\j&=(0,0,1,0),\\k&=(0,0,0,1),\end{aligned}}}$

ve toplam ve çarpım için formüller:

${\displaystyle (a_{1},\ b_{1},\ c_{1},\ d_{1})+(a_{2},\ b_{2},\ c_{2},\ d_{2})=(a_{1}+a_{2},\ b_{1}+b_{2},\ c_{1}+c_{2},\ d_{1}+d_{2}).}$

${\displaystyle {\begin{aligned}(a_{1},\ b_{1},\ c_{1},\ d_{1})&(a_{2},\ b_{2},\ c_{2},\ d_{2})=\\&=(a_{1}a_{2}-b_{1}b_{2}-c_{1}c_{2}-d_{1}d_{2},\\&{}\qquad a_{1}b_{2}+b_{1}a_{2}+c_{1}d_{2}-d_{1}c_{2},\\&{}\qquad a_{1}c_{2}-b_{1}d_{2}+c_{1}a_{2}+d_{1}b_{2},\\&{}\qquad a_{1}d_{2}+b_{1}c_{2}-c_{1}b_{2}+d_{1}a_{2}).\end{aligned}}}$

Üstel, logaritma ve kuvvet

bir dördey veriliyor,

q = a + bi + cj + dk = a + v,

üstel

${\displaystyle \exp(q)=\sum _{n=0}^{\infty }{\frac {q^{n}}{n!}}=e^{a}\left(\cos \|\mathbf {v} \|+{\frac {\mathbf {v} }{\|\mathbf {v} \|}}\sin \|\mathbf {v} \|\right)}$

olarak hesaplanıyor ve

${\displaystyle \ln(q)=\ln \|q\|+{\frac {\mathbf {v} }{\|\mathbf {v} \|}}\arccos {\frac {a}{\|q\|}}}$.

bir dördeyin kutupsal çözülümünü aşağıdaki gibi yazabiliriz

${\displaystyle q=\|q\|e^{{\hat {n}}\theta }=\|q\|\left(\cos(\theta )+{\hat {n}}\sin(\theta )\right),}$

burada açı θ ve birim vektör ${\displaystyle {\hat {n}}}$ ile tanımlanıyor:

${\displaystyle a=\|q\|\cos(\theta )}$

ve

${\displaystyle \mathbf {v} ={\hat {n}}\|\mathbf {v} \|={\hat {n}}\|q\|\sin(\theta ).}$

Herhangi birim dördey ${\displaystyle e^{{\hat {n}}\theta }}$.olan kutupsal biçim içinde ifade edilebilir

bir keyfi (gerçek) üstel ${\displaystyle \alpha }$ için bir yükselen dördeyin kuvveti ile veriliyor:

${\displaystyle q^{\alpha }=\|q\|^{\alpha }e^{{\hat {n}}\alpha \theta }=\|q\|^{\alpha }\left(\cos(\alpha \theta )+{\hat {n}}\sin(\alpha \theta )\right).}$

Kitaplar ve yayınlar

• Hamilton, William Rowan. On quaternions, or on a new system of imaginaries in algebra. Philosophical Magazine. Vol. 25, n 3. p. 489–495. 1844.
• Hamilton, William Rowan (1853), "Lectures on Quaternions 11 Mayıs 2006 tarihinde Wayback Machine sitesinde arşivlendi.". Royal Irish Academy.
• Hamilton (1866) Elements of Quaternions 12 Temmuz 2014 tarihinde Wayback Machine sitesinde arşivlendi. University of Dublin Press. Edited by William Edwin Hamilton, son of the deceased author.
• Hamilton (1899) Elements of Quaternions volume I, (1901) volume II. Edited by Charles Jasper Joly; published by Longmans, Green & Co..
• Tait, Peter Guthrie (1873), "An elementary treatise on quaternions". 2d ed., Cambridge, [Eng.]: The University Press.
• Michiel Hazewinkel, Nadiya Gubareni, Nadezhda Mikhaĭlovna Gubareni, Vladimir V. Kirichenko. Algebras, rings and modules. Volume 1. 2004. Springer, 2004. ISBN 1-4020-2690-0
• Maxwell, James Clerk (1873), "A Treatise on Electricity and Magnetism". Clarendon Press, Oxford.
• Tait, Peter Guthrie (1886), "Quaternion; Wayback Machine". M.A. Sec. R.S.E. Encyclopaedia Britannica, Ninth Edition, 1886, Vol. XX, pp. 160–164. (bzipped PostScript file)
• Joly, Charles Jasper (1905), "A manual of quaternions". London, Macmillan and co., limited; New York, The Macmillan company. LCCN 05036137 //r84
• Macfarlane, Alexander (1906), "Vector analysis and quaternions", 4th ed. 1st thousand. New York, J. Wiley & Sons; [etc., etc.]. LCCN es 16000048
• 1911 encyclopedia: "Quaternions 7 Mayıs 2012 tarihinde Wayback Machine sitesinde arşivlendi.".
• Finkelstein, David, Josef M. Jauch, Samuel Schiminovich, and David Speiser (1962), "Foundations of quaternion quantum mechanics". J. Mathematical Phys. 3, pp. 207–220, MathSciNet.
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• Crowe, Michael J. (1967), A History of Vector Analysis: The Evolution of the Idea of a Vectorial System, University of Notre Dame Press. Surveys the major and minor vector systems of the 19th century (Hamilton, Möbius, Bellavitis, Clifford, Grassmann, Tait, Peirce, Maxwell, Macfarlane, MacAuley, Gibbs, Heaviside).
• Altmann, Simon L. (1986), "Rotations, quaternions, and double groups". Oxford [Oxfordshire]: Clarendon Press; New York: Oxford University Press. LCCN 85013615 ISBN 0-19-855372-2
• Altmann, Simon L. (1989), "Hamilton, Rodrigues, and the Quaternion Scandal". Mathematics Magazine. Vol. 62, No. 5. p. 291–308, December 1989.
• Adler, Stephen L. (1995), "Quaternionic quantum mechanics and quantum fields". New York: Oxford University Press. International series of monographs on physics (Oxford, England) 88. LCCN 94006306 ISBN 0-19-506643-X
• Trifonov, Vladimir (1995), "A Linear Solution of the Four-Dimensionality Problem", Europhysics Letters, 32 (8) 621–626, DOI:10.1209/0295-5075/32/8/001
• Ward, J. P. (1997), "Quaternions and Cayley Numbers: Algebra and Applications", Kluwer Academic Publishers. ISBN 0-7923-4513-4
• Kantor, I. L. and Solodnikov, A. S. (1989), "Hypercomplex numbers, an elementary introduction to algebras", Springer-Verlag, New York, ISBN 0-387-96980-2
• Gürlebeck, Klaus and Sprössig, Wolfgang (1997), "Quaternionic and Clifford calculus for physicists and engineers". Chichester; New York: Wiley (Mathematical methods in practice; v. 1). LCCN 98169958 ISBN 0-471-96200-7
• Kuipers, Jack (2002), "Quaternions and Rotation Sequences: A Primer With Applications to Orbits, Aerospace, and Virtual Reality" (reprint edition), Princeton University Press. ISBN 0-691-10298-8
• Conway, John Horton, and Smith, Derek A. (2003), "On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry", A. K. Peters, Ltd. ISBN 1-56881-134-9.
• Kravchenko, Vladislav (2003), "Applied Quaternionic Analysis", Heldermann Verlag ISBN 3-88538-228-8.
• Hanson, Andrew J. 27 Aralık 2007 tarihinde Wayback Machine sitesinde arşivlendi. (2006), "Visualizing Quaternions", Elsevier: Morgan Kaufmann; San Francisco. ISBN 0-12-088400-3
• Trifonov, Vladimir (2007), "Natural Geometry of Nonzero Quaternions", International Journal of Theoretical Physics, 46 (2) 251–257, DOI:10.1007/s10773-006-9234-9
• Ernst Binz & Sonja Pods (2008) Geometry of Heisenberg Groups American Mathematical Society, Chapter 1: "The Skew Field of Quaternions" (23 pages) ISBN 978-0-8218-4495-3.
• Vince, John A. (2008), Geometric Algebra for Computer Graphics, Springer, ISBN 978-1-84628-996-5.
• For molecules that can be regarded as classical rigid bodies molecular dynamics computer simulation employs quaternions. They were first introduced for this purpose by D.J. Evans, (1977), "On the Representation of Orientation Space", Mol. Phys., vol 34, p 317.
• Zhang, Fuzhen (1997), "Quaternions and Matrices of Quaternions", Linear Algebra and its Applications, Vol. 251, pp. 21–57.

Bağlantılar ve uzman yazıları

• Hazewinkel, Michiel, (Ed.) (2001), "Quaternion", Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104 
• Matrix and Quaternion FAQ v1.21 16 Aralık 2007 tarihinde Wayback Machine sitesinde arşivlendi. Frequently Asked Questions
• "Geometric Tools documentation" (frame 28 Ekim 2007 tarihinde Wayback Machine sitesinde arşivlendi.; body) includes several papers focusing on computer graphics applications of quaternions. Covers useful techniques such as spherical linear interpolation.
• Patrick-Gilles Maillot 12 Temmuz 2014 tarihinde Wayback Machine sitesinde arşivlendi. Provides free Fortran and C source code for manipulating quaternions and rotations / position in space. Also includes mathematical background on quaternions.
• "Geometric Tools source code" (frame; body 11 Ocak 2010 tarihinde Wayback Machine sitesinde arşivlendi.) includes free C++ source code for a complete quaternion class suitable for computer graphics work, under a very liberal license.
• Doug Sweetser, Doing Physics with Quaternions 4 Aralık 2003 tarihinde Wayback Machine sitesinde arşivlendi.
• Quaternions for Computer Graphics and Mechanics (Gernot Hoffman)
• The Physical Heritage of Sir W. R. Hamilton (PDF)
• D. R. Wilkins, Hamilton’s Research on Quaternions 24 Aralık 2007 tarihinde Wayback Machine sitesinde arşivlendi.
• Quaternion Julia Fractals 15 Aralık 2007 tarihinde Wayback Machine sitesinde arşivlendi. 3D Raytraced Quaternion Julia Fractals by David J. Grossman
• Quaternion Math and Conversions 27 Eylül 2007 tarihinde Wayback Machine sitesinde arşivlendi. Great page explaining basic math with links to straight forward rotation conversion formulae.
• John H. Mathews, Bibliography for Quaternions.
• Quaternion powers on GameDev.net
• Andrew Hanson, Visualizing Quaternions home page.
• Diebel, James. "Representing Attitude with Euler Angles and Quaternions: A Reference". Stanford University. 5 Temmuz 2008 tarihinde kaynağından arşivlendi. Erişim tarihi: 22 Aralık 2007.  - Technical report and Matlab toolbox summarizing all common attitude representations, with detailed equations and discussion on features of various methods.
• Charles F. F. Karney, Quaternions in molecular modeling, J. Mol. Graph. Mod. 25(5), 595–604 (January 2007); DOI:10.1016/j.jmgm.2006.04.002; E-print arxiv:0506177 4 Eylül 2015 tarihinde Wayback Machine sitesinde arşivlendi..
• Johan E. Mebius, A matrix-based proof of the quaternion representation theorem for four-dimensional rotations. 4 Eylül 2015 tarihinde Wayback Machine sitesinde arşivlendi., arXiv General Mathematics 2005.
• Johan E. Mebius, Derivation of the Euler–Rodrigues formula for three-dimensional rotations from the general formula for four-dimensional rotations. 4 Eylül 2015 tarihinde Wayback Machine sitesinde arşivlendi., arXiv General Mathematics 2007.
• NUI Maynooth Department of Mathematics, Hamilton Walk 22 Kasım 2007 tarihinde Wayback Machine sitesinde arşivlendi..
• OpenGL:Tutorials:Using Quaternions to represent rotation
• David Erickson, Defence Research and Development Canada (DRDC), Complete derivation of rotation matrix from unitary quaternion representation in DRDC TR 2005-228 paper. Drdc-rddc.gc.ca
• Alberto Martinez, University of Texas Department of History, "Negative Math, How Mathematical Rules Can Be Positively Bent",Utexas.edu
• D. Stahlke, Quaternions in Classical Mechanics Stahlke.org 15 Eylül 2012 tarihinde Wayback Machine sitesinde arşivlendi. (PDF)
• Morier-Genoud, Sophie, and Valentin Ovsienko. "Well, Papa, can you multiply triplets?", arxiv.org 4 Eylül 2015 tarihinde Wayback Machine sitesinde arşivlendi. describes how the quaternions can be made into a skew-commutative algebra graded by Z/2 × Z/2 × Z/2.
• Curious Quaternions 9 Temmuz 2014 tarihinde Wayback Machine sitesinde arşivlendi. by Helen Joyce hosted by John Baez.
• Luis Ibanez "Tutorial on Quaternions" Part I Part II (PDF)
• R. Ghiloni, V. Moretti, A. Perotti (2013) "Continuous slice functional calculus in quaternionic Hilbert spaces, 4 Eylül 2015 tarihinde Wayback Machine sitesinde arşivlendi." Rev.Math.Phys. 25 1350006. An expository paper about continuous functional calculus in quanternionic Hilbert spaces useful in rigorous quaternionic quantum mechanics.
• Visualizing Quaternions 27 Aralık 2007 tarihinde Wayback Machine sitesinde arşivlendi. by Andrew J. Hanson at Indiana University in Bloomington.

Şablon:Number Systems