William Rowan Hamilton
Dalam matematika, **Kuaternion** adalah perluasan dari bilangan-bilangan kompleks yang tidak komutatif, dan diterapkan dalam mekanika tiga dimensi. Kuaternion ditemukan oleh ahli matematika dan astronomi Inggris, William Rowan Hamilton, yang memperpanjang aritmetika kompleks nomor ke kuaternion.

Segera setelah itu penemuan Hamilton, matematikawan Jerman Hermann Grassmann mulai menyelidiki vektor. Meskipun karakter abstrak, fisikawan Amerika JW Gibbs diakui dalam aljabar vektor sistem utilitas besar bagi fisikawan, seperti Hamilton mengakui kegunaan kuaternion. Pengaruh luas dari pendekatan abstrak yang dipimpin George Boole untuk menulis Hukum Thought (1854), perawatan aljabar dasar logika.

Definisi
--------

Sebagai himpunan, kuaternion, berlambang **H**, sama dengan **R**4 yang merupakan ruang vektor bilangan riil empat dimensi. **H** memiliki tiga macam operasi: pertambahan, perkalian skalar dan perkalian kuaternion. Elemen-elemen kuaternion ditandakan sebagai *1*, *i*, *j* dan *k* (*i*, *j* dan *k* adalah komponen imaginer), dan dapat ditulis sebagai kombinasi linear, *a* + *bi* + *cj* + *dk* (*a*, *b*, *c*, dan *d* adalah bilangan riil).

Kuaternion {\displaystyle p=a+bi+cj+dk} bisa dituliskan sebagai {\displaystyle p=a+{\vec {u}}} di mana {\displaystyle {\vec {u}}} adalah vektor 3 bilangan imaginer, {\displaystyle {\vec {u}}=\{bi+cj+dk\}}.

### Perkalian elemen dasar

Persamaan elemen kuaternion *i*, *j*, dan *k* adalah:

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

{\displaystyle -1=ijk,\ }
jika dua sisi dikalikan dengan *k*, maka

{\displaystyle {\begin{aligned}-k&=ijkk=ij(k^{2})=ij(-1),\\k&=ij.\end{aligned}}}
Persamaan-persamaan yang lainnya juga bisa didapatkan dengan tahap aljabar:

{\displaystyle {\begin{alignedat}{2}ij&=k,&\qquad ji&=-k,\\jk&=i,&kj&=-i,\\ki&=j,&ik&=-j,\end{alignedat}}}
Persamaan-persamaan ini lalu bisa ditampilkan dengan tabel di bawah ini:

Perkalian kuaternion| × | 1 | i | j | k |
| --- | --- | --- | --- | --- |
| 1 | 1 | *i* | *j* | *k* |
| *i* | *i* | −1 | *k* | −*j* |
| *j* | *j* | −*k* | −1 | *i* |
| *k* | *k* | *j* | −*i* | −1 |

### Pertambahan

{\displaystyle {\begin{aligned}&p\_{1}+p\_{2}=(a\_{1}+b\_{1}i+c\_{1}j+d\_{1}k)+(a\_{2}+b\_{2}i+c\_{2}j+d\_{2}k)\\&=(a\_{1}+a\_{2})+(b\_{1}+b\_{2})i+(c\_{1}+c\_{2})j+(d\_{1}+d\_{2})k\end{aligned}}}

### Pengurangan

{\displaystyle {\begin{aligned}&p\_{1}-p\_{2}=(a\_{1}+b\_{1}i+c\_{1}j+d\_{1}k)-(a\_{2}+b\_{2}i+c\_{2}j+d\_{2}k)\\&=(a\_{1}-a\_{2})+(b\_{1}-b\_{2})i+(c\_{1}-c\_{2})j+(d\_{1}-d\_{2})k\end{aligned}}}

### Perkalian

{\displaystyle {\begin{aligned}&p\_{1}\times p\_{2}\\&=(a\_{1}a\_{2}-b\_{1}b\_{2}-c\_{1}c\_{2}-d\_{1}d\_{2})+(b\_{1}a\_{2}+a\_{1}b\_{2}-d\_{1}c\_{2}+c\_{1}d\_{2})i+(c\_{1}a\_{2}+d\_{1}b\_{2}+a\_{1}c\_{2}-b\_{1}d\_{2})j+(d\_{1}a\_{2}-c\_{1}b\_{2}+b\_{1}c\_{2}+a\_{1}d\_{2})k\end{aligned}}}

Bila kuaternion dituliskan dengan bentuk {\displaystyle p=a+{\vec {u}}}, maka:

{\displaystyle {\begin{aligned}&p\_{1}\times p\_{2}\\&=(a\_{1}+{\vec {u\_{1}}})\times (a\_{2}+{\vec {u\_{2}}})\\&=(a\_{1}a\_{2}-{\vec {u\_{1}}}\cdot {\vec {u\_{2}}})+(a\_{1}{\vec {u\_{2}}}+a\_{2}{\vec {u\_{1}}}+{\vec {u\_{1}}}\times {\vec {u\_{2}}})\end{aligned}}}

### Pembagian

{\displaystyle {\begin{aligned}&p\_{1}/p\_{2}\\&={\frac {a\_{1}a\_{2}+b\_{1}b\_{2}+c\_{1}c\_{2}+d\_{1}d\_{2}}{m}}+{\frac {b\_{1}a\_{2}-a\_{1}b\_{2}-d\_{1}c\_{2}+c\_{1}d\_{2}}{m}}i+{\frac {c\_{1}a\_{2}+d\_{1}b\_{2}-a\_{1}c\_{2}-b\_{1}d\_{2}}{m}}j+{\frac {d\_{1}a\_{2}-c\_{1}b\_{2}+b\_{1}c\_{2}-a\_{1}d\_{2}}{m}}k\end{aligned}}}
di mana
{\displaystyle m=a\_{2}^{2}+b\_{2}^{2}+c\_{2}^{2}+d\_{2}^{2}}

### Konjugat

Suatu kuaternion *p* = *a* + *bi* + *cj* + *dk* memiliki konjugat *p\**, dan didapatkan dengan rumus berikut:

{\displaystyle {\begin{alignedat}{2}p\*=a-bi-cj-dk\end{alignedat}}}

Persamaan-persamaan konjugasi kuaternion adalah:

{\displaystyle {\begin{matrix}(p^{\*})^{\*}&=&p\\(pq)^{\*}&=&q^{\*}p^{\*}\\(p^{-1})^{\*}&=&{\frac {p}{\|p\|^{2}}}\\(p^{\*})^{-1}&=&{\frac {p}{\|p\|^{2}}}\\(p^{-1})^{-1}&=&p\\(p\_{1}+p\_{2})^{\*}&=&p\_{1}^{\*}+p\_{2}^{\*}\\\end{matrix}}\,}

### Satuan

Dengan fungsi Norma {\displaystyle N()}, bila {\displaystyle N(p)=1}, maka:

{\displaystyle {\begin{matrix}p&=&\cos(\theta )+{\vec {u}}\sin(\theta )\\p&=&\cos(\theta )+{\hat {u}}\sin(\theta )\end{matrix}}\,}

di mana

{\displaystyle \left\|{\vec {u}}\right\|=1}

Bentuk matriks
--------------

Kuaternion, seperti bilangan kompleks, bisa ditulis dalam bentuk matriks, yaitu matriks kompleks 2x2 atau matriks riil 4x4.

Bentuk matriks kompleks 2x2 untuk kuaternion *a* + *bi* + *cj* + *dk* adalah:

{\displaystyle {\begin{bmatrix}a+bi&c+di\\-c+di&a-bi\end{bmatrix}}=a{\begin{bmatrix}\;\;1&0\\0&1\end{bmatrix}}+b{\begin{bmatrix}\;\;i&0\\0&-i\end{bmatrix}}+c{\begin{bmatrix}\;\;0&1\\-1&0\end{bmatrix}}+d{\begin{bmatrix}\;\;0&i\\i&0\end{bmatrix}}}
Bentuk matriks riil 4x4 untuk kuaternion *a* + *bi* + *cj* + *dk* adalah:

{\displaystyle {\begin{bmatrix}a&b&c&d\\-b&a&-d&c\\-c&d&a&-b\\-d&-c&b&a\end{bmatrix}}=a{\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{bmatrix}}+b{\begin{bmatrix}0&1&0&0\\-1&0&0&0\\0&0&0&-1\\0&0&1&0\end{bmatrix}}+c{\begin{bmatrix}0&0&1&0\\0&0&0&1\\-1&0&0&0\\0&-1&0&0\end{bmatrix}}+d{\begin{bmatrix}0&0&0&1\\0&0&-1&0\\0&1&0&0\\-1&0&0&0\end{bmatrix}}}
Selain itu juga terdapat bentuk matriks 3x3 yang digunakan dalam grafika komputer. Berikut adalah bentuk matriks kolom-utama (column-major) yang digunakan di OpenGL. (Matriks baris-utama (row-major) yang digunakan di DirectX sama dengan transposa matriks kolom-utama)

{\displaystyle {\begin{bmatrix}1-2(c^{2}+d^{2})&2(bc-da)&2(bd+ca)\\2(bc+da)&1-2(b^{2}+d^{2})&2(cd-ba)\\2(bd-ca)&2(cd+ba)&1-2(b^{2}+c^{2})\end{bmatrix}}}

Fungsi
------

### Norma

{\displaystyle N(p)=N(a+bi+cj+dk)=a^{2}+b^{2}+c^{2}+d^{2}}

Dan juga,

{\displaystyle {\begin{matrix}N(p^{\*})&=&N(p)\\N(pq)&=&N(p)N(q)\end{matrix}}}

### Kebalikan

{\displaystyle p^{-1}={\frac {p^{\*}}{N(p)}}}

Dan juga,

{\displaystyle {\begin{matrix}pp^{-1}&=&p^{-1}p\\pp^{-1}&=&1\\(p^{-1})^{-1}&=&p\\(pq)^{-1}&=&q^{-1}p^{-1}\end{matrix}}}

### Pemilihan riil

Meskipun tertetap sangat sederhana, fungsi yang hasilnya adalah bagiannya bilangan riil kuaternion ini memiliki kegunaannya tersendiri.
{\displaystyle W(p)=W(a+bi+cj+dk)=a}

Dan juga,

{\displaystyle {\begin{matrix}W(p)&=&(p+p^{\*})/2\end{matrix}}}

### Skalar

Dari kuaternion {\displaystyle p\_{2}={\frac {p+(p^{\*})}{2}}}

Maka: {\displaystyle Scalar(p)=a\_{2}}

### Signum

{\displaystyle \operatorname {sgn}(p)={\frac {p}{|p|}}}

### Argumen

{\displaystyle \arg(p)=\arccos({\frac {Scalar(p)}{|p|}})}

### Pangkat dan Logaritma

Fungsi ekponensial: {\displaystyle \exp(p)=\exp(a)(\cos(|{\vec {u}}|)+\operatorname {sgn}({\vec {u}})\sin(|{\vec {u}}|))}

Logaritma natural: {\displaystyle \ln(|p|)=\ln(|p|)+\operatorname {sgn}({\vec {u}})\arg(p)}

Pangkat: {\displaystyle p^{q}=e^{q\ln(p)}}

Trigonometri
------------

### Fungsi trigonometris

{\displaystyle \sin(p)=\sin(a)\cosh(|{\vec {u}}|)+\cos(a)\operatorname {sgn}({\vec {u}})\sinh(|{\vec {u}}|)}
{\displaystyle \cos(p)=\cos(a)\cosh(|{\vec {u}}|)-\sin(a)\operatorname {sgn}({\vec {u}})\sinh(|{\vec {u}}|)}
{\displaystyle \tan(p)={\frac {\sin(p)}{\cos(p)}}}
### Fungsi hiperbolik

{\displaystyle \sinh(p)=\sinh(a)\cos(|{\vec {u}}|)+\cosh(a)\operatorname {sgn}({\vec {u}})\sin(|{\vec {u}}|)}
{\displaystyle \cosh(p)=\cosh(a)\cos(|{\vec {u}}|)+\sinh(a)\operatorname {sgn}({\vec {u}})\sin(|{\vec {u}}|)}
{\displaystyle \tanh(p)={\frac {\sinh(p)}{\cosh(p)}}}
### Fungsi hiperbolik invers

{\displaystyle \operatorname {arcsinh} (p)=\ln(p+{\sqrt {p^{2}+1}})}
{\displaystyle \operatorname {arccosh} (p)=\ln(p+{\sqrt {p^{2}-1}})}
{\displaystyle \operatorname {arctanh} (p)={\frac {\ln(1+p)-\ln(1-p)}{2}}}

Satuan
------

Kuaternion satuan: {\displaystyle p=\cos(\theta )+{\hat {u}}\sin(\theta )}

### Pangkat

{\displaystyle {\begin{aligned}&p^{t}=(\cos(\theta )+{\hat {u}}\sin(\theta ))^{t}\\&=\exp({\hat {u}}t\theta )\\&=\cos(t\theta )+{\hat {u}}\sin(t\theta )\end{aligned}}}

### Logaritma

{\displaystyle {\begin{aligned}&\log(p)=\log(\cos(\theta )+{\hat {u}}\sin(\theta ))\\&=\log(\exp({\hat {u}}\theta ))\\&={\hat {u}}\theta \end{aligned}}}

### Kalkulus

{\displaystyle {\frac {d}{dt}}p^{t}=p^{t}\log(p)}

Penerapan
---------

### Rotasi vektor grafika 3D

Fungsi rotasi vektor dapat menggunakan operasi kuaternion daripada operasi matriks riil 4x4, dengan rumus:

{\displaystyle {\begin{aligned}&r=qvq\*\\\end{aligned}}}
di mana

{\displaystyle {\begin{aligned}&v=1+x\_{A}i+y\_{A}j+z\_{A}k\\&q=\cos {\frac {\alpha }{2}}+\sin {\frac {\alpha }{2}}x\_{v}i+\sin {\frac {\alpha }{2}}y\_{v}j+\sin {\frac {\alpha }{2}}z\_{v}k\\&r=1+x\_{A}'i+y\_{A}'j+z\_{A}'k\\\end{aligned}}}
dan *A* adalah posisi benda yang dirotasikan, *v* adalah vektor poros rotasi, dan *α* adalah sudut rotasi berlawanan arah jarum jam.

Referensi
---------

Pranala luar
------------

* 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*". Royal Irish Academy.
* Hamilton (1866) *Elements of Quaternions* University of Dublin Press. Edited by William Edwin Hamilton, son of the deceased author.
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* 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
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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).
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| * l
* b
* s
Sistem bilangan |
| --- |
| Himpunan terhitung | 
* Bilangan asli ({\displaystyle \scriptstyle \mathbb {N} })
* Bilangan bulat ({\displaystyle \scriptstyle \mathbb {Z} })
* Bilangan rasional ({\displaystyle \scriptstyle \mathbb {Q} })
* Bilangan aljabar ({\displaystyle \scriptstyle {\overline {\mathbb {Q} }}})
* Perioda
* Bilangan terkomputasi
* Bilangan aritmetis

 |
| Bilangan riil dancabangan | 
* Bilangan riil ({\displaystyle \scriptstyle \mathbb {R} })
* Bilangan kompleks ({\displaystyle \scriptstyle \mathbb {C} })
* Kuaternion ({\displaystyle \scriptstyle \mathbb {H} })
* Oktonion ({\displaystyle \scriptstyle \mathbb {O} })
* Sedenion ({\displaystyle \scriptstyle \mathbb {S} })
* Aljabar Cayley–Dickson
* Bilangan rangkap
* Bilangan kompleks hiperbolik
* Bilangan hiperkompleks
* Bilangan superreal
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* Bilangan transenden
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* Bilangan surreal

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| Sistem lain | 
* Bilangan kardinal
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