Editing Quaternion (section)

== Definition ==
A ''quaternion'' is an [[expression (mathematics)|expression]] of the form

<math display=block>a + b\,\mathbf{i} +  c\,\mathbf{j} + d\,\mathbf{k},</math>

where {{mvar|a}}, {{mvar|b}}, {{mvar|c}}, {{mvar|d}}, are real numbers, and {{math|'''i'''}}, {{math|'''j'''}}, {{math|'''k'''}}, are [[symbol (mathematics)|symbols]] that can be interpreted as unit-vectors pointing along the three spatial axes. In practice, if one of {{mvar|a}}, {{mvar|b}}, {{mvar|c}}, {{mvar|d}} is 0, the corresponding term is omitted; if {{mvar|a}}, {{mvar|b}}, {{mvar|c}}, {{mvar|d}} are all zero, the quaternion is the ''zero quaternion'', denoted 0; if one of {{mvar|b}}, {{mvar|c}}, {{mvar|d}}  equals 1, the corresponding term is written simply {{math|'''i''', '''j'''}}, or {{math|'''k'''}}.

Hamilton describes a quaternion <math>q = a + b\,\mathbf{i} + c\,\mathbf{j} + d\,\mathbf{k}</math>, as consisting of a [[Scalar (mathematics)|scalar]] part and a vector part. The quaternion <math> b\,\mathbf{i} + c\,\mathbf{j} + d\,\mathbf{k}</math> is called the ''vector part'' (sometimes ''imaginary part'') of {{mvar|q}}, and {{mvar|a}} is the ''scalar part'' (sometimes ''real part'') of {{mvar|q}}. A quaternion that equals its real part (that is, its vector part is zero) is called a ''scalar'' or ''real quaternion'', and is identified with the corresponding real number. That is, the real numbers are ''embedded'' in the quaternions. (More properly, the [[field (mathematics)|field]] of real numbers is isomorphic to a subset of the quaternions. The field of complex numbers is also isomorphic to three subsets of quaternions.)<ref>{{harvtxt|Eves|1976|page=391}}</ref> A quaternion that equals its vector part is called a ''vector quaternion''.

The set of quaternions is a 4-dimensional [[vector space]] over the real numbers, with <math>\left\{ 1, \mathbf i, \mathbf j, \mathbf k\right\}</math> as a [[basis (linear algebra)|basis]], by the component-wise addition

<math display=block>\begin{align}
&(a_1 + b_1\mathbf i + c_1\mathbf j + d_1\mathbf k) + (a_2 + b_2\mathbf i + c_2\mathbf j + d_2\mathbf k) \\[3mu]
&\qquad = (a_1 + a_2) + (b_1 + b_2)\mathbf i + (c_1 + c_2)\mathbf j + (d_1 + d_2)\mathbf k,
\end{align}</math>

and the component-wise scalar multiplication

<math display=block>\lambda(a + b\,\mathbf i + c\,\mathbf j + d\,\mathbf k) = \lambda a + (\lambda b)\mathbf i + (\lambda c)\mathbf j + (\lambda d)\mathbf k.</math>

A multiplicative group structure, called the ''Hamilton product'', denoted by juxtaposition, can be defined on the quaternions in the following way:
*The real quaternion {{math|'''1'''}} is the [[identity element]].
*The '''real''' quaternions commute with all other quaternions, that is {{math|''aq'' {{=}} ''qa''}} for every quaternion {{mvar|q}} and every real quaternion {{mvar|a}}. In algebraic terminology this is to say that the field of real quaternions are the [[center (ring theory)|''center'']] of this quaternion algebra.
*The product is first given for the basis elements (see next subsection), and then extended to all quaternions by using the [[distributive property]] and the center property of the real quaternions. The Hamilton product is not [[commutative property|commutative]], but is [[associative property|associative]], thus the quaternions form an associative algebra over the real numbers.
*Additionally, every nonzero quaternion has an inverse with respect to the Hamilton product: <math display=block>(a + b\,\mathbf i + c\,\mathbf j + d \,\mathbf k)^{-1} = \frac{1}{a^2 + b^2 + c^2 + d^2}\,(a - b\,\mathbf i - c\,\mathbf j- d\,\mathbf k).</math>
Thus the quaternions form a division algebra.

=== Multiplication of basis elements ===
{|class="wikitable" align="right" style="text-align:center"
|+[[Multiplication table]]
|-
!width=15|×
!width=15|{{math|1}}
!width=15|{{math|'''i'''}}
!width=15|{{math|'''j'''}}
!width=15|{{math|'''k'''}}
|-
!{{math|1}}
|{{math|1}}
|{{math|'''i'''}}
|{{math|'''j'''}}
|{{math|'''k'''}}
|-
!{{math|'''i'''}}
|{{math|'''i'''}}
|{{math|−1}}
|style="background: #FF9999;"|{{math|'''k'''}}
|style="background: #9999FF;"|{{math|−'''j'''}}
|-
!{{math|'''j'''}}
|{{math|'''j'''}}
|style="background: #FF9999;"| {{math|−'''k'''}}
|{{math|−1}}
|style="background: #99FF99;"|{{math|'''i'''}}
|-
!{{math|'''k'''}}
|{{math|'''k'''}}
|style="background: #9999FF;"|{{math|'''j'''}}
|style="background: #99FF99;"|{{math|−'''i'''}}
|{{math|−1}}
|-
|+Non commutativity is emphasized by colored squares 
|}
The multiplication with {{math|1}} of the basis elements {{math|'''i''', '''j'''}}, and {{math|'''k'''}} is defined by the fact that {{math|1}} is a [[multiplicative identity]], that is,

<math display=block>\mathbf i \, 1 = 1 \, \mathbf i = \mathbf i, \qquad \mathbf j \, 1 = 1 \, \mathbf j = \mathbf j, \qquad \mathbf k \, 1 = 1 \, \mathbf k= \mathbf k .</math>

The products of other basis elements are

<math display=block>\begin{align}
\mathbf i^2 &= \mathbf j^2 = \mathbf k^2 = -1, \\[5mu]
\mathbf{i\,j} &= - \mathbf{j\,i} = \mathbf k, \qquad
\mathbf{j\,k} = - \mathbf{k\,j} = \mathbf i, \qquad
\mathbf{k\,i} = - \mathbf{i\,k} = \mathbf j.
\end{align}</math>

Combining these rules,

<math display=block>\begin{align}
\mathbf{i\,j\,k}&=-1.
\end{align}</math>

=== Center ===
The [[center (ring theory)|''center'']] of a [[noncommutative ring]] is the subring of elements {{mvar|c}} such that {{math|1=''cx'' = ''xc''}} for every {{mvar|x}}. The center of the quaternion algebra is the subfield of real quaternions. In fact, it is a part of the definition that the real quaternions belong to the center. Conversely, if {{math|''q'' {{=}} ''a'' + ''b'' '''i''' + ''c'' '''j''' + ''d'' '''k'''}} belongs to the center, then

<math display=block>0 = \mathbf i\,q - q\,\mathbf i = 2c\,\mathbf{ij} + 2d\,\mathbf{ik} = 2c\,\mathbf k - 2d\,\mathbf j,</math>

and {{math|''c'' {{=}} ''d'' {{=}} 0}}. A similar computation with {{math|'''j'''}} instead of {{math|'''i'''}} shows that one has also {{math|''b'' {{=}} 0}}. Thus {{math|''q'' {{=}} ''a''}} is a ''real'' quaternion.

The quaternions form a division algebra. This means that the non-commutativity of multiplication is the only property that makes quaternions different from a field. This non-commutativity has some unexpected consequences, among them that a [[polynomial equation]] over the quaternions can have more distinct solutions than the degree of the polynomial. For example, the equation {{nowrap|{{math|''z''<sup>2</sup> + 1 {{=}} 0}},}} has infinitely many quaternion solutions, which are the quaternions {{math|''z'' {{=}} ''b'' '''i''' + ''c'' '''j''' + ''d'' '''k'''}} such that {{math|''b''<sup>2</sup> + ''c''<sup>2</sup> + ''d''<sup>2</sup> {{=}} 1}}. Thus these "roots of –1" form a [[unit sphere]] in the three-dimensional space of vector quaternions.

=== Hamilton product ===
For two elements {{math|''a''<sub>1</sub> + ''b''<sub>1</sub>'''i''' + ''c''<sub>1</sub>'''j''' + ''d''<sub>1</sub>'''k'''}} and {{math|''a''<sub>2</sub> + ''b''<sub>2</sub>'''i''' + ''c''<sub>2</sub>'''j''' + ''d''<sub>2</sub>'''k'''}}, their product, called the  '''Hamilton product''' ({{math|''a''<sub>1</sub> + ''b''<sub>1</sub>'''i''' + ''c''<sub>1</sub>'''j''' + ''d''<sub>1</sub>'''k'''}}) ({{math|''a''<sub>2</sub> + ''b''<sub>2</sub>'''i''' + ''c''<sub>2</sub>'''j''' + ''d''<sub>2</sub>'''k'''}}), is determined by the products of the basis elements and the [[distributive law]]. The distributive law makes it possible to expand the product so that it is a sum of products of basis elements. This gives the following expression:

<math display=block>\begin{alignat}{4}
        &a_1a_2  &&+ a_1b_2 \mathbf i   &&+ a_1c_2 \mathbf j   &&+ a_1d_2 \mathbf k\\
  {}+{} &b_1a_2 \mathbf i &&+ b_1b_2 \mathbf i^2 &&+ b_1c_2 \mathbf{ij} &&+ b_1d_2 \mathbf{ik}\\
  {}+{} &c_1a_2 \mathbf j &&+ c_1b_2 \mathbf{ji} &&+ c_1c_2 \mathbf j^2 &&+ c_1d_2 \mathbf{jk}\\
  {}+{} &d_1a_2 \mathbf k &&+ d_1b_2 \mathbf{ki} &&+ d_1c_2 \mathbf{kj}  &&+ d_1d_2 \mathbf k^2
\end{alignat}</math>

Now the basis elements can be multiplied using the rules given above to get:<ref name="SeeHazewinkel" />

<math display=block>\begin{alignat}{4}
          &a_1a_2 &&- b_1b_2 &&- c_1c_2 &&- d_1d_2\\
   {}+{} (&a_1b_2 &&+ b_1a_2 &&+ c_1d_2 &&- d_1c_2) \mathbf i\\
   {}+{} (&a_1c_2 &&- b_1d_2 &&+ c_1a_2 &&+ d_1b_2) \mathbf j\\
   {}+{} (&a_1d_2 &&+ b_1c_2 &&- c_1b_2 &&+ d_1a_2) \mathbf k
\end{alignat}</math>

=== Scalar and vector parts ===
A quaternion of the form {{math|''a'' + 0 '''i''' + 0 '''j''' + 0 '''k'''}}, where {{mvar|a}} is a real number, is called '''scalar''', and a quaternion of the form {{math|0 + ''b'' '''i''' + ''c'' '''j''' + ''d'' '''k'''}}, where {{mvar|b}}, {{mvar|c}}, and {{mvar|d}} are real numbers, and at least one of {{mvar|b}}, {{mvar|c}}, or {{mvar|d}} is nonzero, is called a '''vector quaternion'''.  If {{math|''a'' + ''b'' '''i''' + ''c'' '''j''' + ''d'' '''k'''}} is any quaternion, then {{mvar|a}} is called its '''scalar part''' and {{math|''b'' '''i''' + ''c'' '''j''' + ''d'' '''k'''}} is called its '''vector part'''. Even though every quaternion can be viewed as a vector in a four-dimensional vector space, it is common to refer to the '''vector''' part as vectors in three-dimensional space.  With this convention, a vector is the same as an element of the vector space <math>\mathbb R^3.</math>{{efn|The vector part of a quaternion is an "axial" vector or "[[pseudovector]]", ''not'' an ordinary or "polar" vector, as was formally proved by Altmann (1986).<ref>{{cite book |first=S.L. |last=Altmann |at=Ch.&nbsp;12 |title=Rotations, Quaternions, and Double Groups}}</ref> A polar vector can be represented in calculations (for example, for rotation by a quaternion "similarity transform") by a pure imaginary quaternion, with no loss of information, but the two should not be confused. The axis of a "binary" (180°) rotation quaternion corresponds to the direction of the represented polar vector in such a case.}}

Hamilton also called vector quaternions '''right quaternions'''<ref>{{cite book |url=https://archive.org/details/bub_gb_fIRAAAAAIAAJ |title=Elements of Quaternions |publisher=Longmans, Green, & Company |article=Article 285 |page=[https://archive.org/details/bub_gb_fIRAAAAAIAAJ/page/n381 310] |author=Hamilton, Sir William Rowan |year=1866}}</ref><ref>{{cite journal |url=http://dlxs2.library.cornell.edu/cgi/t/text/pageviewer-idx?c=math;cc=math;q1=right%20quaternion;rgn=full%20text;idno=05140001;didno=05140001;view=image;seq=81 |author=Hardy |title=Elements of Quaternions |journal=Science |year=1881 |volume=2 |issue=75 |page=65 |publisher=library.cornell.edu|doi=10.1126/science.os-2.75.564 |pmid=17819877 }}</ref> and real numbers (considered as quaternions with zero vector part) '''scalar quaternions'''.

If a quaternion is divided up into a scalar part and a vector part, that is,

<math display=block>
\mathbf q = (r,\,\vec{v}),\ \mathbf q \in \mathbb{H},\ r \in \mathbb{R},\ \vec{v}\in \mathbb{R}^3,
</math>

then the formulas for addition, multiplication, and multiplicative inverse are

<math display=block>\begin{align}
(r_1,\,\vec{v}_1) + (r_2,\,\vec{v}_2)
&= (r_1 + r_2,\,\vec{v}_1 + \vec{v}_2), \\[5mu]
(r_1,\,\vec{v}_1) (r_2,\,\vec{v}_2)
&= (r_1 r_2 - \vec{v}_1\cdot\vec{v}_2,\,r_1\vec{v}_2+r_2\vec{v}_1 + \vec{v}_1\times\vec{v}_2), \\[5mu]
(r,\,\vec{v})^{-1}
&= \left(\frac{r}{r^2 + \vec{v}\cdot\vec{v}},\ \frac{-\vec{v}}{r^2 + \vec{v}\cdot\vec{v}}\right),
\end{align}</math>

where "<math>{}\cdot{}</math>" and "<math>\times</math>" denote respectively the [[dot product]] and the [[cross product]].