View source for Quaternion

A [[Hypercomplex number|hypercomplex number]], geometrically realizable in four-dimensional space. The system of quaternions was put forward in 1843 by W.R. Hamilton (1805–1865). Quaternions were historically the first example of a hypercomplex system, arising from attempts to find a generalization of complex numbers. Complex numbers are depicted geometrically by points in the plane and operations on them correspond to the simplest geometric transformations of the plane. It is not possible to  "organize"  a number system similar to the field of real or complex numbers from the points of a space of three or more dimensions. However, if one drops the requirement of commutativity of multiplication, then it is possible to construct a number system from the points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q0767701.png" />-dimensional space. (In 3, 5 or higher-dimensional space it is not even possible to do this.)

The quaternions form a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q0767702.png" />-dimensional algebra over the field of real numbers with basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q0767703.png" /> ( "basic units" ) and the following multiplication table of the  "basic units" :

<table border="0" cellpadding="0" cellspacing="0" style="background-color:black;"> <tr><td> <table border="0" cellspacing="1" cellpadding="4" style="background-color:black;"> <tbody> <tr> <td colname="1" style="background-color:white;" colspan="1"></td> <td colname="2" style="background-color:white;" colspan="1">1</td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q0767704.png" /></td> <td colname="4" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q0767705.png" /></td> <td colname="5" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q0767706.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q0767707.png" /></td> <td colname="2" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q0767708.png" /></td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q0767709.png" /></td> <td colname="4" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677010.png" /></td> <td colname="5" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677011.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677012.png" /></td> <td colname="2" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677013.png" /></td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677014.png" /></td> <td colname="4" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677015.png" /></td> <td colname="5" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677016.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677017.png" /></td> <td colname="2" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677018.png" /></td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677019.png" /></td> <td colname="4" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677020.png" /></td> <td colname="5" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677021.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677022.png" /></td> <td colname="2" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677023.png" /></td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677024.png" /></td> <td colname="4" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677025.png" /></td> <td colname="5" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677026.png" /></td> </tr> </tbody> </table>

</td></tr> </table>

Every quaternion can be written in the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677027.png" /></td> </tr></table>

or (since 1 plays the role of ordinary identity and in writing a quaternion it can be omitted) in the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677028.png" /></td> </tr></table>

One distinguishes the scalar part <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677029.png" /> of the quaternion and its vector part

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677030.png" /></td> </tr></table>

so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677031.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677032.png" />, then the quaternion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677033.png" /> is called a vector and can be identified with an ordinary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677034.png" />-dimensional vector, since multiplication in the algebra of quaternions of two such vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677035.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677036.png" /> is related to the scalar and vector products <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677037.png" /> (cf. [[Inner product|Inner product]]) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677038.png" /> (cf. [[Vector product|Vector product]]) of the vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677039.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677040.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677041.png" />-dimensional space by the formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677042.png" /></td> </tr></table>

This shows the close relationship between quaternions and [[Vector calculus|vector calculus]]. Historically, the latter arose from the theory of quaternions.

Corresponding to each quaternion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677043.png" /> is the conjugate quaternion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677044.png" />, and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677045.png" /></td> </tr></table>

This real number is called the norm of the quaternion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677046.png" /> and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677047.png" />. This norm satisfies the relation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677048.png" /></td> </tr></table>

Any rotation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677050.png" />-dimensional space about the origin can be defined by means of a quaternion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677051.png" /> with norm 1. The rotation corresponding to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677052.png" /> takes the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677053.png" /> to the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677054.png" />.

The algebra of quaternions is the unique associative non-commutative finite-dimensional normed algebra over the field of real numbers with an identity. The algebra of quaternions is a [[Skew-field|skew-field]], that is, division is defined in it, and the quaternion inverse to a quaternion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677055.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677056.png" />. The skew-field of quaternions is the unique finite-dimensional real associative non-commutative algebra without divisors of zero (see also [[Frobenius theorem|Frobenius theorem]]; [[Cayley–Dickson algebra|Cayley–Dickson algebra]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.A. Kaluzhnin,   "Introduction to general algebra" , Moscow  (1973)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.L. Kantor,   A.S. Solodovnikov,   "Hyperkomplexe Zahlen" , Teubner  (1978)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.G. Kurosh,   "Higher algebra" , MIR  (1972)  (Translated from Russian)</TD></TR></table>



====Comments====
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677057.png" /> be the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677058.png" /> in the algebra of quaternions. The Hurwitz ring of integral quaternions is the ring

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677059.png" /></td> </tr></table>

The Hurwitz ring is a non-commutative ring in which an analogue of the Euclidean division property (cf. [[Euclidean algorithm|Euclidean algorithm]]) holds: For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677060.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677061.png" /> there exist elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677062.png" /> such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677063.png" /></td> </tr></table>

with

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677064.png" /></td> </tr></table>

(This property does not hold for the subring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677065.png" />.) It follows that every left ideal is left principal, and this in turn can be used to give a proof of the Lagrange four-square theorem, to the effect that every positive integer can be written as a sum of four squares of integers.

The Lagrange identity, which plays an important role in the proof of this result,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677066.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677067.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677068.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677069.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677070.png" /></td> </tr></table>

for real numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677071.png" />, is equivalent to the multiplicativity of the norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677072.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677073.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677074.png" /> are the quaternions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677075.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677076.png" />.

Writing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677077.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677078.png" /> and putting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677079.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677080.png" />, one obtains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677081.png" />. It is easily proved that the algebra of quaternions is isomorphic to the algebra of complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677082.png" />-matrices

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677083.png" /></td> </tr></table>

with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677084.png" /> the complex conjugates of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677085.png" />.

When one wishes to retain the multiplicativity of the norm, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677086.png" />, there is only one possible generalization of the quaternions (over the reals): the octaves or octonions, which have 8 instead of 4 components (Hurwitz's theorem, 1898; cf. [[Cayley numbers|Cayley numbers]]).

The centre of the skew-field of quaternions is the field of real numbers. Later the notion of hypercomplex system has been generalized in a theory of skew-fields over arbitrary fields, e.g. the theory of the [[Brauer group|Brauer group]] of a commutative field.

In this connection, a generalized quaternion algebra is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677087.png" />-dimensional algebra over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677088.png" /> generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677089.png" /> with multiplication table <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677090.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677091.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677092.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677093.png" /> are non-zero elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677094.png" />. (The quaternions are the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677095.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076770/q07677096.png" /> the field of real numbers.)

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.A. Albert,   "Structure of algebras" , Amer. Math. Soc.  (1935)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R. Brauer,   E. Noether,   "Über minimale Zerfällungskörper irreducibler Darstellungen"  ''Sitzungsber. Akad. Berlin'' , '''27'''  (1927)  pp. 221–226</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J.H.M. Wedderburn,   "On hypercomplex numbers"  ''Proc. London Math. Soc. Ser. 2'' , '''6'''  (1907)  pp. 77–118</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  R. Brauer,   E. Weiss,   "Non-commutative rings" , Harvard Univ. Press  (1950)  pp. Part I</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  H. Behnke,   F. Bachmann,   "Grundzüge der Mathematik" , '''I''' , Göttingen  (1962)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  S. Maclane,   G. Birkhoff,   "Algebra" , Macmillan  (1979)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  M. Crowe,   "A history of vector analysis, the evolution of the idea of a vectorial system" , Univ. Notre Dame  (1967)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  R.J. Stephenson,   "Development of vector analysis from quaternions"  ''Amer. J. Physics'' , '''34'''  (1966)  pp. 194–201</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  B.L. van der Waerden,   "Hamiltons Entdeckung der Quaternionen" , Vandenhoeck &amp; Ruprecht  (1973)</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  I.N. Herstein,   "Topics in algebra" , Wiley  (1975)  pp. Sect. 7.4</TD></TR></table>