Quaternion
A hypercomplex number, geometrically realizable in fourdimensional 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 $ 4 $
dimensional space. (In 3, 5 or higherdimensional space it is not even possible to do this.)
The quaternions form a $ 4 $ dimensional algebra over the field of real numbers with basis $ 1 , i , j , k $( "basic units" ) and the following multiplication table of the "basic units" :
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Every quaternion can be written in the form
$$ X = x _ {0} \cdot 1 + x _ {1} \cdot i + x _ {2} \cdot j + x _ {3} \cdot k $$
or (since 1 plays the role of ordinary identity and in writing a quaternion it can be omitted) in the form
$$ X = x _ {0} + x _ {1} i + x _ {2} j + x _ {3} k . $$
One distinguishes the scalar part $ x _ {0} $ of the quaternion and its vector part
$$ V = x _ {1} i + x _ {2} j + x _ {3} k , $$
so that $ X = x _ {0} + V $. If $ x _ {0} = 0 $, then the quaternion $ V $ is called a vector and can be identified with an ordinary $ 3 $ dimensional vector, since multiplication in the algebra of quaternions of two such vectors $ V _ {1} $ and $ V _ {2} $ is related to the scalar and vector products $ ( V _ {1} , V _ {2} ) $( cf. Inner product) and $ [ V _ {1} , V _ {2} ] $( cf. Vector product) of the vectors $ V _ {1} $ and $ V _ {2} $ in $ 3 $ dimensional space by the formula
$$ V _ {1} V _ {2} = \  ( V _ {1} , V _ {2} ) + [ V _ {1} , V _ {2} ] . $$
This shows the close relationship between quaternions and vector calculus. Historically, the latter arose from the theory of quaternions.
Corresponding to each quaternion $ X = x _ {0} + V $ is the conjugate quaternion $ \overline{X}\; = x _ {0}  V $, and
$$ X \cdot \overline{X}\; = \overline{X}\; \cdot X = \ x _ {0} ^ {2} + x _ {1} ^ {2} + x _ {2} ^ {2} + x _ {3} ^ {2} . $$
This real number is called the norm of the quaternion $ X $ and is denoted by $ N ( X) $. This norm satisfies the relation
$$ N ( XY ) = N ( X) N ( Y) . $$
Any rotation of $ 3 $ dimensional space about the origin can be defined by means of a quaternion $ P $ with norm 1. The rotation corresponding to $ P $ takes the vector $ X = x _ {1} i + x _ {2} j + x _ {3} k $ to the vector $ Y = y _ {1} i + y _ {2} j + y _ {3} k = P X P ^ {} 1 $.
The algebra of quaternions is the unique associative noncommutative finitedimensional normed algebra over the field of real numbers with an identity. The algebra of quaternions is a skewfield, that is, division is defined in it, and the quaternion inverse to a quaternion $ X $ is $ \overline{X}\; / N ( X) $. The skewfield of quaternions is the unique finitedimensional real associative noncommutative algebra without divisors of zero (see also Frobenius theorem; Cayley–Dickson algebra).
References
[1]  L.A. Kaluzhnin, "Introduction to general algebra" , Moscow (1973) (In Russian) 
[2]  I.L. Kantor, A.S. Solodovnikov, "Hyperkomplexe Zahlen" , Teubner (1978) (Translated from Russian) 
[3]  A.G. Kurosh, "Higher algebra" , MIR (1972) (Translated from Russian) 
Comments
Let $ \zeta $ be the element $ ( 1+ i+ j+ k)/2 $ in the algebra of quaternions. The Hurwitz ring of integral quaternions is the ring
$$ H = \{ {m _ {0} \zeta + m _ {1} i + m _ {2} j + m _ {3} k } : { m _ {0} , m _ {1} , m _ {2} , m _ {3} \in \mathbf Z } \} . $$
The Hurwitz ring is a noncommutative ring in which an analogue of the Euclidean division property (cf. Euclidean algorithm) holds: For any $ a, b \in H $ with $ b \neq 0 $ there exist elements $ q, r \in H $ such that
$$ a = qb + r $$
with
$$ N( r) < N( b) . $$
(This property does not hold for the subring $ \{ {n _ {0} + n _ {1} i + n _ {2} j + n _ {3} k } : {n _ {0} , n _ {1} , n _ {2} , n _ {3} \in \mathbf Z } \} $.) It follows that every left ideal is left principal, and this in turn can be used to give a proof of the Lagrange foursquare 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,
$$ ( a _ {0} ^ {2} + a _ {1} ^ {2} + a _ {2} ^ {2} + a _ {3} ^ {2} ) ( b _ {0} ^ {2} + b _ {1} ^ {2} + b _ {2} ^ {2} + b _ {3} ^ {2} ) = $$
$$ = \ ( a _ {0} b _ {0}  a _ {1} b _ {1}  a _ {2} b _ {2}  a _ {3} b _ {3} ) ^ {2} + $$
$$ + ( a _ {0} b _ {1} + a _ {1} b _ {0} + a _ {2} b _ {3}  a _ {3} b _ {2} ) ^ {2} + $$
$$ + ( a _ {0} b _ {2}  a _ {1} b _ {3} + a _ {2} b _ {0} + a _ {3} b _ {1} ) ^ {2} + $$
$$ + ( a _ {0} b _ {3} + a _ {1} b _ {2}  a _ {2} b _ {1} + a _ {3} b _ {0} ) ^ {2} $$
for real numbers $ a _ {0,\ } a _ {1} , a _ {2} , a _ {3} , b _ {0} , b _ {1} , b _ {2} , b _ {3} $, is equivalent to the multiplicativity of the norm $ N( XY) = N( X) N( Y) $, where $ X $, $ Y $ are the quaternions $ X = a _ {0} + a _ {1} i + a _ {2} j + a _ {3} k $, $ Y = b _ {0} + b _ {1} i + b _ {2} j + b _ {3} k $.
Writing $ X $ as $ ( a _ {0} + a _ {1} i) + ( a _ {2} + a _ {3} i ) j $ and putting $ \alpha = a _ {0} + a _ {1} i $, $ \beta = a _ {2} + a _ {3} i $, one obtains $ X = \alpha + \beta j $. It is easily proved that the algebra of quaternions is isomorphic to the algebra of complex $ ( 2 \times 2) $ matrices
$$ \left ( \begin{array}{cc} \alpha &\beta \\ \overline \beta \; &\overline \alpha \; \\ \end{array} \right ) , $$
with $ \overline \alpha \; , \overline \beta \; $ the complex conjugates of $ \alpha , \beta \in \mathbf C $.
When one wishes to retain the multiplicativity of the norm, $ N( XY) = N( X) N( Y) $, 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).
The centre of the skewfield of quaternions is the field of real numbers. Later the notion of hypercomplex system has been generalized in a theory of skewfields over arbitrary fields, e.g. the theory of the Brauer group of a commutative field.
In this connection, a generalized quaternion algebra is a $ 4 $ dimensional algebra over a field $ F $ generated by $ 1, x, y , xy $ with multiplication table $ x ^ {2} = a $, $ y ^ {2} = b $, $ yx =  xy $, where $ a, b $ are nonzero elements of $ F $. (The quaternions are the case $ a = b =  1 $, and $ F $ the field of real numbers.)
References
[a1]  A.A. Albert, "Structure of algebras" , Amer. Math. Soc. (1935) 
[a2]  R. Brauer, E. Noether, "Über minimale Zerfällungskörper irreducibler Darstellungen" Sitzungsber. Akad. Berlin , 27 (1927) pp. 221–226 
[a3]  J.H.M. Wedderburn, "On hypercomplex numbers" Proc. London Math. Soc. Ser. 2 , 6 (1907) pp. 77–118 
[a4]  R. Brauer, E. Weiss, "Noncommutative rings" , Harvard Univ. Press (1950) pp. Part I 
[a5]  H. Behnke, F. Bachmann, "Grundzüge der Mathematik" , I , Göttingen (1962) 
[a6]  S. Maclane, G. Birkhoff, "Algebra" , Macmillan (1979) 
[a7]  M. Crowe, "A history of vector analysis, the evolution of the idea of a vectorial system" , Univ. Notre Dame (1967) 
[a8]  R.J. Stephenson, "Development of vector analysis from quaternions" Amer. J. Physics , 34 (1966) pp. 194–201 
[a9]  B.L. van der Waerden, "Hamiltons Entdeckung der Quaternionen" , Vandenhoeck & Ruprecht (1973) 
[a10]  I.N. Herstein, "Topics in algebra" , Wiley (1975) pp. Sect. 7.4 
Hurwitz number. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hurwitz_number&oldid=42370