A 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 -dimensional space. (In 3, 5 or higher-dimensional space it is not even possible to do this.)
The quaternions form a -dimensional algebra over the field of real numbers with basis ( "basic units" ) and the following multiplication table of the "basic units" :
Every quaternion can be written in the form
or (since 1 plays the role of ordinary identity and in writing a quaternion it can be omitted) in the form
One distinguishes the scalar part of the quaternion and its vector part
so that . If , then the quaternion is called a vector and can be identified with an ordinary -dimensional vector, since multiplication in the algebra of quaternions of two such vectors and is related to the scalar and vector products (cf. Inner product) and (cf. Vector product) of the vectors and in -dimensional space by the formula
This shows the close relationship between quaternions and vector calculus. Historically, the latter arose from the theory of quaternions.
Corresponding to each quaternion is the conjugate quaternion , and
This real number is called the norm of the quaternion and is denoted by . This norm satisfies the relation
Any rotation of -dimensional space about the origin can be defined by means of a quaternion with norm 1. The rotation corresponding to takes the vector to the vector .
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, that is, division is defined in it, and the quaternion inverse to a quaternion is . The skew-field of quaternions is the unique finite-dimensional real associative non-commutative algebra without divisors of zero (see also Frobenius theorem; Cayley–Dickson algebra).
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Let be the element in the algebra of quaternions. The Hurwitz ring of integral quaternions is the ring
The Hurwitz ring is a non-commutative ring in which an analogue of the Euclidean division property (cf. Euclidean algorithm) holds: For any with there exist elements such that
(This property does not hold for the subring .) 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,
for real numbers , is equivalent to the multiplicativity of the norm , where , are the quaternions , .
Writing as and putting , , one obtains . It is easily proved that the algebra of quaternions is isomorphic to the algebra of complex -matrices
with the complex conjugates of .
When one wishes to retain the multiplicativity of the norm, , 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 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 of a commutative field.
In this connection, a generalized quaternion algebra is a -dimensional algebra over a field generated by with multiplication table , , , where are non-zero elements of . (The quaternions are the case , and the field of real numbers.)
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Quaternion. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Quaternion&oldid=35148