# Quaternion

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" :

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 1                       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).

How to Cite This Entry:
Quaternion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quaternion&oldid=35148
This article was adapted from an original article by N.N. Vil'yams (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article