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The norm of a quaternion is defined bywhereis the quaternion conjugate.

The quaternions are members of a noncommutative division algebra first invented by William Rowan Hamilton. The idea for quaternions occurred to him while he was walking along the Royal Canal on his way to a meeting of the Irish Academy, and Hamilton was so pleased with his discovery that he scratched the fundamental formula of quaternion algebra,(1)into the stone of the Brougham bridge (Mishchenko and Solovyov 2000). The set of quaternions is denoted , , or , and the quaternions are a single example of a more general class of hypercomplex numbers discovered by Hamilton. While the quaternions are not commutative, they are associative, and they form a group known as the quaternion group.By analogy with the complex numbers being representable as a sum of real and imaginary parts, , a quaternion can also be written as a linear combination(2)The quaternion is implemented as Quaternion[a, b, c, d] in the Wolfram Language package Quaternions` where..

The set of octonions, also sometimes called Cayley numbers and denoted , consists of the elements in a Cayley algebra. A typical octonion is of the formwhere each of the triples , , , , , , behaves like the quaternions . Octonions are not associative. They have been used in the study of eight-dimensional space, in which a general rotation can be written as

There are at least two definitions of hypercomplex numbers. Clifford algebraists call their higher dimensional numbers hypercomplex, even though they do not share all the properties of complex numbers and no classical function theory can be constructed over them.According to van der Waerden (1985), a hypercomplex number is a number having properties departing from those of the real and complex numbers. The most common examples are biquaternions, exterior algebras, group algebras, matrices, octonions, and quaternions.One type of hypercomplex number due to Davenport (1996) and sometimes called "the" hypercomplex numbers are defined according to the multiplication table(1)(2)(3)and therefore satisfy(4)Note that these are not quaternions, and that the multiplication of these hypercomplex numbers is commutative. Unlike real and complex numbers, not all nonzero hypercomplex numbers have a multiplicative inverse...

Let be an -dimensional linear space over a field , and let be a quadratic form on . A Clifford algebra is then defined over , where is the tensor algebra over and is a particular ideal of .Clifford algebraists call their higher dimensional numbers hypercomplex even though they do not share all the properties of complex numbers and no classical function theory can be constructed over them.When is Euclidean space, the Clifford algebra is generated by the standard basis vectors with the relations(1)(2)for . The standard Clifford algebra is then generated additively by elements of the form , where , and so the dimension is , where is the dimension of .The defining relation in the general case with vectors is(3)where denotes the quadratic form, or equivalently,(4)where is the symmetric bilinear form associated with .Clifford algebras are associative but not commutative.When , the Clifford algebra becomes exterior algebra.Clifford algebras are..

The only nonassociative division algebra with real scalars. There is an 8-square identity corresponding to this algebra.The elements of a Cayley algebra are called Cayley numbers or octonions, and the multiplication table for any Cayley algebra over a field with field characteristic may be taken as shown in the following table, where , , ..., are a bases over and , , and are nonzero elements of (Schafer 1996, pp. 5-6).

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