Grassmann Algebra

ExploringCliffordAlgebra.nb 47
Historical Note
This complex sum of two quaternions was called a biquaternion by Hamilton [Hamilton,
Elements of Quaternions, p133] but Clifford in a footnote to his Preliminary Sketch of
Biquaternions [Clifford, Mathematical Papers, Chelsea] says 'Hamilton's biquaternion is a
quaternion with complex coefficients; but it is convenient (as Prof. Pierce remarks) to suppose
from the beginning that all scalars may be complex. As the word is thus no longer wanted in its
old meaning, I have made bold to use it in a new one."
Hamilton uses the word biscalar for a complex number and bivector [p 225 Elements of
Quaternions] for a complex vector, that is, for a vector x + � y, where x and y are vectors; and
the word biquaternion for a complex quaternion q 0 + � q 1 , where q 0 and q 1 are quaternions. He
emphasizes here that "… � is the (scalar) imaginary of algebra, and not a symbol for a
geometrically real right versor …"
Hamilton introduces his biquaternion as the quotient of a bivector (his usage) by a (real) vector.
12.16 Clifford Algebras of a 4-Space
� The Clifford product table in 4-space
In this section we explore the Clifford algebras of 4-space. First we declare a (not necessarily
orthogonal) basis for the 4-space, and generate the associated Clifford product table. Because of
the size of the table, only the first few columns are shown in the print version.
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