it seems convenient to enlarge by definition the signification of the new word tensor, so as to render it capable of including also those other cases in which we operate on a line by diminishing instead of increasing its length; and generally by altering that length in any definite ratio. We shall thus (as was hinted at the end of the article in question) have fractional and even tensors, which will simply be numerical multipliers, and will all be positive or (to speak more properly) SignLess Numbers, that is, unclothed with the algebraic signs of positive and negative; because, in the operation here considered, we abstract from the directions (as well as from the situations) of the lines which are compared or operated on.

incommensurable

Other operators in detail[edit]

Scalar and vector[edit]

Two important operations in two the classical quaternion notation system were S(q) and V(q) which meant take the scalar part of, and take the imaginary part, what Hamilton called the vector part of the quaternion. Here S and V are operators acting on q. Parenthesis can be omitted in these kinds of expressions without ambiguity. Classical notation:

Biquaternions[edit]

Geometrically real and geometrically imaginary numbers[edit]

In classical quaternion literature the equation

Cayley–Dickson construction

Octonions

Frobenius theorem

W.R. Hamilton (1853), Lectures on Quaternions at Google Books Dublin: Hodges and Smith

W.R. Hamilton (1866), Elements of Quaternions at Google Books, 2nd edition, edited by Charles Jasper Joly, Longmans Green & Company.

A.S. Hardy (1887), Elements of Quaternions

P.G. Tait (1890), An Elementary Treatise on Quaternions, Cambridge: C.J. Clay and Sons

Herbert Goldstein(1980), Classical Mechanics, 2nd edition, Library of congress catalog number QA805.G6 1980