## Abstract

An E-number is an unusual mathematical form, constructed from sixteen algebraic variables and sixteen E-symbols (square roots of - 1 obeying a non-commutative algebra), which has distinctive properties different from those of the square matrix of the variables employed in tensor theory. When the latter is considered to be a tensor, the corresponding E-number cannot be a standard tensor, and tensor calculus has made no use of it. Without loss of generality the E-symbols can always be envisaged as transforms of a certain set of numerical matrices, and it is shown here that when this is done every E-number can be expressed in a simplest form, called an $\scr{E}$-number. The reduction of an E- to an $\scr{E}$-number takes the form of a transformation analogous to that by which a physical magnitude is expressed in a different unit, the variables and the symbols being transformed oppositely. When evaluated the $\scr{E}$-number shows itself as a square matrix, which is characterized by having the data expressed by the sixteen variables arranged in another systematic way than that employed in the standard matrix of the variables. It may be obtained from this by a novel kind of transposition, in which the transposed elements are not the matrix terms themselves, but the symmetric and anti-symmetric parts of the terms. Alternatively, it may be obtained by submitting the standard matrix of the variables to a special type of transformation. The transformations employed in these operations differ from those normally used in matrix theory, as they take the form of sums of sets of matrix transformations. General formulae for the products of $\scr{E}$-numbers are derived.