  The square root of Not is a logical operator of importance in quantum
computing theory and of interest as a mathematical object in its own right. In
physics, it is a square complex matrix of dimension 2. In the present work it
is a complex square matrix of arbitrary dimension. The introduction of linear
algebra into logical theory has been enhanced in recent decades by the
researches in the field of neural networks and quantum computing. Here we will
make a brief description of the representation of logical operations through
matrices and we show how general expressions for the two square roots of the
Not operator are obtained. Then, we explore two topics. First, we study an
extension to a non-quantum domain of a short form of Deutsch's algorithm. Then,
we assume that a root of Not is a matrix extension of the imaginary unit i, and
under this idea we obtain fully matrix versions for the Euler expansions and
for the representations of circular functions by complex exponentials.
