01 January 1970 0 10K Report

This is indended for a wide discussion of anything to do with elements as i

which are not real, but has assisted algebra since long ago. Or anything pretended to

to be an infinitesimal which is complicated by the fact of their inexistance in the real nos.

Things like i have been shown to vanish like phantoms when discussing binary or matrix elements, whereupon thier existance becomes tangible, like a (0,1) bracket expresion.

Thus the expectation may be that others are really more complex entities than simply

real nos, or simple nos. A hint is that i is really both i or -i, minus specify their rotational

operator property.

We start this off by giving a nilpotent matrix M in parametrized form

1/(1+tt) times the matrix

t -tt

1 -t

which is singular, traceless, nilpotent MM=0.

We use this together with the transpose M(T): M(T)M(T)=0

Properties

M-M(T) = Matrix expression for i

as

0 -1

1 0

MM(T)+M(T)M= I

MM(T) or M(T)M contribute just a 1 along the diagonal, both needed, indempotent operators.

with PP=P property. The 1/(1+tt) is a normalization factor.(otherwise we get a CI instead of I)

Using a 1,t , sqrt(1+tt) triangle , M is entirely expressible as sin(theta), cos(theta) entity.

sin(theta) = 1/sqrt(1+tt)

Replacing this language with e, e(T) one can

replace i(or C as a ring), or use he as proper infinitesimal, no second order d(x)d(y) residuals, and

f(x+ey) = f(x) +ey Df(x)/dx

The whole thing of course beggs for generalizatins as

e(T) ee +e e(T) e + ee e(T) =I , for some eee=0 and e not zero.

The wider conception of ring versus field is threatning many traditional conceptionslike

beyond Complex nos. one cannot go, the only respectable stuff in Calculus are fields,

etc,etc.

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