Hi,

I believe you have made a mistake in doing Z + Z = 2Z = Z

for 2Z to equal Z you have to assume that one of the Z sums ends at -1 so that the other can conveniently start at 1. This is a misconception as these are infinite series that have no end.

The only conclusion you can draw from Z + Z is as follows

Z = 1 - 1 + 1 - 1 + 1 - .......
+
Z = 1 - 1 + 1 - 1 + 1 - .......

which is equal to 2 - 2 + 2 - 2 + 2 - .......

Christian Whittaker email me at cpwhittaker@hotmail.co.uk if you are unsure (or disagree) with this

take out a factor of 2 you get 2( 1 - 1 + 1 - 1 + 1 - .....) = 2Z in which case you are back where you started

So because 2Z is not infact Z, your theory is irrelevant