> ... to get 0.5 for the sum of
> 1-1+1-1+.... requires real numbers.
Actually it doesn't, it only requires the rationals.
But it's clear now that you're not really talking about maths at all, so the comment about existing, established theory about modular arithmetic doesn't really help. You seem to be doing something, well, different.
And regardless, in the long-established theory of modular arithmetic, 1.2 is equal to -0.8 mod 2, regardless of you claiming that it's nonsense.
So at this point I have no idea what you're talking about.
meh, I tend to skip thinking about the rational numbers when I move onto this hardcore theory stuff :p
There are two ways of constructing a number line from -inf to +inf
The first, is that "nothing exists" to the left of -inf or to the right of +inf
The other, more useful, is that -inf=+inf+1 and +inf=-inf-1, (or -inf=+inf, never remember which is the more useful) and they form a loop.
Such as that constructed by a signed integer.
e.g.
e.g. with an 8 bit number
127+1 = -127
This has nothing to do with modulo afaik. (but all the basic construct stuff is related) and is more to do with every dimension being curved in another (meaning they always form such loops)
The point of the construct is it applies to a circle of arbitrary/unknown length 14,12.2, 500,000 million billion point 6 light years.
and the average of the sum of 1+2+3+4+... will be/tend to -1/12
and its easy to test, just use a signed x bit number.