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zerology
The mathematics of zero may seem to be much ado about nothing, but it is related to the larger field of surreal number theory. It concerns those numbers in the gap between 1/00 and -1/00, where e = G(2, e, w) = {G(3, 1, {0, 1, 2, 3, ...|}), G(3, 2, {0, 1, 2, 3, ...|}), G(3, 3, {0, 1, 2, 3, ...|}}, ...}, w = {0, 1, 2, 3, ...|} and 00 = G(2, 1/e, w) which might be called "subreal" numbers, by analogy with "surreal numbers".
More common numbers can be formed from zero:
1 = G(3, 2, 0) = G(2, 0, 0)
2 = G(0, G(3, 2, 0), G(3, 2, 0))
3 = G(0, G(3, 2, 0), G(0, (G(3, 2, 0), G(3, 2, 0)))
4 = G(2, 2, G(0, G(3, 2, 0), G(3, 2, 0)))
The entity John Horton Conway refered to in On Numbers and Games as "star", * = {0|0} = {{|}|{|}}, can according to his terminology also be called , ± 0. What he named "up", {0|*} = 0|0||0 = {{|}|{{|}|{|}}} = {0|± 0} = +0, and "down", {*|0} = 0||0|0 = {{{|}|{|}}|{|}} = -0. From these comes:
(+0)(+0)(+0)
(+0)(+0)
(+0)(± 0)
(+0)(-0)
+0 +0 +0
+0 +0 ± 0
+0 +0 -0
+0 +0
+0 ± 0
+0 -0
-0 -0
-0 ± 0
-0 -0 +0
-0 -0 ± 0
-0 -0 -0
(-0)(± 0)
(-0)(-0)
(-0)(-0)(-0)
and all their additions, multiplications, combinations and extrapolations.
1/h= G(2, -1, h)
1/z = G(2, -1, z)
1/e = G(2, -1, e)
1/w = G(2, -1, w)
1/00 = G(2, -1, 00)
-1/00 = G(2, -1, -00)
-1/w = G(2, -1, -w)
-1/e = G(2, -1, -e)
-1/z = G(2, -1, -z)
-1/h= G(2, -1, -h)