What would happen if we created this:

x {0} y = x + y

x {0.5} y = x @ y

x {1} y = x * y

x {2} y = x ^ y

And then {0.25} will be the same arithmetic-geometric algorithm of {0} and {0.5}; {0.75} will be the arith-geo-algo of {1} and {0.5}, so on and so forth.

We could then solve for x {1.5} n, n E N, since:

x {1.5} 2 = x {0.5} x

Perhaps Taylor series will be derivable giving us complex arguments.

It'd also be very interesting to see what happens with logs, i.e:

log(x {1.5} 2) = ? since normal operators undergo a transformation I wonder if something happens for these.

x {0} y = x + y

x {0.5} y = x @ y

x {1} y = x * y

x {2} y = x ^ y

And then {0.25} will be the same arithmetic-geometric algorithm of {0} and {0.5}; {0.75} will be the arith-geo-algo of {1} and {0.5}, so on and so forth.

We could then solve for x {1.5} n, n E N, since:

x {1.5} 2 = x {0.5} x

Perhaps Taylor series will be derivable giving us complex arguments.

It'd also be very interesting to see what happens with logs, i.e:

log(x {1.5} 2) = ? since normal operators undergo a transformation I wonder if something happens for these.