The usual complex numbers and their connections with the circular functions cos, sin, tan etc have relativistic analogs, which are crucial in understanding both the corresponding hyperbolic functions cosh, sinh, tanh etc as well as the log and exp functions. The former are associated to the red complex numbers, and the latter to the green complex numbers. Surprisingly these three different versions of complex numbers are naturally connected. In the green geometry, the unit circle becomes the standard hyperbola xy=1, which we argue is the most important curve in mathematics. A parametrization of this is exactly parallel to the usual parametrization of the usual unit circle. But now rotations are replaced by green multiplications by green unit vectors!
This story is a crucial backdrop for a correct understanding of "log" and "exp", as signed areas for the standard hyperbola become values of log. In particular we will see how to adapt the computation for Leibniz's formula for "pi/4" to get Mercator's formula for "log 2".
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