A Geometric Interpretation of ii
Algebraically and numerically:
but geometrically the two expressions can be interpreted as two points in two slightly different sets of coordinates lying in two different locations.
In the graph below, the natural exponential function is in violet and we introduce and define the imaginary exponential function which is in red.
Just as with x taking on different values and forming the natural exponential, z taking on different values forms the imaginary exponential.
The two graphs are versions of ‘4Dii Functions’ wherein each of four variables, a complex number input and a complex number output, has a precise geometric interpretation.
For the natural exponential/log function:
and the rotation may be seen here.
The coordinates are:
and the yellow dot is the point:
For the imaginary exponential/imaginary log function:
The imaginary log function logi is derived by change of base.
So, the coordinates are:
and the black dot is the point:
For the natural exponential function the x-axis is real and the rotation is imaginary. For the imaginary exponential function the x-axis is imaginary and the rotation is real.
These two sets of coordinates are different forms of ‘4Dii Coordinates’. See eBook section 11.1 “Functions in 4Dii” for discussion.
This post illustrates the topic ‘A Geometric Interpretation of i^i’ on the MathIsFun – This is Cool forum.