How can you make rational of i power i?

Answer 1

It is NOT rational, but it IS real.

Start with Euler's formula: e^ix = cos(x) + i*sin(x) for all x.

When x = pi/2,

e^(i*pi/2) = cos(pi/2) + i*sin(pi/2) = 0 + i*1 = i

or i = e^(i*pi/2)

Raising both sides to the power i gives

i^i = e^[i*(i*pi/2)] = e^[i*i*pi/2]

and since i*i = -1,

i^i = e^(-pi/2) = 0.20788, approx.