i to the i power = .207…, but why?

If you take i to the i power on your calculator you will discover that the “imaginary unit” taken to the power of the “imaginary unit”, equals a very real number. Students are always surprised by that, and in fact unbelieving. I always demonstrate this phenomenon on the calculator, the TI-­‐84 for instance, when the imaginary unit is first used in Algebra, and promise it will be proven to them in Precalculus.
Here is the proof. From Euler’s Identity we know, e^(iθ) =cosθ+i(sinθ) Since this is true for all values of θ, it is true for       θ=π /2.
Substitute π /2 for θ. e^(i(π/2)) =cos(π/2)+i(sin(π/2)) e^(i(π/2)) =0+i=i
Taking the natural log, ln( e^(i(π /2) ) = ln(i) iπ / 2 = ln(i)
Multiplying both sides by i, −π /2=i(ln(i)) −π /2=ln(i^(i))
Take the antilog, e^(−π/2) =i^(i)
Q.E.D. e^(−π/2) =.207…

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