(or a toddler) Why does “i^i” result in a real number “e^-0.5pi”? How did they figure those out in the first place?

Question

(or a toddler) Why does “i^i” result in a real number “e^-0.5pi”? How did they figure those out in the first place?

In: 9

Answer 1

It comes from

i = e^(i*pi / 2)

If you raise both sides to the power of i, you get

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

Since i*i is -1, that turns into

i^i = e^(-pi/2)

Proving the original formula is a little more complicated. It comes from Euler’s identity which is

e^ix = cos(x) + i*sin(x)

If you plug pi/2 in for x, cos(x) becomes 0 and sin(x) becomes 1, so it changes to

e^(i*pi/2) = i

As for why Euler’s identity works, I’m not sure I can explain that.