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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.
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∙ 9y ago
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