The first fact: A complex number whose modulo equals 1 lies on the unit circle, and finding the square root is equivalent to halving its angle on the unit circle. The angle of -1 is 180 degrees, the square root of -1 means that the angle of $i$ is 90 degrees, the angle of the square root of $i$ is 45 degrees, and the angle of the square root of $i$ is 22.5 degrees ... which is described as the following picture
The second fact
$(1+i/n)^n=e^i \ when \ n \to \infty $ which means a rotation of one radian,
To a summary, $i$ multiplying a real number means a $\pi/2$ rotation, $i$ at the power of $e$ means a 1 radian rotation.
I have a strange idea, could use -1 of the form in the fist fact to represent 1 radian rotation.
i.e. how to use $\sqrt{-1} \ \sqrt{\sqrt{-1}} \ \sqrt{\sqrt{\sqrt{-1}}} \ ...$ to represent an arbitray radian rotation by combination or some what you could think
A $\pi$-radian rotation from $1$ gives $-1$; that is, $$-1=e^{\pi i}$$ Taking $\pi$-th roots, $$(-1)^{\frac{1}{\pi}}=e^i$$
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