## Geometric interpretation of sqrt(-1)

I wrote a post titled Euler rule is almost mystical where I mentioned the Cotes-Euler identity

$\displaystyle e^{i \phi } = cos(\phi) + i sin(\phi)$

Euler rule

$\displaystyle e^{i \pi } + 1 = 0$

is a special case of the Cotes-Euler identity.

$\displaystyle e^{i \phi }$ represents the rotations of the complex unit vector.

This provides a natural interpretation for $\displaystyle i=\sqrt{-1}$. Multiplication by $i$ can be interpreted as anticlockwise rotation by 90 degress ($\displaystyle \frac{\pi}{2}$ radians).

If we plug-in $\displaystyle \phi=\frac{\pi}{2}$ into the Cotes-Euler identity we get $\displaystyle e^{i \pi /2 } = i$ which demonstrates the consistency of the $\displaystyle e^{i \phi }$ representation as the rotations of the complex unit vector.