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.

i_rotationThis 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.


About Suresh Emre

I have worked as a physicist at the Fermi National Accelerator Laboratory and the Superconducting Super Collider Laboratory. I am a volunteer for the Renaissance Universal movement. My main goal is to inspire the reader to engage in Self-discovery and expansion of consciousness.
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