/sci/ - Science & Math

Prepare for the proof of all proofs .

Given: [math]{e^{i\theta }} = \cos \theta + i\sin \theta [/math]

Through trivial calculation we can see:

[math]{e^{i\pi /2}} = \cos \left( {\pi /2} \right) + i\sin \left( {\pi /2} \right) = 0 + i\left( 1 \right) = i[/math]

and
[math]{e^{2\pi i}} = \cos \left( {2\pi } \right) + i\sin \left( {2\pi } \right) = 1[/math]


However using the basic properties of exponents we can clearly see:

[math]{e^{i\theta }} = {e^{2\pi i\theta /2\pi }} = {\left( {{e^{2\pi i}}} \right)^{\theta /2\pi }} = {1^{^{\theta /2\pi }}} = 1[/math]


Therefore:

[math]1 = i[/math]
QED

There is only one most beautiful formula...