Proof of Euler's Identity | Complex Numbers

Given any introduction to complex numbers, one sooner or later is exposed to Euler's formula (or Euler's identity), which expresses an exponential of an imaginary number in terms of the sum of two trigonometric functions. Many proofs are either technical or unenlightening and in most cases both. In this video, we give an elementary proof of this result, using nothing more than the product rule from calculus.

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Hi, my name is Kyle and I'm currently doing my doctoral mathematics degree in complex differential geometry under the supervision of Professor Gang Tian and Professor Ben Andrews.

[KyleBroder]

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[Daeqius_Money]

By far the best proof I've found, awesome

[elliotcarmo]

Can you get intuition for what Euler's identity says from this proof?

[gavincourts]

What does it mean to raise a number to the power of a complex number?