What's the simplest way to prove Euler's formula? Complex number retard here

you can get to the result from the third line. the rest is a bit of unnecessary wankery

For a more intuitive approach, observe that the derivative of y=e^(i*theta) is iy, which is just y rotated 90 degrees, and that the initial condition e^(i*0) is 1, so we can conclude that the function draws out a circle (because the tangent is perpendicular to the radius) with radius 1.
Then we can conclude that since |y| is always equal to 1, |dy/dx| is always equal to 1, since it's just y rotated.
Thus, as theta increases linearly, y draws out a circular arc with a constant rate of 1, so the distance y travels around this arc is equal to theta. Setting theta to pi, we see y travels a distance of pi around the unit circle in the complex plane, thus placing it at -1.

what's * ?

>Calling other proofs lazy when you abuse notation with "dy/dx = 1 implies dy = dx"
You can't make this shit up

but dy=(dy/dx)dx so if dy/dx=1 then it does imply that dy=dx

No, that's laziness and abuse of notation

You might be able to create a less nice geometric proof if you want to aviod using tayler-series. Not sure how basic you want this proof to be or what assumtions you would start with, but it should be easy enough to show that e^{i\pi} * complex conjugate (e^{-i\pi}) is a norm anb thus that |e^{i\pi}|=1 (so that e^{i\pi} lies on the unit circle around 0 in the complex plane). From there on ist just some geometry to define the real and the complex part of the triangle (with the line from 0 to e^{i\pi} as hypotenuse) as Sin and Cos or maybe just check that the real and complex part are in fact cos and sin...
Its kinda late right now so I cut it short: construc a geometric proof that e^{i\phi} = Cos(\phi) + i Sin(\phi) and then use that (Cos(\phi)=-1)
But maybe this doenst work this simple and I wasted the last 3 minutes... in any case: good night guys.

geometrically in the complex plane

>take out graphing calculator
>enter e^(I*π)+1
>result is 0

Why do mathfags over think everything? Serious question. Were you hoping for some kind of proof? Give me a real world scenario outside of a school assignment where writing a formal proof for that equation is of any use.

>abuse of notation
Well meme'd my friend