Pi is irrational

After writing my MathML demo, I wanted to make a quick note of something that I found kinda interesting. First, it uses Euler's Identity (e +1 = 0), written in terms of pi, which results in a fraction of two number which simply don't exist. Then, if you assume that pi can be expressed as a fraction if integers (a/b), it leads to what seems to be a contradiction.

Pi has already been proven to be irrational in many ways, but this one might be the simplest, if it qualifies as a proof at all. The conclusion seems impossible to me, but might not actually be a contradiction.

e i π + 1 = 0 e i π = 1 ln ( e i π ) = ln ( 1 ) i π = ln ( 1 ) π = ln ( 1 ) i π = ln ( 1 ) 1 a , b π = a b ln ( 1 ) 1 = a b a × 1 = b × ln ( 1 ) e^{i %pi } +1 = 0 newline e^{i %pi } = -1 newline ln( e^{i %pi}) = ln( -1 ) newline i %pi = ln( -1 ) newline %pi = {ln(-1)} over {i} newline %pi = { ln(-1) } over { sqrt{-1} } newline newline a in setZ , b in setZ newline %pi = { a } over { b } newline { ln(-1) } over { sqrt{-1} } = a over b newline ∴ a times sqrt{-1} = b times ln( -1 ) newline