>> Okay, let's put these two things together: You cannot just put these two together without making your proof cyclic. AFAIK, the second one, e ^ x = lim ((1

>> Okay, let's put these two things together: You cannot just put these two together without making your proof cyclic. AFAIK, the second one, e ^ x = lim ((1 + x/N) ^ N), can be proven only for real x without using Euler's formula or almost proving it as an intermediate step. Thus substitution from #1 using complex numbers is not legal without proof. Here's an analogy to consider: 1. Boyle's law: P1.V1 = P2.V2 2. Charles law: V1/T1 = V2/T2 Multiply LHS and RHS of the above two. 3. P1.(V1^2)/T1 = P2.(V2^2)/T2. Yet, the gas equation is ... 4. P1.V1/T1 = P2.V2/T2 ... and not #3. While there is nothing wrong with the multiplication used for computing #3, this process adds the assumptions used for #1 and #2. Here's a more precise version then: 1'. Boyle's law: P1.V1 = P2.V2 when T1 = T2 2'. Charles law: V1/T1 = V2/T2 when P1 = P2 Multiply LHS and RHS of the above two. 3'. P1.(V1^2)/T1 = P2.(V2^2)/T2 when T1=T2 and P1=P2. Derivation for #4 avoids those two assumptions. In your proof of Euler's identity, the assumptions for your #1 and #2 cannot be considered to be compatible without proof.