Let's "prove" that every complex number

*z* is real.

If *z*=0 it's obvious. For all other complex numbers *z*=r*e^(θi), where r is a real number, and i=√-1.

Now, *z*= r*e^(θi)= r*(e^(2πi))^(θ/2π). Now as we know that e^(2πi)=1 we can write *z* =r*(1)^(θ/2π) → *z*=r.

What's wrong with this?

exp^(ab)=(exp^a)^b does not apply to complex no.it is only for real numbers.if it apply to complex no then say b is complex

(exp^(a))^b=(cos a +sin a)^b=cos ab + sin ab

quantity ab is complex, so cosine or sine of it is meaningless

exp^(θ*i)=cos θ + i sin θ is not equal to

cos (θ*i) + sin (θ*i) as your reasoning apply