Question

What is wrong with the following line of thinking?

Answer 1

\[e^{2\pi i}=1 \text{ } \checkmark\]\[\left(e^{2\pi}\right)^i=1\]\[e^{2\pi}=\sqrt[i]{1}\]\[e^{2\pi}\ne 1\]

Answer 2

it doesn't seem wrong!!!!!!

Answer 3

\[e^{2\pi}\] is most certainly not a true statement while \[e^{2\pi i}=1\]is a true statement. D:

Answer 4

Sorry, for the first equation, I meant \[e^{2\pi}=1\]...that is the false result.

Answer 5

ya because i is a complex number i=-1

Answer 6

I know, and I think you mean\[i^2=-1\]But by Euler's formula \[e^{ix}=\cos x + i \sin x\]Which derives the first formula. Then I rearranged things by simple means and out popped a false statement.

Answer 7

i dont think it makes sense to say the i'th root of 1?

Answer 8

I would agree with adbermie, I've never seen the i'th root of something... that's not to say it doesn't seem to make sense...

Answer 9

To say the \(i\)th root would mean to raise to the power \(\frac{1}{i}\).

Answer 10

is the ith root of 1 really 535.49...?