complex numbers

[ibjazzo]

hey there!

I'm doing my exploration on imaginary exponents and Euler's formula etc. One thing that I'm investigating is that what happens when you change the base number which is raised to the imaginary exponent. for positive numbers it is pretty straight forward as you can express them in terms of e. But when considering negative base numbers it gets a lot trickier. Basically the problem understanding (-1)^i , as you can then multiply it by any positive number raised to imaginary exponent. I still haven't figured out how it work. And the most curios thing about is that it gives a real number! (app. 0,043214). I know getting that from a calculator doesn't prove anything but I got that from various calculators (including google!) so I think there must some way to calculate it

[Vioh]

hey there!

I'm doing my exploration on imaginary exponents and Euler's formula etc. One thing that I'm investigating is that what happens when you change the base number which is raised to the imaginary exponent. for positive numbers it is pretty straight forward as you can express them in terms of e. But when considering negative base numbers it gets a lot trickier. Basically the problem understanding (-1)^i , as you can then multiply it by any positive number raised to imaginary exponent. I still haven't figured out how it work. And the most curios thing about is that it gives a real number! (app. 0,043214). I know getting that from a calculator doesn't prove anything but I got that from various calculators (including google!) so I think there must some way to calculate it

 

Well substitute -1 = cis(Ï€) = e^(i Ï€) into the equation, we get e^(i^2 * Ï€) = e^(-Ï€) = 1/e^Ï€ = 0.043214.

 

That's basically the derivation. Good luck with your IA