how do i raise to the i power? For example, 4 to the power of i. (SQRT -1)
+122 Answer,then I'll explain.
4^i =cos(ln4)+isin(ln4).
Why? Recall that I can write any number I please,let's say X as always,as
X=e^lnX because e^x and lnX are inverses. So I will write 4^i as e^ln(4^i).
Using laws of logs,ln(4^i)=i(ln4).Now I will use Euler's formula e^ix =cosx +isinx.
We now have 4^i= e^ln(4^i) = e^i(ln4)= cos (ln4)=isin(ln4). This is a deep result which should show you that the set of complex numbers ( the all-encompassing number set) is fundamentally sinusoidal. If you have understood this,now look at the Riemann Zeta function and you will be able to appreciate the beauty of it.Me,I'm off for a beer......
