Question

Can someone explain how come \(\Large 1^i = 1\)?

And how to calculate \(\Large 3^i\) ?

Answer 1

\[1=e^{i \cdot 0}\] so \[1^{i}=(e^{i\cdot 0})^{i}=e^{i^{2}\cdot 0}=e^{0}=1\]

Answer 2

I see. What about \(\Large3^i\)?

Answer 3

\[3=e^{\ln 3 +i\cdot 0}\] so \[3^{i}=(e^{\ln 3 + i\cdot 0})^{i}=e^{i\ln 3}\] and therefore this is a complex number with magnitude 1 and argument \[\ln 3\]. If you wish to represent this as \[a + ib\] you can do so with the standard equations, \[a=r\cos\theta, b=r\sin\theta\]