While analysing AC circuits, we write voltage, current etc all with complex numbers namely "phasors". While studying the same, I wondered if Kirchhoff's laws held good with current and voltage in their phasor form. And the internet said they did! They argued somewhat as follows:

$I=Re[\vec{I}]=Re[I\_{max}e^{j(\omega t+\phi)}]$

Now, $\Sigma I=0$
[By normal Kirchhoff's law]

Or, $\Sigma Re[I\_{max}e^{j(\omega t+\phi)}]=0$

Or, $Re[\Sigma I\_{max}e^{j\omega t} e^{j\phi}]=0$

Now, $e^{j\omega t}\neq 0$

Therefore, $\Sigma I\_{max}e^{j\phi}=0$

i.e. $\Sigma\vec{I}=0$

Here $I$ stands for scalar current and $\vec{I}$ for phasor current. Similar argument went on for the voltage law.

**I didn't get what they did in the sixth step. The fact that real part of a complex number is zero doesn't always imply that the number itself is zero.** Can anyone please explain (if this is correct at all!)? And I would be glad if anyone kindly provides any argument, appropriate and more lucid, for the same. Thanks.

P.S. [Here](https://books.google.co.in/books?id=9j1_0tqrKNQC&pg=PA485&lpg=PA485&dq=kirchhoff%20current%20law%20in%20current%20and%20voltage%20phasor&source=bl&ots=83XySJXc3O&sig=CP7jhkMwpyUxRqwFM8a5i6tIt3g&hl=en&sa=X&ved=0ahUKEwit0JX1pZzMAhUFj44KHZXrCmsQ6AEIJDAI#v=onepage&q=kirchhoff%20current%20law%20in%20current%20and%20voltage%20phasor&f=false) is the link to what I found on the internet.