On the other hand, we do divide complex numbers by each other

That's true. The impedance of a circuit element is the phasor voltage divided the phasor current

$$Z = \frac{\vec V}{\vec I} $$

But note that the impedance is *not* a phasor - it does not represent the amplitude and phase of a sinusoid like the voltage and current phasors.

Similarly, the complex power is the product of the (rms) phasor voltage and (rms) conjugate phasor current

$$S = \tilde V \cdot \tilde I^*$$

And again, the complex power is not a phasor, it is just a complex number.

That fact is that products and ratios of phasors are not phasors. Thus, we can't apply phasor analysis to non-linear circuits.

For example, let a circuit element voltage be proportional to the current squared:

$$v = ki^2 $$

If the current is a sinusoid of frequency \$\omega\$, the voltage is a constant plus a sinusoid of frequency \$2\omega\$.

$$v = k(I\cos\omega t)^2 = \frac{kI^2}{2}(1 + \cos2\omega t)$$

But, for phasor analysis, *we depend on the fact that all the circuit voltages and currents are of the same form, i.e., are sinusoids of the same frequency, only differing in amplitude and phase*.

Moreover, as pointed earlier, the square of a phasor is not a phasor thus we cannot square the current phasor and hope to get a voltage phasor.

Go to the "end game" when all the caps can be assumed to have charged up - what will be the current through a capacitor when the voltage is not changing across it? This you should know for the formula I = C dV/dt.

Once you have concluded correctly you should be able to see what the voltages are across all the resistors. It's not a big leap of faith (given the "special" value of the resistor ratios) to make definite argument that this state of affairs also happens right at the moment when the switch closes.

Here's another tip - think wheatstone bridge.

## Best Answer

If you consider Real Power and Imaginary Power we are talking about resistive power and reactive power with energy stored in inductors, and capacitors. The vector sum of both is called "apparent power"

Even in mechanical systems there are complex reciprocal devices with stored energy in flywheels or springs. Inductors and Capacitors are similar in that they can store energy , in math called imaginary value.

But when an inductor opens current and arcs, it turns in to real energy similar to shorting out a capacitor into some resistance. Although this is a crude example like putting a crowbar brake across a flywheel.