His method is a bit more complex, but you eventually need to describe how different frequencies change the characteristics of a system. Initially, it's nice to see impedance in terms of a + bi, but that's not easy to manipulate when you eventually want to know a circuit's behavior at various frequencies.

This is because (a + bi) does not inherently describe the sinusoidal behavior of AC. By changing things into the frequency domain (where writing a phasor implies a driving rotational force at some variable frequency), we can now see how the impedance of an entire system varies as frequency changes.

http://betterexplained.com/articles/intuitive-arithmetic-wit...

http://en.wikipedia.org/wiki/Phasor

It's really just a math abstraction. Maybe this analogy is a stretch, but you interface with stacks using push and pop, and there's an inherent size to the stack that you keep in the back of your mind. You don't care about the individual memory addresses and their values, you just want to manipulate the stack such that it doesn't overflow/underflow. Just like AC - you don't care where in the complex plane an impedance is at a given frequency (which would be in terms of a+bi), you'd rather just see how its behavior changes as frequencies vary.

And why do you care about impedances for a whole range of frequencies? One example: http://www.ecoustics.com/electronics/products/articles/13106...