Complex Numbers Question

I'm afraid I don't completely understand your question for two reasons.

When I rearrange z+2+i=-pi/6 when z=x+iy I get
y=-1+y(x+2+pi/6) not the relation expressed.

In addition I saw the expressed relationship as the line y=Ax+B with A=-1/sqrt(3) and B=-[(1+sqrt(3))/sqrt(3)]

Back to complex and the cartesian coordinates.

With z=x+iy look at the composite number z as having a real and an imaginary part, with the real number as the x (horizontal) axis and the imaginary part as the y (vertical) axis. z then is a line that goes through the origon (x intercept is 0).

If you look at the magnitude of z (|z|) the z=x+iy can be viewed as a vector of the magnitude of z on a unit circle on the x y plane where

z=|z|(cos(phi)+isin(phi)) where cos(phi)=x/|z| and sin(phi)=y/|z| and cos(phi)+isin(phi)=|1| is a unit circle.

Mathworld gives a much better explanation of complex numbers than can be done in the limited html of this forum. I would recommend looking at this page.

Rap