Straight Line Equation in Complex Plane

Question

I'm confused about the straight line equation in complex plane: how does $0 = Re((m+i)z + b)$ come from $y = mx + b$? I mean when I see $y = mx + b$, I can draw a graph in my mind, but when I see $0 = Re((m+i)z + b)$, there is nothing on my mind.

How can I connect the two equatinos?

Does anyone could help me, thanks!

Answer 1

Let $z=x+iy$. Then, $(m+i)(x+iy)+b=mx+ix+imy-y+b$. Thus, looking at the real part, we get $mx-y+b=0$, also known as $y=mx+b$.

Answer 2

$mx-y+b=0\rightarrow Re(mz)+Re(iz)+Re(b)=0\rightarrow Re[(m+i)z+b]=0$

Answer 3

When you multiply complex numbers you add the angles and multiply the lengths.

Think of $m+i$ as a vector that forms and angle of $\cot^{-1} m$ with the x axis.

The function of the line intersects the x axis at and angle of $\tan^{-1} m$.

Adding these two angles together equals $\pi/2$

When we multiply by $(m+i)$ it rotates the solutions set to a vertical line that intersects the $x-$axis at $-b.$ Adding $b,$ and it intersects at 0.