Complex form of line

Question

I'm not really sure how to form a line in the complex numbers.

I know that in these questions, we use $z$ and $\overline z$, but how do we transform that from $x$ and $y$???

For example, say I want to graph the line $y=2x+1$.

Answer 1

Well, $x=\dfrac{z+\overline z}2$ and $y=\dfrac{y-\overline y}{2i}$. Thus, for two $a,b\in\Bbb R$, the line $$ax+by+c=0$$ becomes $$(ai+b)z+(ai-b)\overline z+2ic=0$$

Or, equivalently $$\omega z-\overline{\omega z}+2ic=0 $$ for some $c\in\Bbb R$ and $\omega\in\Bbb C\setminus\{0\}$. More concisely: $\Im(\omega z)=c$

Notice that $a=\Im \omega$ and $b=\Re\omega$.

Answer 2

Notice that $z=x+yi$ and $\overline{z}=x-yi$.

We know that $z+\overline z = 2x$, and $z - \overline{z} = 2yi$.

Therefore, we have that $\displaystyle x = \frac{1}{2}(z+\overline z)$, and $\displaystyle y=\frac{1}{2i}(z-\overline z)$.

Rewriting the line $y=2x+1$ as $y-2x=1$ gives $\displaystyle \frac{1}{2i}(z-\overline z)-(z+\overline z)=1 \implies \boxed{(z-\overline z)-2i(z+\overline z)=2i}$