A complex line is the image of a linear function, L : $\mathbb{C}$ $\rightarrow$ $\mathbb{C}^{n}$.
In real sense, a real-line is an equation given by $Ax+By+C$ = $0$. Why does complex line not an equation but a function? I am having a hard time for understanding this.
Thanks!
In mathematics there is often more than one way to represent the same data. For example the line $y=mx+b$ is the same set of points as the image of $L(t) = (t, mt+b)$. However an equation $Ax+By+c=0$ can only represent a line in $\mathbb{C}^2$ or $\mathbb{R}^2$. The higher dimensional analog of this equation would represent a plane (with 3 variables) or a hyper plane (with more than 3 variables).