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.


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    $\begingroup$ Who says a real line is an equation? A line is a point set in any case, real or complex $\endgroup$ – MPW Feb 6 '18 at 2:20
  • $\begingroup$ oh i see.. thanks.. somehow that comment did it for me $\endgroup$ – HumbleStudent Feb 6 '18 at 2:22

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).


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