How to rotate a line in the complex plane? How do I rotate the the line $arg(z) = 0$ by $\frac{\pi}{4}$ radians counter-clockwise about the origin in the complex plane. 
The general transformation is $z\mathrm{e}^{\frac{\pi}{4}\mathrm{i}}$ however how do I algebraically find the image of the rotation of the line?
Thanks.

If I understand both the OP's desire and gimusi's response correctly, then an alternative approach is to algebraically prove that for complex z and w, arg(zw) = arg(z) + arg(w).
Thus, when |w| = 1, rotating z by arg(w) is equivalent to multiplying z by w.  Therefore, rotating z by $\;\pi/4\;$ is equivalent to multiplying z by $\;\frac{1}{\sqrt{2}}(1 + i).$
