How to rotate a line in the complex plane?

Question

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.

Answer 1

The line $\arg(z)=0$ correspond to $x$ real axis therefore by $z=t\ge 0, t\in \mathbb R$

$$t\mathrm{e}^{\frac{\pi}{4}\mathrm{i}}=t\left(\frac{\sqrt 2}2+i\frac{\sqrt 2}2\right)$$

which represents indeed the bisector ray of the first quadrant .

Answer 2

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 $\;\frac{1}{\sqrt{2}}(1 + i).$