I am having trouble understanding a passage in Stein and Shakarchi's complex analysis textbook. Here is the passage for reference. (It earlier wrote $z = re^{i \theta}$ and then wrote Euler's identity.)
Since $|e^{i \theta}| = 1$ we observe that $r = |z|$, and $\theta$ is simply the angle (with positive counterclockwise orientation) between the positive real axis and the half-line starting at the origin and passing through $z$.
I think I have a very minimal understanding of what an "orientation" is. I have seen this defined as the order of basis vectors. In $\mathbb{R}^3$, this is the order of the three standard basis vectors, I believe, and it tells me what the "positive" direction for $x$, $y$, and $z$ are. I don't think I know what this means with respect to an angle, even though I'm aware that $\theta$ is measured counterclockwise and, as I move counterclockwise around the complex plane, $\theta$ increases from $0$ to $2\pi$.
So my problem is that my understanding of orientation is extremely superficial at best. Can someone help to grasp this passage?