# Complex number loci and min/max argument

The Question:

I have some gaps in this chapter, and I would like some clarifications. What does arg(z) represent and what does $${\displaystyle \arg \left( z+2-2\, \sqrt{3}i \right) }$$ represent? (the angle from which point to which point)

Also how can I find the minimum and maximum angle?

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Are you familiar with the representation of complex numbers as points in the plane? How about polar coordinates? – J. M. Apr 28 '12 at 17:21
Yes I am familiar with representation of complex numbers as points in the plane but not in polar coordinates – εν οίδα ότι ουδέν οίδα Apr 28 '12 at 17:23

Hint: Rewrite as $$\left|z-(-2+2\sqrt{3}i)\right|=2.$$ This "says" that the distance from $z$ to $-2+2\sqrt{3}i$ is $2$. So the locus is the circle with a certain centre, a certain radius.

Draw that circle (crucial). You are interested in lines from the origin to points on your circle. The maximum, minimum angles are at points of tangency. One of them will be obvious from the picture. You can work out the other using once familiar geometry/trigonometry. Note that it is $\text{arg}(z)$ that you are being asked about.

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Ok, I managed to draw the circle and I understood that the arg(z) is at the tangent to every point on the line, but is it the angle that the tangent makes with the center or origin or something else? – εν οίδα ότι ουδέν οίδα Apr 28 '12 at 17:45
The term arg is not completely defined, although one often uses the value between $0$ and $2\pi$. You can think of it as the angle throgh which the positive $x$-axis (the positive real axis) has to be rotated (counterclockwise) in order to pass through $z$. – André Nicolas Apr 28 '12 at 17:49
I understood thanks – εν οίδα ότι ουδέν οίδα Apr 28 '12 at 18:00
One more thing arg(z) starts from the origin, so arg(z-1-2i) lets say starts from (1,2) right? – εν οίδα ότι ουδέν οίδα Apr 28 '12 at 18:02
@panayiotis: Good. In complex variables, there is a great deal of interplay between the algebra and the geometry. In principle one could work out the answers without a picture, but that would miss the main point. And about the one more thing, if you think of $z$ as a vector, it would be the angle the vector joining $(1,2)$ to $z$ makes with the positive $x$-axis. – André Nicolas Apr 28 '12 at 18:03