In complex numbers, does the loci of arg(z) extend in both directions?

Question

For the arg(z-i) = pi/4 on an angard diagram does the line extend before i (into 2nd quad) as well as after?

If so where would you shade if they asked to shade the area where arg(z-1)= pi/4?

Answer 1

Let see what we have:

The argument of a complex number z, denoted as arg⁡(z), is the angle that the line connecting the origin to

the point z makes with the positive real axis.

Locus of arg⁡(z−i)=π/4:

For arg⁡(z−i)=π/4, we are considering the set of points z=x+iy such that the argument of the complex number z−i is π/4

1. Geometrical Interpretation: The equation arg⁡(z−i)=π/4 represents a line in the complex plane passing through the point i (which is (0,1) on the Argand diagram). This line makes an angle of π/4 with the positive real axis.
2. Extension of the Line: This line extends infinitely in both directions: it goes into the 2nd quadrant and the 4th quadrant. Specifically:
3. In the 4th quadrant, the line extends below the point i, moving toward the positive real axis.
4. In the 2nd quadrant, the line extends above the point i, moving toward the negative imaginary axis.

Region for arg⁡(z−1)=π/4:

Now, consider the argument arg⁡(z−1)=π/4. Here, z−1 shifts the point of reference from 0 to 1 on the real axis.

1. Geometrical Interpretation: The equation arg⁡(z−1)=π/4 represents a ray starting from the point 1+0i (which is (1,0) on the Argand diagram) and making an angle of π/4 with the positive real axis.
2. Region: To find the region where arg⁡(z−1)<π/4​, you would shade the area below this ray. This is because any point below the ray would have an argument less than π/4 relative to the point 1+0i.

In summary:

1. The line representing arg⁡(z−i)=π/4 does extend both before and after the point i, going into both the 2nd and 4th quadrants.
2. For arg⁡(z−1)=π/4, the ray starts from the point 1 on the real axis, and the region below this ray (in the 1st quadrant) is shaded.

I hope this helps somehow.