Solving Complex Number Questions: Arg(z) & Arg(w) and More

[vorcil]

quick note, i am not allowed to use a calculator when doing these questions!

1: if Z = -i and W = 3+3i
find arg(z) and arg(w)

2: 2Z + z^{-}(that's the conjugate symbol) = a+2i
then z =

3: |z-i| <=1, what would it look like?, describe it's position from the point 0,0

my attempts

1:
z=well i'd use tan^-1 (-1/0) but that's un defined so I'm not quite sure and I am not allowed to use a calculator

w=tan^-1(3/3) = 45 degrees,(tan inverse of 1 is 45 degrees?

2: no idea hwo to solve this

3: well because it's -i, i know the centre of the circle is bellow y=0, not sure for the rest

 

[rock.freak667]

1:
z=well i'd use tan^-1 (-1/0) but that's un defined so I'm not quite sure and I am not allowed to use a calculator[/tex]

w=tan^-1(3/3) = 45 degrees,(tan inverse of 1 is 45 degrees?

Draw z=-i on an argand diagram and the answer should be clear. Your answers should be in radians so convert 45 degrees to radians.

2: no idea hwo to solve this

Put Z=x+iy, then find the conjugate of this. Sub into the equation, equate the real and imaginary terms

3: well because it's -i, i know the centre of the circle is bellow y=0, not sure for the rest

Yes it is a circle. If you aren't sure how to figure it out from the complex form put z=x+iy and form the Cartesian equation

|X+iY|=\sqrt{X^2+Y^2}

 

[HallsofIvy]

|a- b| represents the distance from a to b. |z- i| is the distance from the point z to i. |z-i|= 1 represents all points whose distance from i is 1 and so is a circle with center i (NOT -i) and radius 1. Finally |z-i|< 1 is the disk of points inside that circle.