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Math Help - Sketching region in the complex plane -- is this correct?

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    Sketching region in the complex plane -- is this correct?

    "In the complex plane, sketch the region:
    A = { z ∈C | |z - 1| < 3 and Re z ≥ 0 }"

    From what I have attempted to work out, I'm stuck between two answers: http://i56.tinypic.com/zodlko.jpg
    (Basically, the difference is in terms of the centre of the circle.)
    However, I'm still unsure as to whether that is correct or not. I figured the circle, with radius 3 for |z-1| < 3 -- was that correct and which centre would be right? I know that since Re z >= 0, then the region needs to be within the first and fourth quadrants (right?) but is there any way that I should show this or is shading the defined region the right thing to do? Plus, are my graph and axes drawn and labelled correctly? I'm unsure as to whether labelling "Im" for the y-axis was the right thing to do and now, I'm doubting myself with that...
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  2. #2
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    Quote Originally Posted by cottontails View Post
    "In the complex plane, sketch the region:
    A = { z ∈C | |z - 1| < 3 and Re z ≥ 0 }"

    From what I have attempted to work out, I'm stuck between two answers: http://i56.tinypic.com/zodlko.jpg
    (Basically, the difference is in terms of the centre of the circle.)
    However, I'm still unsure as to whether that is correct or not. I figured the circle, with radius 3 for |z-1| < 3 -- was that correct and which centre would be right? I know that since Re z >= 0, then the region needs to be within the first and fourth quadrants (right?) but is there any way that I should show this or is shading the defined region the right thing to do? Plus, are my graph and axes drawn and labelled correctly? I'm unsure as to whether labelling "Im" for the y-axis was the right thing to do and now, I'm doubting myself with that...
    The second circle you had is correct. The first circle you had corresponds to |z - i| = 3. (But since you show no working the source of your confusion and why you think the first circle is possibly correct will remain a mystery).
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    Is the shaded region correct as well? My friend just told me that I've shaded Im z ≥ 0 instead of Re z ≥ 0 and that I have to shade in the region to the right of the y-axis.
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    Quote Originally Posted by cottontails View Post
    Is the shaded region correct as well? My friend just told me that I've shaded Im z ≥ 0 instead of Re z ≥ 0 and that I have to shade in the region to the right of the y-axis.
    That is correct.
    The region \{z|\mathbf{Re}(z)\ge 0\} is just the right closed half-plane the edge of which is the imaginary axis.
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  5. #5
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    Quote Originally Posted by mr fantastic View Post
    The second circle you had is correct. The first circle you had corresponds to |z - i| = 3. (But since you show no working the source of your confusion and why you think the first circle is possibly correct will remain a mystery).
    I actually had mistaken the '1' that I originally wrote in this question as it was supposed to be an 'i' (in the original question). So, that is where my confusion does come from. So, the first circle was actually right but with the 1st and 4th quadrants shaded within the circle. I've fixed up those mistakes in my final answer now so, thank you.
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