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    "Solve the equation 2*z = |z| + 2*i , for any complex number z "

    My working out thus far consists of writing out z in terms of its imaginary/real parts(A + B*i) and |z| as sqrt(A^2 + B^2) and then equating the real and imaginary parts of both sides. Although, I think this is the entirely wrong direction anyway.

    Any help would be appreciated .
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    (Original post by Mr.bob)
    "Solve the equation 2*z = |z| + 2*i , for any complex number z "

    My working out thus far consists of writing out z in terms of its imaginary/real parts(A + B*i) and |z| as sqrt(A^2 + B^2) and then equating the real and imaginary parts of both sides. Although, I think this is the entirely wrong direction anyway.

    Any help would be appreciated .
    Equating imaginary parts is sufficient to get you B, since |z| is real.

    What have you got, thus far?
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    (Original post by ghostwalker)
    Equating imaginary parts is sufficient to get you B, since |z| is real.

    What have you got, thus far?
    yeah i found B to be 1, then substituted this into the expression "2*A = sqrt(A*2 + B*2)" to find A(which i found to be sqrt(1/3)).
    The problem is i dont really understand what the question is looking for :dontknow:
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    (Original post by Mr.bob)
    yeah i found B to be 1, then substituted this into the expression "2*A = sqrt(A*2 + B*2)" to find A(which i found to be sqrt(1/3)).
    The problem is i dont really understand what the question is looking for :dontknow:
    You're looking for the value of z, and since you have z=A+iB, and you know what A,B are, ...
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    (Original post by ghostwalker)
    You're looking for the value of z, and since you have z=A+iB, and you know what A,B are, ...
    so Z = +/-sqrt(1/3) + i
    This is what i had for my answer but im just uncertain how this satisfies what the question was asking
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    (Original post by Mr.bob)
    so Z = +/-sqrt(1/3) + i
    This is what i had for my answer but im just uncertain how this satisfies what the question was asking
    You initially said A=sqrt(1/3), but now you've put +/- that.

    I presume that at one point in your working you squared an equation. This can introduce additional solutuions that don't satisfy the original equation. So, you need to check each solution to make sure it does.

    The original question is poorly worded, IMO. If it had just said "solve the equation ..." and left it at that, it would have made more sense.
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    (Original post by ghostwalker)
    You initially said A=sqrt(1/3), but now you've put +/- that.

    I presume that at one point in your working you squared an equation. This can introduce additional solutuions that don't satisfy the original equation. So, you need to check each solution to make sure it does.

    The original question is poorly worded, IMO. If it had just said "solve the equation ..." and left it at that, it would have made more sense.
    i know such as stupidly phrased question..cheers for the help though!
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    (Original post by Mr.bob)
    "Solve the equation 2*z = |z| + 2*i , for any complex number z "

    My working out thus far consists of writing out z in terms of its imaginary/real parts(A + B*i) and |z| as sqrt(A^2 + B^2) and then equating the real and imaginary parts of both sides. Although, I think this is the entirely wrong direction anyway.

    Any help would be appreciated .
    One approach is to note that 2z=|z| +2i \Rightarrow |z| = 2(z-i) so that 2(z-i) is a real number since it is equal to |z|, which is a real number.

    Edit: easier: note that z=\frac{|z|}{2}+i allows you to write down a quadratic equation for |z|, which then immediately gives you the solutions for z
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    (Original post by atsruser)
    One approach is to note that 2z=|z| +2i \Rightarrow |z| = 2(z-i) so that 2(z-i) is a real number since it is equal to |z|, which is a real number.

    Edit: easier: note that z=\frac{|z|}{2}+i allows you to write down a quadratic equation for |z|, which then immediately gives you the solutions for z
    yeah, the approach i used results in the same imaginary part being found as this method(that being equal to 1). What about the real part though?
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    (Original post by Mr.bob)
    yeah, the approach i used results in the same imaginary part being found as this method(that being equal to 1). What about the real part though?
    Since z=\frac{|z|}{2}+i} then the real part is |z|/2. So, find |z|, and you're done.
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    (Original post by atsruser)
    Since z=\frac{|z|}{2}+i} then the real part is |z|/2. So, find |z|, and you're done.
    but how do i find |z|
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    (Original post by Mr.bob)
    but how do i find |z|
    You know that z=\frac{|z|}{2}+i} but since |z| is real, z is already in the form a+ib

    So we have z=\frac{|z|}{2}+i = a+ib with a=\frac{|z|}{2}, b=1

    Now write down |z| in terms of a and b, and hence in terms of what they equal. You will then have an equation in |z| to solve.
 
 
 
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