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Math Help - Absolute Value of Complex Numbers

  1. #1
    Member thaopanda's Avatar
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    Absolute Value of Complex Numbers

    I've been staring at this problem for quite some time...

    Let z belong in C such that |z| = 1. Compute |1+z|^2 + |1-z|^2

    If it was like |z + w|^2 where w was also a complex number, then that would be a lot like multiplying vectors using the dot product, correct? So it is much simpler where w is a real number? I'm not sure if I understand how to go about this...

    please help

    - Nicole
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  2. #2
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    Quote Originally Posted by thaopanda View Post
    I've been staring at this problem for quite some time...

    Let z belong in C such that |z| = 1. Compute |1+z|^2 + |1-z|^2

    If it was like |z + w|^2 where w was also a complex number, then that would be a lot like multiplying vectors using the dot product, correct? So it is much simpler where w is a real number? I'm not sure if I understand how to go about this...

    please help

    - Nicole
    Let z = x + iy such that |z| = 1.

    Therefore x^2 + y^2 = 1.


    1 + z = 1 + x + iy and 1 - z = 1 - x - iy.

    Therefore

    |1 + z|^2 = (1 + x)^2 + y^2

    |1 - z|^2 = (1 - x)^2 + (-y)^2 = (1 - x)^2 + y^2.


    Therefore

    |1 + z|^2 + |1 - z|^2 = (1 + x)^2 + y^2 + (1 - x)^2 + y^2

     = 1 + 2x + x^2 + 1 - 2x + x^2 + 2y^2

     = 1 + 2x^2 + 2y^2

     = 1 + 2(x^2 + y^2)

     = 1 + 2 since x^2 + y^2 = 1

     = 3.
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  3. #3
    Member thaopanda's Avatar
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    wow... so it was as straightforward as I thought it was.. I feel dumb now

    Thank you very much!!
    Nicole
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    Quote Originally Posted by thaopanda View Post
    I've been staring at this problem for quite some time...

    Let z belong in C such that |z| = 1. Compute |1+z|^2 + |1-z|^2

    If it was like |z + w|^2 where w was also a complex number, then that would be a lot like multiplying vectors using the dot product, correct? So it is much simpler where w is a real number? I'm not sure if I understand how to go about this...

    please help

    - Nicole
    Remember that |z|^2 = z\bar z.
    Therefore, |1+z|^2+|1-z|^2 = (1+z)(1+\bar z) + (1-z)(1-\bar z).
    Thus, 1+z+\bar z + z\bar z + 1 - z - \bar z + z\bar z = 1 + |z|^2 + |z|^2 = 1+1+1=3.
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