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Math Help - Complex unit circle questions

  1. #1
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    Complex unit circle questions

    Hi there,

    I have these complex unit Cicle questions.


    Let O = {w \mathbb{C}||z|=1}


    Let the sets alpha, beta, delta and gamma be defined as the following.


    alpha = {w \mathbb{C}| \exists z_1 \in O \exists z_2 \in O : w = z_1 + z_2}

    beta = {w \mathbb{C}| \forall z_1 \in O \forall z_2 \in O : w = z_1 + z_2}

    delta = {w \mathbb{C}| \exists z_1 \in O \forall z_2 \in O : w = z_1 + z_2}

    gamma = {w \mathbb{C}| \forall z_1 \in O \exists z_2 \in O : w = z_1 + z_2}

    Solution(alpha)

    If e^{i \theta} + e^{-i \theta} for 0 \leq \theta \leq \pi/2 belongs to alpha

    hence [0,2] \subseteq alpha

    if z \in [0,2] then e^{i \theta}z \in alpha

    for all 0 \leq \theta \leq 2 \pi

    hence {z: |z| \leq 2} \subseteq alpha.

    Therefore no complex larger than two belongs to alpha.

    No I am in shady waters here with the remaining set. i

    beta solution

    Isn't this the same as alpha?

    delta solution

    if z_1 belongs to [0,1], and if z_2 belong to [0,2], then e^{i \theta} z_1 belongs to delta and e^{i \theta} z_2 belongs to delta.

    Then as long as {w:|z_1 + z_2| \leq 2}

    No complex number larger than 2 belongs to delta.

    How do this sound?

    Best Regards

    Billy
    Last edited by Billy2007; December 12th 2006 at 03:37 PM.
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