# Thread: Complex unit circle questions

1. ## 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