Let C* be group of nonzero complex numbers under multiplication, and let H= {z∈C∗ :∣z∣=1} (the unit circle). Remember that for a complex number z=a+ib, the norm of z is ∣z∣= √a^2 +b^2. Furthermore, if z,w are complex numbers, ∣zw∣=∣z∣∣w∣, and ∣ z−1∣ =1/∣z∣ if z ≠0. (a) Find at least 4 different complex numbers in the coset (3+4i)H. (b) Determine if 2− √21i ∈ (3+4i)H. (c) Give a geometric description of the coset (3+4i)H. That is, what is the shape of the set (3+4i)H inside C∗ ?
