I need help converting the complex number z = (-1 + sqrt(1 + 4i))/2i to the form rei^(theta).

Question

I found what the number in the square root is in the form rei^(theta) is which is sqrt(1 + 4i) = sqrt(17) e^i(tan-1(4) + pi*k) , but I am having trouble dealing with i in the denominator of z.

Answer 1

1+4i = √17 cisθ where tanθ=4
so, √(1+4i) = ∜17 cis(θ/2) = 2.0305 e^0.6629i = 1.6005+1.2496i

So, we have
-1+√(1+4i) = 0.6005+1.2496i

(-1+√(1+4i))/2i = 0.3002/i + 0.6248
= 0.6248-0.3002i

Now just convert that to polar form.

Answer 2

Thank you!