Question

(i) Express the complex number z = 4i/(1-i√3)in Cartesian form.
(ii) Determine the modulus and argument of z, and hence express z in
polar form.
(b) Solve the equation z4 − z = 0, expressing your solutions in Cartesian form.

Answer 1

4i/(1-√3 i)

= 4i(1+√3 i)/(1+3) = -√3 + i
Now, tanθ = -1/√3, so z = (2,-π/6)

z^4-z = 0
z(z^3-1) = 0
z(z-1)(z^2+z+1) = 0
and so on

Answer 2

b) form a) z = -√3 + i
which is in quadrant II if graphed in the Argand plane
so if tanθ = -1/√3 , θ = 5π/6
modulus of z = √(3 + 1) = 2
argument of z = 5π/5
z = 2cos 5π/6 + 2sin 5π/6

check: 2cos 5π/6 + 2sin 5π/6 i = -1.73205.. + (1)i or -√3 + i