3.9.4—Complex numbers

z=a+bi is rectangular form. In polar form z=r(cosθ+i sinθ)=r cisθ, where r=|z| and θ is an argument measured from the positive real axis.

Use rectangular form for addition and polar form for multiplication, division and powers. Choose an argument branch consistently and identify the quadrant.

1+i=√2 cis(π/4); its square is 2 cis(π/2)=2i.

The argument is not determined by tan⁻¹(b/a) alone; the quadrant may require adding π.