Roots Of Complex Numbers

Examples:

to obtain that r = 1 and θ = 9π/6 = 3π/2.

In summary, the three cube roots of i are

Argument of a Complex Number:

The (non-unique) angle θ associated to a complex number z ≠ 0 is called its argument, and is written arg z. For instance, arg(1 i) = π/4. But it is also correct to write arg(1 i) = 9π/4, 17π/4,−7π/4, etc. We generally choose the argument  to satisfy 0 ≤ θ < 2π. This is the principal branch of the argument:

Figure: The third root of i.

Under multiplication of complex numbers, arguments are additive and moduli multiply. That is, if z = re^iθ and w = se^iΨ  then