Let a complex number be = z = x + i y
This is also defined in the complex plane using polar coordinates.
r = |z| = √(x²+y²)
tan Ф = y/x = Imaginary part of complex number / real part of complex number.
here, Ф is called argument of the complex number. r is the modulus of complex number.
argument = tan⁻¹ y/x
In case of quadratic equations, if the determinant is negative, then we have a complex number as the solution.
a x² + b x + c = 0
Determinant = b² - 2 ac < 0, then let -Δ = (4ac - b²) = positive
x1 = [ - b + j √(4ac-b²) ] / 2 a and x2 = [ -b - j √(4ac - b²) ] / 2 a
argument will be (for x1) : tan Ф = - √(4ac-b²) / b
Ф = tan⁻¹ [ - √(4ac-b²) / b ]