Argument of a Complex Number

If a complex number is given by z = x + iy, then the argument of z i.e. θ is defined as \[\theta=\arg\left(z\right)=\tan^{-1}\left(\frac{y}{x}\right),0\leq\theta<2\pi\]

• arg (any +ve real no.) = 0, arg (any +ve imaginary no.) = π/2

• arg (any −ve real no.) = π, arg (any −ve imaginary no.) = −π/2

• arg (z₁z₂) = arg(z₁) + arg(z₂)

• arg (z₁ / z₂) = arg(z₁) − arg(z₂)

• arg (z̄) = − arg(z) = arg (1/z)

• arg (+z) = π ± arg(z) and arg (−z) = arg(z) ± π
• arg (z) + arg (z̄) = 0

• arg(z₁z₂z₃…zₙ) = arg(z₁) + arg(z₂) + … + arg(zₙ)

• arg(zⁿ) = n arg(z)

• arg(z̄) = −arg(z)

• arg(z²) = 2 arg(z)

• If arg(z) = 0 ⇒ z is real