For any integer \( k \), \( i^{4k}=1,\)\( i{4k + 1}=i\), \(i^{4k + 2}= â€“ 1,\) \(i{4k + 3} = â€“ i \)

The conjugate of the complex number \( z = a + ib\), denoted by z , is given by \( z = a â€“ ib\).

The polar form of the complex number \( z = x + iy\) is \(r (\cos Î¸ + i \sin Î¸)\), where \(r = \sqrt{x^2 + y^2} \) (the modulus of z) and \( \cos Î¸ = \frac{x}{r} \), \(\sin Î¸ = \frac{y}{r}\). (Î¸ is known as the argument of z. The value of Î¸, such that â€“ Ï€ < Î¸ â‰¤ Ï€, is called the principal argument of z.

A polynomial equation of *n* degree has *n* roots.

The solutions of the quadratic equation \(ax^2 + bx + c = 0\), where a, b, c âˆˆ R, a â‰ 0, \( b^2 â€“ 4ac < 0 \), are given by \( x = \frac{âˆ’ b \pm \sqrt{4ac – b2} i}{2a} \).