Square Roots & Geometrical representation of Conjugate of a Complex Number

Square Roots of a Complex Number:

Let a + ib be a complex number such that √ (a + ib) = x + iy, where x and y are real numbers.

Then

If b is positive 

If b is negative 

Geometrical representation of Conjugate of a Complex Number:

Let z = x + iy be a complex number. Clearly. z = x + iy is represented by a point P (x, y) in the argand plane.

Now, z = x + iy ⇒ z̄ = x - iy = x - iy = x + (-y). So, z̄ is represented by a point Q (x, -y) in the argand plane.

Clearly, Q is the image of point P in the real axis.

Thus, if a point P represents a complex number z, then its conjugate z̄ is represented by the image of P in the real axis.

|z| = |z̄| and arg (z̄) = -arg (z)

The general value of arg (z̄) is 2nπ + arg (z).