# 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. If b is positive $$\sqrt{a+ib}=\pm \left\{ \sqrt{\frac{1}{2}\left\{ \sqrt{{{a}^{2}}+{{b}^{2}}}+a \right\}+i\left\{ \sqrt{{{a}^{2}}+{{b}^{2}}}-a \right\}} \right\}$$.

If b is negative $$\sqrt{a+ib}=\pm \left\{ \sqrt{\frac{1}{2}\left\{ \sqrt{{{a}^{2}}+{{b}^{2}}}+a \right\}-i\left\{ \sqrt{{{a}^{2}}+{{b}^{2}}}-a \right\}} \right\}$$.

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).