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Conjugate of a Complex Numbers


Conjugate of a complex number z = a + ib is denoted and defined by conjugate-z = a-ib.

In a complex number if we replace i by -i, we get conjugate of the complex number. conjugate-z is the mirror image of z about real axis on Argand's Plane.

Geometrical representation of conjugate of complex number -

           |z| = |conjugate-z|

        arg (conjugate-z) = - arg (z)

        General value of arg (conjugate-z)= 2nΠ P.V. arg(z)

                geometrical-representation-of-conjugate-of-complex-number

Properties:

(i) If z = x+y, then x= z+conjugate-z/2, y = z+conjugate-z/2

(ii) z = conjugate-z = z is purely real

(iii) z + conjugate-z    = 0 = j is purely imaginary

(iv) |conjugate-z|2 = z conjugate-z

(v)  conjugate-z = z

properties-in-conjugate 

(ix) Imaginary roots of polynomial equations with real coefficient occur is conjugate pairs.

(x) If w=f(z), then  conjugate-w   = f(conjugate-z)

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