A complex number z is purely imaginary if and only if z Proof Let z = x iy . Then by definition z iy Therefore, z = − z ⇔ iy = − iy ⇔ x = ⇔ z is purely imaginary. Similarly, we can verify the other properties of conjugate of complex numbers. Example .
Write i in the x iy form, hence find its real and imaginary parts. To find the real and imaginary parts of i , first it should be expressed in the rectangular form iy .To simplify the quotient of two complex numbers, multiply the numerator and denominator by the conjugate of the denominator to eliminate i in the denominator. i = = - - Complex Numbers Therefore, i = . This is in the x iy form.
Hence real part is − and imaginary part is . Example . Simplify . into rectangular form We consider i = i , and i = i .
Therefore, = i Example . If z , find the complex number z in the rectangular form We have z = + i ⇒ z + = ⇒ z + = i z ⇒ i z = ⇒ z = i . Example . If z and z , find z in the rectangular form Using the given value for z and z the value of z = = ( = i .
- - Example . Find z − , if z We have z