Conjugate of Complex Number
Let z = a+ any complex number, then the conjugate of z is a-. It is generally denoted by or z', thus
It follows from the definition, that the conjugate of a complex number is obtained by replacing by . Thus the conjugate of a purely real number is the number itself.
Properties of Conjugate
Let and be any three complex numbers, then
P-2 z+ = 2 Re(z)
Reasoning: Since, z = a+
Reasoning: Since, z + a+
P-4 If then Im (z) = 0 or z is purely real
2lm(z) = 0+0
lm(z) = 0 and hence z is a Purely Real Number.
P-5 If then Re (z) = 0 or z is purely imaginary
2Re (z) = 0
z is purely imaginary.
Reasoning: Here, =
P-13 Re Re
P-14 Im - Im =