Where, (C – Di) and (C + Di) are called as conjugate pairs. In case of purely imaginary numbers as real part is zero, we have C = 0. So, imaginary complex conjugate pairs can be written as (0 - Di) and (0 + Di). Similarly, while writing real complex conjugates we omit imaginary part as: (C – 0i) and (C + 0i).
Arithmetic operations that we are much familiar with like addition, subtraction, multiplication and division can be applied in same way to complex conjugate pairs also. Let’s understand this with some formulae and examples:
(C + Di) + (C – Di) = 2 C, addition is purely real.
(C + Di) - (C – Di) = 2 Di, subtraction is purely imaginary.
Sine function of Trigonometry can be represented as:
Sin (Z) = 1 / 2 (eiZ – e-iZ),
Where, 'z' is any angle of which we have expanded sine function.
Conjugate of sine function can be written as follows:
Suppose conjugate of Z = Z* = V,
Sin (Z*) = 1 / 2 (eiZ – e-iZ)*,
Sin (V) = 1 / 2 (- eiV + e-iV)*,
Sin (V) = - (1 / 2 (eiV - e-iV)),
Sin (V) = - Sin (Z*).