Complex Conjugate

We say that a Complex Number is formed by the Combination of the real and the imaginary part. Before learning about complex numbers, let us first look at the imaginary numbers. If we have any number, whose Square is a negative number, then we say that the number is Imaginary Number, which is represented by I, called iota. So we write root ( -9) = 3i, where root (-1) = i. So we say that the complex number is of the form of Z = ( a + ib), where a and b are the Real Numbers and  I = root (-1 ). The Set of the complex numbers is represented by the alphabet C.

 Now we will learn about complex conjugate,  where we say that  if the  complex  number z is represented as  a + bi then its  mathematical complex conjugate is represented as Ż = a – ib. To find the complex conjugates of the complex number z= 2 + 3i, we will simply write it as conj (Z) = 2-3i
 Another important thing to be remembered is that if the complex number z and its conjugate are added, then the imaginary part of the two numbers cancels out, as one   is a positive number and the other is a negative. Let us try to add the above given z and its conjugate, then we have:
 Z + conjugate ( Z) = 2 + 3i +  2 – 3i
= 4 + 3i – 3i
= 4 Ans

Let us look at a special situation of finding the conjugate:
If we have z = i^3, the to find its conjugate we precede as follows:
We know that z = i^3 = -I  [ as we know that  i^2 = 1]
 So z = 0 – i
Thus we write conjugate ( z) = 0 + i = i Ans