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Introduction
In this topic of the chapter we will look at the conjugate more closely and we will also look at the properties of the conjugate in detail. The conjugate of the complex number z would be represented byz. z and z are called as the complex conjugate pair.
There is a complex number x+ yi and it’s conjugate would be x- yi
Q Find the conjugate of the complex number 22-13i.
Sol- To find the conjugate of the given complex number, we have to change the sign of the imaginary part. So the conjugate will be 22+13i
z+z=2*a=2a
Therefore the sum of a complex number and its conjugate will be equal to twice the magnitude of the real part.
If we have to write in words, then we can say that the product of a complex number and its conjugate will yield the sum of the square of the magnitude of the real part and the square of the magnitude of the imaginary part.
ADDITION OF TWO COMPLEX NUMBERS CONJUGATE-
Let us take an example of two complex numbers z1 and z2
Let z1=p+qi and let z2=c+di
- What happens when take the conjugate of the sum of these two complex numbers?
Sol:-
So this property proves that the conjugate of a sum of two complex numbers is equal to the sum of the individual conjugates of those two complex numbers and vice-versa.
DIFFERENCE OF TWO COMPLEX NUMBERS CONJUGATE-
Let us see what happens if we try to find the conjugate of the difference of two complex numbers.
Let z1=p+qi and let z2=c+di
This time we will do it the opposite way, that is from RHS of LHS
This property proves that the difference of the conjugate of a complex number and the conjugate of another complex number is equal to the conjugate of the difference of those two complex numbers and vice-versa.
PRODUCT OF TWO COMPLEX NUMBERS CONJUGATES-
So to summarize, this property states that the product of the conjugate of one complex number and the conjugate of another complex number is equal to the conjugate of the product of those two complex numbers and vice-versa.
DIVISION OF TWO COMPLEX NUMBERS CONJUGATE-
The conjugate of the division of one complex number by another is equal to the division of the conjugate of one complex number by the conjugate of another complex number and vice-versa.
Conclusion
In this chapter, first we saw a brief description of The Conjugate of Complex Numbers. Then in the Introduction we looked at the 8 Properties of The Conjugate of Complex Numbers. We saw the proof of all those 8 properties. We also solved some sample questions related to The Conjugate. For this chapter you need to be very thorough with the proofs of the properties of The Conjugate as these proofs are very important and will most likely be asked in your examinations.
