the conjugate of a sum is the sum of the conjugates of the individual complex numbers

Question

please help I have no idea what this means and I need the work to show how to do it

Answer 1

For a complex number a + bi, the conjugate is a - bi.  OK let's choose two complex numbers, 2 + 3i and 5 - 2i and unravel this mystery.

 

1)  The "conjugate of the sum" means to add the two complex numbers, then take the conjugate of the sum, as follows:

 

     2 + 3i

 + (5  - 2i)

     -------

     7 + i

 

     The sum is 7 + i.  The "conjugate of the sum" is is 7 - i. 

 

2)  The "sum of the conjugates" means take the conjugates of the two numbers we started with, then add them.  

     The conjugate of 2 + 3i is 2 - 3i.  The conjugate of 5 - 2i is 5 + 2i.  Now let's sum the conjugates:

 

     2  - 3i

+ (5 + 2i)

     ------

     7  - i

 

Which is the same answer we got in part 1 above.  Hence the "conjugate of the sum" equals the "sum of the conjugates".  Now you too can baffle your friends with your knowledge of complex numbers!

Answer 2

Let Z₁ = a + bi and Z₂ = c + di be any two complex numbers, then the sum of these 2 complex numbers is

 

Z = Z₁ + Z₂

   = (a + bi) + (c + di)

   = (a + c) + (b + d)i

 

The complex conjugate Z* of Z is

 

Z* = (a + c) − (b + d)i

     = (a − bi) + (c − di)

     = Z₁* + Z₂*

 

So the conjugate of a sum is the sum of the conjugates of the individual complex numbers.