1. ## Complex Conjugation Confusion

Hi

I seem to have contradicting rules on how to treat complex conjugates

where we have

a = x + iy
a*= x - iy

but in an example showing how you can find whether a complex expression is real or imaginery we have

b = i(cd* - c*d)
b* = -i(c*d - cd*)

so to compare to my previous expression

if we say x = 0 and y = (cd* - c*d)

we have the first example saying

a = x + iy
a*= x - iy

and the second saying

b = x + iy
b*= x - iy*

is it simply that the * is dropped when a variable is known to be real ?

I am new to this subject and any insights would be most welcome !

2. I'm not quite sure what you're asking, but I'll try to answer you anyway.

If a is a complex number, then a can be written as a=x+iy where x and y are real numbers. The conjugate of a is a*=x-iy. A complex number a is real if and only if a=a* (or equivalently, y=0).

If you want to make sense of what b is in terms of real numbers, then you need to let c=x+iy and d=z+iw for real numbers x,y,z,w and then simplify those expressions.
Saying y=(cd*-c*d) doesn't make sense until you are sure that cd*-c*d is real (which I believe it is, but it must be checked first).

I'm not sure if I understood your question but I hope some of this helps.

3. Hi, thanks for your awnser

Yes sorry I seem to have left out some details! I will try and phrase it again, thanks for bearing with me!

I have 2 examples of conjugates taken of 2 different complex numbers and they seem to be create different definitions.

Where a and b are complex numbers I have the definitions of
a = x + iy
a*= x - iy
and
b = i(cd* - c*d)
b* = -i(c*d - cd*)

If I substitue (cd* - c*d) for y and 0 for x in the second equation then the conjugate of b looks like

b*= x - iy*
not
b*= x - iy

which is defined by the first equation where

a*= x - iy

I am asking why the y is turned into the conjugate of y on the second example and not on the first where i is simply reversed and y is left alone.

Thanks again

4. y is a real number. So y*=y

5. Of course ...... sorry I was being stupid !

thanks I could have gone round in circles for days over that