Prove: If Z and W are complex numbers, then the conjugate of (Z+W) is equal to the conjugate of Z plus the conjugate of W.

Question

My thought is that this is kind of like the distributive property, but I'm not sure. It doesn't help that I haven't written a proof in over 10 years. Help? Perhaps point me in the direction of a site that specializes in proofs?

Thanks,
Amy :)

Answer 1

Z = zx + zy i
W = wx + wy i
so
Z+W = (zx+wx) + (zy+wy)i
conjugate of Z+W = (zx+wx) - (zy+wy)i
= zx + wx - zy i - wy i
= (zx - zy i) + (wx - wy i)
which is of course conjugate of Z + conjugate of W

conjugate of

Answer 2

Thanks! One more question:

Prove: Let z be a complex number. Show that z is an element of the real number set if and only if the conjugate of z is equal to z.

My teacher said there would be two things to prove from this since it was an "if and only if" problem. So my question is, what am I proving? I think one proof is going to be that the conjugate of z is equal to z. I don't know what the other part is.
Help?

Thanks again,
Amy :)

Answer 3

idk man