Proof

TheWrathOfMath

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z1 and z2 are two complex numbers such that z1*z2 is a real and non-zero number.
Prove that z1= x* (z2)bar, where x=real number.
 
Let z1=a+biz_1 = a + bi and z2=c+diz_2 = c + di. Multiply z1z2z_1^* z_2. What are the imaginary terms?

-Dan
 
z1 and z2 are two complex numbers such that z1*z2 is a real and non-zero number.
Prove that z1= x* (z2)bar, where x=real number.
First lets us cleanup the notation: z1=z=(a+bi) & z2=w=(c+di)z_1=z=(a+bi)~\&~z_2=w=(c+di)
Then zw=(acbd)+(ad+bc)iz\cdot w=(ac-bd)+(ad+bc)i.
If that product is real non-zero then (ad+bc)=0 & (acbd)0(ad+bc)=0~\&~(ac-bd)\not=0
Here is the part that makes no sense whatsoever. The conjugate is w=cdi\overline{\,w\,}=c-di
You say that you are asked to prove that for some real tt, z=twz=t\cdot\overline{\,w\,}.
But how would that work??
 
It means that z1*z2=ac-bd, and ad = -bc
(or c/d = −a/b).

I highly appreciate your assistance, by the way.
Since you want to relate z1=a+biz_1 = a+bi to z2ˉ=cdi\bar{z_2} = c-di, how about using, not c/d = -a/b, but c/a = -d/b?
 
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