Complex Number

Let z₁, z₂be two complex numbers with non-zero imainary parts such that z₁z₂= r where r is real.
Then z₁=r/z₂
⇒z₁=r/z₂× conjugate(z₂)/conjugate(z₂)
⇒z₁=r/[z₂× conjugate(z₂)]×conjugate(z₂)
⇒z₁=(r/|z₂|²)×conjugate(z₂)
As r/|z₂|² is real, ∴ z₁is a real multiple of conjugate of z₂

Thus the required 2 complex numbers do not exist.

2012-07-15 12:17:56 補充：
2i=-2(-i)=-2×(conjugate(i)).
We may conclude 2i is a multiple of conjugate of i.

一樓的答案錯，因為ri=(-r)(-i)=(-r)conjugate(i)，不附合題目。

二樓對，但難明，我提出另一個解釋。

設兩個complex numbers，一個叫z₁=(a+bi)，一個叫z₂=(c+di)。
(a+bi)(c+di)=(ac-bd)+(ad+bc)i
⇒ad+bc=0
⇒ad=-bc
⇒a/b=-c/d
所以z₁、z₂是倍數關係，並是conjugate。

所以題目是無解。

i and 2i
i x 2i = -2 which is real
i and 2i are not conjugate or multiple of conjugate of each other.