complex numbers

Question: Consider the set H = \{z \in C: z=x-1+xi ; x \in R\}.

Prove that there is a unique number z \in H such that |z| \leq |w| for all w \in H.


Let w=y-1+yi with y \in R. It suffices to prove that there is unique number x \in R such that

(x-1)^{2}+x^{2} \leq (y-1)^{2}+y^{2} for all y \in R.

In other words, x is the minimum point of the function

f: R \longrightarrow R, f(y)=(y-1)^{2}+y^{2}=2y^{2}-2y+1=2(y-1/2)^{2}+1/2.

Hence, x=1/2 and z=-1/2+i/2.

Crisp and clear …right?

More later…

Nalin Pithwa