# 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 $w \in H$.

Solution:

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

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