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If $a$ is a complex number $a\neq 0$. Under that conditions $|a|\leq |a+z|$ with $z$ complex number?

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    @Andres http://meta.stackexchange.com/questions/5234/how-does-accepting-an-answer-work2012-10-02

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That condition is equivalent to $d(a,0) \le d(a,-z)$. Time to draw a picture and see what is happening.