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Mathematics
Topology and Analysis
Complex Numbers Moduli Problem
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[QUOTE="andrewkirk, post: 5443350, member: 265790"] I think you may have copied the wrong equation for (9). The triangle inequality says that ##|a+b|\leq|a|+|b|##. Applying that to your equation we get: $$|z^3+3z^2-2z+1|\leq|z^3+3z^2-2z|+|1|\leq|z^3+3z^2|+|-2z|+|1|\leq|z^3|+|3z^2|+|-2z|+|1|$$ [applying the triangle inequality three times in succession] $$=|z|^3+3|z|^2+|-2||z|+1$$ [applying (8) ] $$=|z|^3+3|z|^2+2|z|+1<2^3+3\cdot 2^2+2\cdot 2+1=25$$ EDIT: Just saw Samy's post. I don't have the book but, based on that picture, it looks more likely a typo in that the ref to (9) should be to (10), rather than you miscopying it. [/QUOTE]
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Complex Numbers Moduli Problem
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