Complex analysis (triangle inequality)












1












$begingroup$


I have to prove: $$|z|leq|Re(z)|+|Im(z)|$$



Work done so far:
I know how to prove the triangle inequality for two vectors but I am not sure how to show it holds for two components of a single vector. Any help would be appreciated.






complex-analysis







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$endgroup$








  • 1




    $begingroup$
    Consider the vectors $(x,y)$, $(x,0)$ and $(0,y)$.
    $endgroup$
    – Robert Z
    Jan 24 at 6:09










  • $begingroup$
    @RobertZ You are right, thanks!
    $endgroup$
    – Bertrand Wittgenstein's Ghost
    Jan 24 at 6:15
















1












$begingroup$


I have to prove: $$|z|leq|Re(z)|+|Im(z)|$$



Work done so far:
I know how to prove the triangle inequality for two vectors but I am not sure how to show it holds for two components of a single vector. Any help would be appreciated.






complex-analysis







share|cite|improve this question









$endgroup$








  • 1




    $begingroup$
    Consider the vectors $(x,y)$, $(x,0)$ and $(0,y)$.
    $endgroup$
    – Robert Z
    Jan 24 at 6:09










  • $begingroup$
    @RobertZ You are right, thanks!
    $endgroup$
    – Bertrand Wittgenstein's Ghost
    Jan 24 at 6:15














1












1








1





$begingroup$


I have to prove: $$|z|leq|Re(z)|+|Im(z)|$$



Work done so far:
I know how to prove the triangle inequality for two vectors but I am not sure how to show it holds for two components of a single vector. Any help would be appreciated.






complex-analysis







share|cite|improve this question









$endgroup$




I have to prove: $$|z|leq|Re(z)|+|Im(z)|$$



Work done so far:
I know how to prove the triangle inequality for two vectors but I am not sure how to show it holds for two components of a single vector. Any help would be appreciated.






complex-analysis




complex-analysis






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Jan 24 at 5:47









Bertrand Wittgenstein's GhostBertrand Wittgenstein's Ghost

483215




483215








  • 1




    $begingroup$
    Consider the vectors $(x,y)$, $(x,0)$ and $(0,y)$.
    $endgroup$
    – Robert Z
    Jan 24 at 6:09










  • $begingroup$
    @RobertZ You are right, thanks!
    $endgroup$
    – Bertrand Wittgenstein's Ghost
    Jan 24 at 6:15














  • 1




    $begingroup$
    Consider the vectors $(x,y)$, $(x,0)$ and $(0,y)$.
    $endgroup$
    – Robert Z
    Jan 24 at 6:09










  • $begingroup$
    @RobertZ You are right, thanks!
    $endgroup$
    – Bertrand Wittgenstein's Ghost
    Jan 24 at 6:15








1




1




$begingroup$
Consider the vectors $(x,y)$, $(x,0)$ and $(0,y)$.
$endgroup$
– Robert Z
Jan 24 at 6:09




$begingroup$
Consider the vectors $(x,y)$, $(x,0)$ and $(0,y)$.
$endgroup$
– Robert Z
Jan 24 at 6:09












$begingroup$
@RobertZ You are right, thanks!
$endgroup$
– Bertrand Wittgenstein's Ghost
Jan 24 at 6:15




$begingroup$
@RobertZ You are right, thanks!
$endgroup$
– Bertrand Wittgenstein's Ghost
Jan 24 at 6:15










2 Answers
2






active

oldest

votes


















1












$begingroup$

Note that $|z| = sqrt{|Re(z)|^2+|Im(z)|^2}$, and the inequality follows immediately after squaring both sides.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    Thank you so much, sometimes a hint is all one needs!!!
    $endgroup$
    – Bertrand Wittgenstein's Ghost
    Jan 24 at 6:14



















1












$begingroup$

Use $z=text{Re }z+i,text{Im }z$ and the conventional triangle inequality:
$$|z|le|text{Re }z|+|i,text{Im }z|=|text{Re }z|+|text{Im }z|.$$






share|cite|improve this answer









$endgroup$













  • $begingroup$
    Thanks for the answer!
    $endgroup$
    – Bertrand Wittgenstein's Ghost
    Jan 24 at 6:15











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2 Answers
2






active

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2 Answers
2






active

oldest

votes









active

oldest

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active

oldest

votes









1












$begingroup$

Note that $|z| = sqrt{|Re(z)|^2+|Im(z)|^2}$, and the inequality follows immediately after squaring both sides.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    Thank you so much, sometimes a hint is all one needs!!!
    $endgroup$
    – Bertrand Wittgenstein's Ghost
    Jan 24 at 6:14
















1












$begingroup$

Note that $|z| = sqrt{|Re(z)|^2+|Im(z)|^2}$, and the inequality follows immediately after squaring both sides.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    Thank you so much, sometimes a hint is all one needs!!!
    $endgroup$
    – Bertrand Wittgenstein's Ghost
    Jan 24 at 6:14














1












1








1





$begingroup$

Note that $|z| = sqrt{|Re(z)|^2+|Im(z)|^2}$, and the inequality follows immediately after squaring both sides.






share|cite|improve this answer









$endgroup$



Note that $|z| = sqrt{|Re(z)|^2+|Im(z)|^2}$, and the inequality follows immediately after squaring both sides.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Jan 24 at 5:52









DeepSeaDeepSea

71.3k54488




71.3k54488












  • $begingroup$
    Thank you so much, sometimes a hint is all one needs!!!
    $endgroup$
    – Bertrand Wittgenstein's Ghost
    Jan 24 at 6:14


















  • $begingroup$
    Thank you so much, sometimes a hint is all one needs!!!
    $endgroup$
    – Bertrand Wittgenstein's Ghost
    Jan 24 at 6:14
















$begingroup$
Thank you so much, sometimes a hint is all one needs!!!
$endgroup$
– Bertrand Wittgenstein's Ghost
Jan 24 at 6:14




$begingroup$
Thank you so much, sometimes a hint is all one needs!!!
$endgroup$
– Bertrand Wittgenstein's Ghost
Jan 24 at 6:14











1












$begingroup$

Use $z=text{Re }z+i,text{Im }z$ and the conventional triangle inequality:
$$|z|le|text{Re }z|+|i,text{Im }z|=|text{Re }z|+|text{Im }z|.$$






share|cite|improve this answer









$endgroup$













  • $begingroup$
    Thanks for the answer!
    $endgroup$
    – Bertrand Wittgenstein's Ghost
    Jan 24 at 6:15
















1












$begingroup$

Use $z=text{Re }z+i,text{Im }z$ and the conventional triangle inequality:
$$|z|le|text{Re }z|+|i,text{Im }z|=|text{Re }z|+|text{Im }z|.$$






share|cite|improve this answer









$endgroup$













  • $begingroup$
    Thanks for the answer!
    $endgroup$
    – Bertrand Wittgenstein's Ghost
    Jan 24 at 6:15














1












1








1





$begingroup$

Use $z=text{Re }z+i,text{Im }z$ and the conventional triangle inequality:
$$|z|le|text{Re }z|+|i,text{Im }z|=|text{Re }z|+|text{Im }z|.$$






share|cite|improve this answer









$endgroup$



Use $z=text{Re }z+i,text{Im }z$ and the conventional triangle inequality:
$$|z|le|text{Re }z|+|i,text{Im }z|=|text{Re }z|+|text{Im }z|.$$







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Jan 24 at 6:08









Lord Shark the UnknownLord Shark the Unknown

106k1161133




106k1161133












  • $begingroup$
    Thanks for the answer!
    $endgroup$
    – Bertrand Wittgenstein's Ghost
    Jan 24 at 6:15


















  • $begingroup$
    Thanks for the answer!
    $endgroup$
    – Bertrand Wittgenstein's Ghost
    Jan 24 at 6:15
















$begingroup$
Thanks for the answer!
$endgroup$
– Bertrand Wittgenstein's Ghost
Jan 24 at 6:15




$begingroup$
Thanks for the answer!
$endgroup$
– Bertrand Wittgenstein's Ghost
Jan 24 at 6:15


















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