Understanding Inequality of Complex Numbers: |z+w|=|z-w|?

Click For Summary

Discussion Overview

The discussion revolves around the inequality involving complex numbers, specifically the relationship between |z+w| and |z-w|. Participants explore the implications of replacing w with -w in the context of the given inequalities.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant references an inequality ||z|-|w||≤|z+w|≤|z|+|w| and questions how replacing w with -w leads to ||z|-|w||≤|z-w|≤|z|+|w|.
  • Another participant asserts that |-w|=|w|, emphasizing that this property holds for both real and complex numbers.
  • A third participant clarifies that |z+w| is not equal to |z-w|, explaining that replacing w with -w transforms |z+w| into |z-w|, but does not imply equality between the two expressions.
  • One participant expresses gratitude for the clarification received from others in the discussion.

Areas of Agreement / Disagreement

Participants generally agree on the properties of absolute values of complex numbers, but there is some debate regarding the implications of the inequalities and the equality of |z+w| and |z-w|.

Contextual Notes

The discussion does not resolve the implications of the inequalities fully, and the relationship between |z+w| and |z-w| remains a point of contention.

mynameisfunk
Messages
122
Reaction score
0
OK, in my book we have an inequality ||z|-|w||[tex]\leq[/tex]|z+w|[tex]\leq[/tex]|z|+|w| then from here it simply states, "Replacing w by -w here shows that ||z|-|w||[tex]\leq[/tex]|z-w|[tex]\leq[/tex]|z|+|w|.

How do we know that?
is |z+w|=|z-w|?? Note that z and w are complex numbers.
 
Physics news on Phys.org
You can choose any number for w. In particular, -w works too. The key is that |-w|=|w|
 
mynameisfunk said:
OK, in my book we have an inequality ||z|-|w||[tex]\leq[/tex]|z+w|[tex]\leq[/tex]|z|+|w| then from here it simply states, "Replacing w by -w here shows that ||z|-|w||[tex]\leq[/tex]|z-w|[tex]\leq[/tex]|z|+|w|.

How do we know that?
is |z+w|=|z-w|?? Note that z and w are complex numbers.
No, |z+w| is not equal to |z- w| and it doesn't say that. "Replacing w by -w" changes |z+ w| to |z+ (-w)|= |z- w|. And, as Office_Shredder said, |-w|= |w| whether w is real or complex.
 
Thanks guys, I get it now. I very much appreciate the help, as always.
 

Similar threads

Undergrad On inverse function theorem in Spivak's CoM
  • · Replies 6 ·
Replies
6
Views
4K
Undergrad Find f(z) given f(x, y) = u(x, y) + iv(x,y)
  • · Replies 8 ·
Replies
8
Views
1K
Graduate Help needed with derivation: solving a complex double integral
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad On sub and super solutions: Teschl and others
  • · Replies 5 ·
Replies
5
Views
4K
Graduate Quadruple Integral in the Lamb Shift
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Intersections between this infinite power tower and a multifactorial
  • · Replies 1 ·
Replies
1
Views
2K
MHB Proving Equation (1): Let r(t) be a Vector in $\mathbb{R^3}$
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Questions about the arg of complex numbers
  • · Replies 4 ·
Replies
4
Views
3K
Graduate Existence of a limit implies that a function can be harmonic extended
  • · Replies 1 ·
Replies
1
Views
3K
Undergrad Apparent counterexample to Cauchy-Goursat theorem (Complex Analysis)
  • · Replies 7 ·
Replies
7
Views
3K