- #1
variety
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When is it true that is |a|=|b|, then either a=b or a=b*, where a and b are complex numbers and b* is the complex conjugate of b?
Absolute value of complex numbers
- Context: Undergrad
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Discussion Overview
The discussion revolves around the conditions under which the absolute values of two complex numbers, |a| and |b|, being equal implies that either a equals b or a equals the complex conjugate of b. The scope includes theoretical exploration of complex numbers and their properties.
Discussion Character
- Exploratory, Technical explanation, Debate/contested
Main Points Raised
- One participant questions the conditions under which |a| = |b| leads to a = b or a = b*.
- Another participant corrects a misconception regarding complex conjugates, stating that -1 is not the complex conjugate of 1.
- A participant suggests that the statement can be rephrased in trivial ways, implying that if it were straightforward, the notation would be unnecessary.
- Several interpretations of |a| = |b| are proposed, including geometric interpretations related to the unit circle and distances in the complex plane.
- Mathematical relationships involving the real and imaginary parts of a and b are mentioned as potential conditions.
- Existence of polar representations for a and b is noted, indicating that they can be expressed in terms of a common radius and angles.
Areas of Agreement / Disagreement
Participants express differing views on the implications of |a| = |b|, with no consensus on the conditions that lead to the stated equality of a and b or their conjugates.
Contextual Notes
The discussion includes various interpretations and mathematical expressions that may depend on specific definitions or assumptions about complex numbers.
- #2
CompuChip
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-1 is not the complex conjugate of 1.
- #3
variety
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What I'm asking is, are there any conditions on a and b such that the above statement is true?
- #4
CompuChip
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Yes, but they are all trivial rewritings of |a| = |b|.
If it were as simple as "a = b or a = b*", for example, we wouldn't need the bar notation, would we
Some of those rewritings / interpretations are:
* a / b lies on the unit circle
* a and b lie on a circle in the complex plane
* the distance of a and b to the origin is the same
* Re(a)^2 - Re(b)^2 = Im(b)^2 - Im(a)^2
* There exists r > 0, \theta_1, \theta_2 \in [0, 2\pi) such that a = r e^{i\theta 1}, b = r e^{i\theta_2} (and r = |a| = |b|)
* There exists \theta \in [0, 2\pi) such that a = e^{i \theta} b.
If it were as simple as "a = b or a = b*", for example, we wouldn't need the bar notation, would we
Some of those rewritings / interpretations are:
* a / b lies on the unit circle
* a and b lie on a circle in the complex plane
* the distance of a and b to the origin is the same
* Re(a)^2 - Re(b)^2 = Im(b)^2 - Im(a)^2
* There exists r > 0, \theta_1, \theta_2 \in [0, 2\pi) such that a = r e^{i\theta 1}, b = r e^{i\theta_2} (and r = |a| = |b|)
* There exists \theta \in [0, 2\pi) such that a = e^{i \theta} b.
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