How Does the Inverse of a Complex Number Affect Its Magnitude and Argument?

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Discussion Overview

The discussion centers on how the multiplicative inverse of a complex number affects its magnitude and argument, exploring theoretical aspects of complex numbers in mathematics.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant states that the magnitude of the inverse of a complex number is the reciprocal of the original magnitude, |z^{-1}| = \frac{1}{|z|}, and that the argument becomes its supplement, arg(z^{-1}) = \pi - arg(z).
  • Another participant challenges the term "supplement," arguing that the reciprocal of 1 is 1, but the supplement of 0 is not 0.
  • A different participant proposes that the argument of the inverse is actually negative, suggesting arg(z^{-1}) = -arg(z).

Areas of Agreement / Disagreement

Participants express differing views on how the argument of the inverse of a complex number is defined, indicating unresolved disagreement on this aspect.

Contextual Notes

There are unresolved definitions regarding the term "supplement" in relation to the argument of complex numbers, and the discussion does not clarify the mathematical steps leading to the claims made.

plexus0208
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How does taking the multiplicative inverse (reciprocal) of a complex number change its magnitude and argument?
 
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A complex number z, when inverted, will have its magnitude also inverted, |z^{-1}| = \frac{1}{|z|} and its argument will become its supplement, arg(z^{-1}) = \pi - arg(z)

Why don't you try proving that?
 
This does not itself have anything to do with "differential equations" so I am moving it to "general mathematics".
 
I think "supplement" is wrong. The reciprocal of 1 is 1, but the supplement of 0 is not 0.
 
Hey,
I think that arg(z^-1) = - arg (z)
 

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