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

AI Thread Summary
Taking the multiplicative inverse of a complex number inverts its magnitude, resulting in |z^{-1}| = 1/|z|, while the argument becomes its negative, expressed as arg(z^{-1}) = -arg(z). There is some debate about terminology, with one participant suggesting that "supplement" is incorrect in this context. The discussion also touches on the relationship between the reciprocal and the argument, with differing opinions on the correct expression for arg(z^{-1}). Overall, the key points focus on how the inverse affects both magnitude and argument. The conversation emphasizes the mathematical properties of complex numbers and invites further exploration of these concepts.
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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