Argument in complex numbers

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Mentallic
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When given a complex number [itex]z=x+iy[/itex] and transforming this into its mod-arg form giving [itex]rcis\theta[/itex] where [itex]r=\sqrt{(x^2+y^2)}[/itex] and [itex]\theta=arctan(y/x)[/itex], we are assuming that [itex]-\pi/2<\theta<\pi/2[/itex].

What if however a student is asked to convert the complex number -1-i into mod-arg form? If they just start to plug-and-chug they'll quickly end up with the result [itex]\theta=\pi/4[/itex] and all of a sudden they've changed the complex number into it's negative, 1+i.

How does one avoid this dilemma?
 

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Mentallic
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Thanks :smile:
 
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mathman
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When given a complex number [itex]z=x+iy[/itex] and transforming this into its mod-arg form giving [itex]rcis\theta[/itex] where [itex]r=\sqrt{(x^2+y^2)}[/itex] and [itex]\theta=arctan(y/x)[/itex], we are assuming that [itex]-\pi/2<\theta<\pi/2[/itex].

What if however a student is asked to convert the complex number -1-i into mod-arg form? If they just start to plug-and-chug they'll quickly end up with the result [itex]\theta=\pi/4[/itex] and all of a sudden they've changed the complex number into it's negative, 1+i.

How does one avoid this dilemma?
Be careful with the angle domain. You have a semicircle, while it should be a full circle. The signs of x and y have to be taken into account so that you end up in the correct quadrant.
 

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