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  1. Two complex numbers are represented by c + di and e-fi, where c, d, e, and f are positive real numbers. In what two quadrants may the product of these complex numbers lie? Explain your answer in complete sentences.

 

If possible please show me how you did it. I do not understand how to do these problems. 

 

Edited to present one question only - Melody.

 Apr 13, 2020
edited by Melody  Apr 13, 2020
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c + di lies on the 1st Quadrant, which means the polar form of c + di is \(re^{i\theta}\), where \(r = \sqrt{c^2 + d^2}\)\(\theta \in \left(0, \dfrac\pi2\right) \).

e - fi lies on the 4th Quadrant, which means the polar form of e - fi is \(Re^{i\phi}\), where \(R = \sqrt{e^2 + f^2}\)\(\phi \in \left(\dfrac{3\pi}2, 2\pi\right) \)

 

Therefore, (c + di)(e - fi) = \(rRe^{i\left(\theta + \phi\right)}\)

For \(\theta \in \left(0, \dfrac\pi2\right) \) and \(\phi \in \left(\dfrac{3\pi}2, 2\pi\right) \)\(\theta + \phi \in \left(\dfrac{3\pi}{2} , \dfrac{5\pi}{2}\right)\)

 

Therefore (c + di)(e - fi) can lie inside the 4th Quadrant and the 1st Quadrant.

 Apr 14, 2020

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