Question

Two complex numbers are represented by c + d$i$ and e-f$i$ , 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.

Answer 1

Answers: First Quadrant and Fourth Quadrant

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Explanation:

Let

• w = c + di
• z = e - fi

be the two complex numbers.

Multiply them out to see what we get

$w*z = (c+di)(e-fi)\\\\w*z = c(e-fi)+di(e-fi)\\\\w*z = ce-cf*i+de*i-df*i^2\\\\w*z = ce-cf*i+de*i-df*(-1)\\\\w*z = ce-cf*i+de*i+df\\\\w*z = (ce+df)+(-cf+de)*i$

The result we get is in the form a+bi where

• a = ce+df = real part
• b = -cf+de = imaginary part

Recall that any complex number of the form a+bi can be plotted on the xy plane with 'a' being treated as the x coordinate and b as the y coordinate. In short, the location of a+bi is at the point (a,b)

With c,d,e,f being positive, this means ce and df are positive, and a = ce+df is also positive.

This places the result of wz in either the first or fourth quadrants (the northeast or southeast quadrants respectively), due to the positive x coordinate.

We don't have enough info to determine whether b = -cf+de is positive or not. So that's why we can't nail down the precise quadrant of wz

If b > 0, then wz is in quadrant 1

If b < 0, then wz is in quadrant 4