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# Consider the complex numbers in the following picture, as well as the line segments connecting them to the origin: [asy] size(200); import

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Consider the complex numbers in the following picture, as well as the line segments connecting them to the origin: Here's a list of pairwise sums of the conjugates of these complex numbers:

\overline{z}_1+\overline{z}_2, \overline{z}_1 + \overline{z}_3, \overline{z}_1 + \overline{z}_4, \overline{z}_2 + \overline{z}_3, \overline{z}_2 + \overline{z}_4, \overline{z}_3 + \overline{z}_4.\
Find the number of the quadrant each of these pairwise sums is in, and answer with the ordered list, such that your first number corresponds to the quadrant that \overline{z}_1+\overline{z}_2 is in, your second number corresponds to the quadrant that \overline{z}_1+\overline{z}_3 is in, etc.

Dec 16, 2018

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"Here's a list of pairwise sums of the conjugates of these complex numbers:

$$\overline{z}_1+\overline{z}_2, \overline{z}_1 + \overline{z}_3, \overline{z}_1 + \overline{z}_4, \overline{z}_2 + \overline{z}_3, \overline{z}_2 + \overline{z}_4, \overline{z}_3 + \overline{z}_4$$

Find the number of the quadrant each of these pairwise sums is in, and answer with the ordered list, such that your first number corresponds to the quadrant that $$\overline{z}_1+\overline{z}_2$$  is in, your second number corresponds to the quadrant that $$\overline{z}_1+\overline{z}_3$$

is in, etc."

Add two of the given complex numbers, $$z_1+z_2$$ say, and reflect the result in the x-axis (i.e. the Real axis).

So, for example, it looks like  $$z_1+z_2$$ when added will result in a complex number in the 2nd quadrant.  Reflect this in the x-axis and the result will be in quadrant 3. i.e. $$\overline{z}_1+\overline{z}_2$$will be in quadrant 3.

Repeat this process for all the specified pairs.

Dec 16, 2018
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One question, alan....why do we reflect across the x axis????   CPhill  Dec 16, 2018
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"why do we reflect across the x axis????"

Let $$z=p+iq$$ , then $$\overline{z}=p-iq$$

So, if z is the resultant obtained by adding two complex numbers, the real part of the complex conjugate of z has the same magnitude and the same sign, but the imaginary part of its complex conjugate has the same magnitude but opposite sign i.e. its complex conjugate is just a reflection in the real (or x) axis.

Of course the problem could be tackled by finding both com plex conjugates of a pair and then simply adding them, but the result would be the same.

Alan  Dec 16, 2018