CVP5

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Show that $\overline{z_1z_2}=\overline{z_1}\,\overline{z_2}\,$

Let $z_1 = x_1 + iy_1\,$ and $z_2 = x_2 + iy_2\,$ . Then

$\overline{z_1z_2}$	$=\overline{(x_1 + iy_1)(x_2 + iy_2)}$
	$= \overline{x_1x_2 + x_1y_2i + x_2y_1i - y_1y_2}$
	$=\overline{(x_1x_2 - y_1y_2) + i(x_1y_2 + x_2y_1)}$
	$=(x_1x_2 - y_1y_2) - i(x_1y_2 + x_2y_1)\,$
	$=x_1x_2 - x_1y_2i - x_2y_1i - y_1y_2\,$
	$=(x_1 - y_1i)(x_2 - y_2i)\,$
	$=\overline{(x_1 + y_1i)}\,\overline{(x_2 + y_2i)}$
	$=\overline{z_1}\,\overline{z_2}$

Main Page : Complex Variables : Proofs

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