If y & z are two complex numbers such that |y|=|z| and arg y=arg z then show that y & z are conjugate complex numbers. Also, if |y|=|z|=1 and arg y=arg z then show that y=1/z.

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If $z_{1}, z_{2}$ are conjugate complex numbers, and $z_{3}, z_{4}$ are also conjugate, then show that

$arg \frac{z_{3}}{z_{2}} + arg \frac{z_{1}}{z_{4}} = 0$ .

If $(z_{1}, z_{2})$ and $(z_{3}, z_{4})$ are two pairs of conjugate complex numbers, then show that

$arg \frac{z_{1}}{z_{3}} + arg \frac{z_{2}}{z_{4}} = 0$ .

z

w

| z w | = 1

a r g (z) - a r g (w) = \frac{π}{2}

z

ω

| z ω | = 1

arg (z) - arg (ω) = \frac{π}{2},

Z

W

¯ z + i ¯ w = 0

a r g (z w) = π

a r g (Z) =

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