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[Solved] Complex number

 
Complex Numbers
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June 15, 2021 2:35 pm
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(@armor)
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Let z\in \mathbb{C} / \left \{ 0 \right \}.  Prove that Re\left \{ \frac{1}{z} \right \}> 0,  if and only if   Re\left \{ z \right \}> 0.

 

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June 15, 2021 2:35 pm
Topic starter
Staff 
(@armor)
Admin
  Honorable Member
317 Posts
156 153 0

\small \frac{1}{z}=\frac{\bar{z}}{z.\bar{z}}=\frac{x-iy}{x^2+y^2}=\frac{x}{x^2+y^2}-i\frac{y}{x^2+y^2}

hence 

\small Re\left \{ \frac{1}{z} \right \}=\frac{z}{x^2+y^2}>0,     if and only if \small x=Re\left \{ z \right \}>0.

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