Let ⁡(𝑧)=θ where 0 lt; θ lt; π ∴⁡(−𝑧)=−π +θ ⇒⁡(𝑧)−⁡(−𝑧) =θ −(−π +θ ) =θ +π −θ =π ∴ If 0 lt; ⁡(𝑧) lt; π, then ⁡(𝑧)−⁡(−𝑧)=π ...

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Algebra of Complex Numbers

If 0< ⁡𝑧< π ...

Question

If $0 < arg (z) < π$ , then $arg (z) - arg (- z) =$ ________

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Solution

Let $arg (z) = θ$

where $0 < θ < π$

$∴ arg (- z) = - π + θ$

$\Rightarrow arg (z) - arg (- z) = θ - (- π + θ) = θ + π - θ = π$

$∴$ If $0 < arg (z) < π$ , then $arg (z) - arg (- z) = π$

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