Argument of a quotient

#argument#quotient#computation

Statement

Compute the argument of $z=\frac{1+i}{1-i}$ .

Hint

Simplify $z$ first.

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Simplification.
$z=\frac{1+i}{1-i}=\frac{(1+i)^2}{1+1}=\frac{2i}{2}=i.$

Argument. $\arg(i)=\pi/2$ .

Conclusion. $\arg z=\pi/2\pmod{2\pi}$ , and $z=i$ .