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Complex Numbers - Arithmetic with Complex Numbers in Cartesian Form

Complex Numbers - Arithmetic with Complex Numbers in Cartesian Form

If \(z=\dfrac{2-i}{1+3i}\), find the real and imaginary parts of \(z-\dfrac1z\).

Hence, find all complex numbers \(w\) in Cartesian form such that \(w-\dfrac1w=\dfrac1{10}+\dfrac{21}{10}i\).

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Answer:\(\operatorname{Re}(z-1/z)=\dfrac1{10},\ \operatorname{Im}(z-1/z)=-\dfrac{21}{10}\). \(w=-\dfrac1{10}+\dfrac7{10}i\) or \(w=\dfrac15+\dfrac75i\).

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