hi i just started class and my sports competition was this weekend so I would really appreciate it if someone could guide me through this
The complex numbers $z$ and $w$ satisfy $|z| = |w| = 1$ and $zw \ne -1.$
(a) Prove that $\overline{z} = \frac{1}{z}$ and $\overline{w} = \frac{1}{w}.$
(b) Prove that $\frac{z + w}{zw + 1}$ is a real number.
even my tutor couldn't figure out how to get around this....
thank you in ad vance :D
The complex numbers \(z\) and \(w\) satisfy \(|z| = |w| = 1\) and \(zw \ne -1.\)
(a) Prove that \(\overline{z} = \frac{1}{z}\) and \(\overline{w} = \frac{1}{w}.\)
(b) Prove that \(\frac{z + w}{zw + 1}\) is a real number.
sorry about that :(
+118608 The complex numbers $z$ and $w$ satisfy $|z| = |w| = 1$ and $zw \ne -1.$
(a) Prove that \(\overline{z} = \frac{1}{z}\quad and \quad \overline{w} = \frac{1}{w}\)
(b) Prove that \(\frac{z + w}{zw + 1}\) is a real number.
\(Let \quad z=e^{i\theta}=cos\theta+ isin \theta\qquad\\ and \quad w=e^{i\alpha}=cos\alpha +isin \alpha\\ \theta+\alpha \ne (2n+1)\pi\\~\\ Prove \;\; \bar z=\frac{1}{z} \\\bar z=cos\theta-isin\theta\\ \frac{1}{z}=(e^{i\theta})^{-1}\\ \frac{1}{z}=e^{-\theta i}\\ \frac{1}{z}=cos(-\theta)+isin(-\theta)\\ \frac{1}{z}=cos(\theta)-isin(\theta)\\ \frac{1}{z}=\bar z \qquad QED \)
