Question

Daca z,u apartin multimii nr complexe cu |z|=|u|=1 ,aratati ca z+u/1+z*u apartine lui R

Answer 1

$\text{Din }|z|=|u|=1 \Rightarrow \overline{z} =\dfrac{1}{z}\text{ si } \overline{u}=\dfrac{1}{u} (1)\\\text{Un numar complex a apartine lui }\mathbb{R},\text{ daca } a=|a| .\\\text{Prin urmare avem de demonstrat ca :}\\\dfrac{z+u}{1+u\cdot u}= \overline{\left(\dfrac{z+u}{1+z\cdot u}\right)}\\\text{Din proprietatile numerelor complexe stim ca :}\\ \overline{\left(\dfrac{z+u}{1+z\cdot u}\right)}=\dfrac{\overline{u}+\overline{z}}{1+\overline{u}\cdot \overline{z}} , \text{ si folosind in continuare (1)}$

$\dfrac{\frac{1}{z}+\frac{1}{u}}{1+\frac{1}{u}\cdot \frac{1}{z}}= \dfrac{\frac{u+z}{u\cdot z}}{\frac{u\cdot z+1}{u\cdot z}}=\dfrac{u+z}{u\cdot z}\cdot \dfrac{u\cdot z}{u\cdot z+1}=\dfrac{u+z}{u\cdot z+1}~~~Q.E.D.$